prolongation in equal and microtonal temperaments
TRANSCRIPT
Prolongation in Equal and Microtonal Temperaments
by
Matthew Barber
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Robert Morris
Department of Composition
Eastman School of Music
University of Rochester
Rochester, New York
2015
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Matthew Barber was born October 19, 1980 in Denver, Colorado. He earned the de-
gree Bachelor of Music in music composition at e Juilliard School, where he stud-
ied composition with Milton Babbitt from 1998-2002. In 2008 he earned the degree
Master of Arts at the Eastman School of Music, and in pursuing his Master of Arts and
Doctor of Philosophy he has studied primarily with Robert Morris and Allan Schindler,
and also with David Liptak and Ricardo Zohn-Muldoon. He has taught composition
and computer music at Eastman, and in 2006 he was awarded a commission for the
Elizabeth Rogers memorial concert. From 2009-2010 he was a visiting instructor in
music composition at Colgate University. He currently serves as an associate editor of
the journal Perspectives of New Music, and his compositions Interface Chapel, Call It
What YouWill, and Bob’s Harmonic Labyrinth have been released on recordings issued by
Perspectives of New Music and Open Space Magazine.
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I am very happy to have this opportunity to thank the people who have made this disser-
tation possible. Robert Morris has been an extraordinary teacher, adviser, and friend.
His invaluable comments made this paper better than I could have hoped without them;
he is more responsible for the shape of my music than any other teacher. Likewise, Al-
lan Schindler deserves many thanks for his comments on this and many other of my
projects, and his boundless support for all of his students. Robert Hasegawa provided
me with many fascinating resources and was an ideal reader given his expertise and in-
terest in microtonal theory and composition. My colleague Steven Rice offered dozens
of corrections and suggestions which improved the argument and style considerably.
I would like to thank the Eastman Composition Department for fielding a diverse
and engaging group of students; my colleagues at Eastman and elsewhere have made
graduate study worthwhile. I must especially thank Paul Coleman, Scott Petersen,
Baljinder Sekhon, Robert Pierzak, Steven Rice, Kevin Ernste, Scott Worthington, and
Erik Carlson, all of whom have influenced my compositional thought in myriad ways.
eUniversity of Rochester deserves my gratitude for awardingme the RaymondN.
Ball Dissertation-Year Fellowship, as does Deanna Phillips for so patiently shepherding
me through the logistical aspects of the process.
My dear friend Caren Armstrong has been there with emotional and intellectual
support throughout, and my little dog Molly saw me through many days of writer’s
block and frustration with superhuman compassion. I extend my love and gratitude to
my parents, Steve and Michele, and to my brothers, David and Timothy, for a lifetime
of support.
To my wife, Christina Crispin, I owe everything; countless are the roads I could not
have traveled without her.
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is dissertation presents a compositional theory of prolongation in equal tempera-
ments, some of which are microtonal. Part 1 is a discussion of problems associated
with generalizing prolongation and an exposition of possible approaches to composing
with prolongation. Using the work of Fred Lerdahl, John Rahn, and Joseph Straus as
a point of departure, a framework for discussing generalized theories of harmony and
voice leading is advanced. Straus’s four conditions for establishing prolongational sys-
tems serve as the primary constraints on the subsequent development of the theory.
David Lewin’s formal theory of generalized tonal functions is extended in order to meet
the scale structure requirements of Straus’s conditions.
Building on the ideas fromPart 1 and drawing from scale theory, the theory proper is
developed in Part 2. Multiplicative operations on cyclic orderings form the basis for the
construction of generalized diatonic systems, while Straus’s conditions provide limits on
the range of possibilities. Part 2 ends by generalizing some properties of generated, well-
formed scales to show how generalized diatonic systems may be embedded in various
equal temperaments.
A “musical interlude” appears in each of the two parts in order to illustrate practical
applications of the theory. e first interlude includes a compositional reconstruction
of a canon from Bach’s Goldberg Variations, employing prolongational synthesis. is
is followed by an exploration of pitch-class transformations of the passage in order to
discover how they do or do not preserve its prolongational structure. e second in-
terlude introduces the “diatonic staff ” concept to show that prolongational composition
in a generalized diatonic system depends more on the structure of that system than
the temperament it is embedded in. It also includes microtonal examples illustrating
microtonal analogues to modal mixture and tonicization.
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is dissertationwas supervised by Professors RobertMorris (advisor) andAllan Schindler
of theDepartment ofComposition at the Eastman School ofMusic, andRobertHasegawa
of the Department of Music Research at McGill University. is research was sup-
ported by the Raymond N. Ball Dissertation-Year Fellowship from the University of
Rochester.
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Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivContributors and Funding Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Introduction 1
1 Prolongational Approaches 5A Note About Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 ree Authors on Generalized Prolongation . . . . . . . . . . . . . . . . . . 6Fred Lerdahl: “Atonal Prolongational Structure” . . . . . . . . . . . . . . . 7John Rahn: “Logic, Set eory, Music eory” . . . . . . . . . . . . . . . . 19Joseph Straus: “e Problem of Prolongationin Post-Tonal Music” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Discussion of the Straus Conditions . . . . . . . . . . . . . . . . . . . . . . . 44
1.2 Musical Interlude I: A Goldberg Canon . . . . . . . . . . . . . . . . . . . . . 47From Middleground Out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47e M5/M7 Transform: [037] and [014] Trichords . . . . . . . . . . . . . . 56e Goldberg Canon Under the M5/M7 Transform . . . . . . . . . . . . . 59
1.3 Prolongation and Other eories of Tonality . . . . . . . . . . . . . . . . . . 65Lewin’s Generalized Triad Space . . . . . . . . . . . . . . . . . . . . . . . . . 66Discussion of Lewin’s Generalized Triad Space . . . . . . . . . . . . . . . . 74Harmonic Function in Neo-Riemannian and Prolongational eories . . 92
2 Prolongation in Equal Temperament 992.1 Constructing Generalized Diatonic Scale Cycles For Prolongation . . . . 100
Multiplication Groups mod n and Multiplicative Permutation . . . . . . 100
TABLE OF CONTENTS vii
Constraints on Scale Cardinality and Multiplicative Permutation Oper-ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.2 Musical Interlude II: Règle De L’Octave . . . . . . . . . . . . . . . . . . . . . 122e Diatonic Staff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A Goldberg Prolongation Revisited . . . . . . . . . . . . . . . . . . . . . . . . 131Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Tonicization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
2.3 From Diatonic to Chromatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Properties of Generated Scales and the Straus Conditions . . . . . . . . . 170Regions, Interval Classes, and Relations Between c and d . . . . . . . . . 178From Temperament to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183From Scale to Temperament . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184From Harmony to Temperament and Scale . . . . . . . . . . . . . . . . . . . 187
Connections and Implications 194
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
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1.1 A -style analysis of a Bach chorale. . . . . . . . . . . . . . . . . . . . . 111.2 Arnold Schoenberg, Klavierstück, Op. 33a, bars 1-9. . . . . . . . . . . . . 131.3 Lerdahl’s prolongational reduction of bars 1-9 of Schoenberg’s Op. 33a. 141.4 Rahn’s K. 331 level analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5 Another K. 331 level analysis which complies with Rahn’s definitions. . 261.6 “Level synthesis” using row forms from Berg’s Lyric Suite. . . . . . . . . 301.7 “Level synthesis” using row forms from Berg’s Lyric Suite, continued. . 321.8 Aria from Bach’s Goldberg Variations, Bars 1-8. . . . . . . . . . . . . . . . 481.9 A. e bass line with functions. B. Chords used most often in the vari-
ations, showing possible canonic relationships. C. An initial sketch forthe canon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.10 Middleground hypothesis for the Goldberg canon. . . . . . . . . . . . . . . 501.11 Goldberg canon background structures. . . . . . . . . . . . . . . . . . . . . . 511.12 First transformation of the Goldberg canon middleground. . . . . . . . . . 521.13 First attempt at a canon based on the transformed middleground. . . . . 531.14 ird species canon based on the previous attempt. . . . . . . . . . . . . . 541.15 A hypothetical foreground for the Goldberg canon. . . . . . . . . . . . . . 551.16 e surface of the Goldberg canon. . . . . . . . . . . . . . . . . . . . . . . . 561.17 A passage from the introduction to Lamento della ninfa by Monteverdi. 571.18 An M7 transform of the dominant prolongation from Lamento della nifna. 581.19 An M7 transform of the entire Lamento della ninfa passage. . . . . . . . . 591.20 M5/M7 transforms of the G-major scale. . . . . . . . . . . . . . . . . . . . . 601.21 M5/M7 transforms of the deep backgrounds from Figure 1.11 . . . . . . 611.22 T8M5 of the Goldberg canon passage. . . . . . . . . . . . . . . . . . . . . . . 611.23 T6M7 of the Goldberg canon passage. . . . . . . . . . . . . . . . . . . . . . . 621.24 T6M7 of the Goldberg canon passage, with clearer voice separation. . . . 621.25 Middleground reductions of figures 1.23 and 1.24. . . . . . . . . . . . . . . 631.26 e “canonical listing” for C-major. . . . . . . . . . . . . . . . . . . . . . . . 671.27 e generalized canonical listing for a Riemann system. . . . . . . . . . . 68
LIST OF FIGURES ix
1.28 e Riemann System (G n, 5, 4) and its transformations. . . . . . . . . . . 711.29 RI -chains in Beethoven, Op. 132 and Webern, Op. 5. . . . . . . . . . . . 721.30 Creating a Tonnetz fragment from the (C , 7, 4) RI -cycle. . . . . . . . . . 731.31 Cohn’s abstract Tonnetz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751.32 Scale and trichord implementation of Riemann system (G n, 5, 4). . . . . 761.33 “Wolf closure” on a Tonnetz derived from Riemann system (C n, 7, 4). . . 781.34 Wolf closure in (C n, 1, 4) and (C n, 5, 4) Tonnetze. . . . . . . . . . . . . . . . 801.35 “Wolf-extended” Riemann system (G n, 5, 4)-WE. . . . . . . . . . . . . . . 821.36 (G n, 5, 4)-WE middleground (cf. Goldberg middleground in figure 1.10). 841.37 Hypothetical canonic surface derived from the first 7 chords of figure
1.36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851.38 e 19 “cognate” of the extended Riemann system from figure 1.35. 861.39 Multiplicative orderings of 12 : (G n, 7, 4) and 19 : (G n, 8, 6)-WE. . . . . . 89
2.1 a) e table of operations for (Z/12Z)×. b) e multiplicative opera-tions acting as automorphisms on all integers mod 12. . . . . . . . . . . . 102
2.2 Multiplication cycles and graphs for (Z/9Z)× and (Z/16Z)× where σ = 1.1042.3 Multiplication groups for integers 3-24. . . . . . . . . . . . . . . . . . . . . 1072.4 Two possible Q systems for d = 13. . . . . . . . . . . . . . . . . . . . . . . . 1122.5 Diatonic prolongation templates from d = 7 to d = 23. . . . . . . . . . . 1222.6 A 7-PC scale on diatonic staves. . . . . . . . . . . . . . . . . . . . . . . . . . 1242.7 Various systems modeled by the 7-PC diatonic staff. . . . . . . . . . . . . 1252.8 A 9-PC scale on 6-line diatonic staves. . . . . . . . . . . . . . . . . . . . . . 1272.9 Normative voice leading for the 9-PC diatonic. . . . . . . . . . . . . . . . . 1282.10 9-PC diatonic tonic-dominant-tonic progressions. . . . . . . . . . . . . . . 1292.11 e middleground level from figure 1.36 (Goldberg analogue). . . . . . . 1312.12 A diatonic-staff transcription of the middleground level in figure 2.11. . 1322.13 Reduction sequence for the diatonic middleground in figure 2.12. . . . . 1332.14 16 chromatic and 9-PC diatonic scales.. . . . . . . . . . . . . . . . . . . 1352.15 16 transcription of the diatonic-staff middleground structure in fig-
ure 2.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362.16 Reduction sequence for the 16 middleground in figure 2.15. . . . . . 1372.17 20 notation and the embedded 11-PC major scale. . . . . . . . . . . . 1412.18 11-PC minor and small scales in 20. . . . . . . . . . . . . . . . . . . . . 1432.19 e major and two minor 11-PC scales in19{3:1}. . . . . . . . . . . . 1452.20 Normative voice leading for d = 11. . . . . . . . . . . . . . . . . . . . . . . . 1462.21 Ursatz paradigms for d = 11. Left: gd = 6. Right: gd = 4. . . . . . . . . 147
LIST OF FIGURES x
2.22 Background hypothesis and development in the d = 11, gd = 6 system,to be embedded in 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.23 Foreground in the d = 11, gd = 6 system, to be embedded in 20. . . 1502.24 20 realization of the foreground from figure 2.23. . . . . . . . . . . . . 1502.25 Background hypothesis and development in the d = 11, gd = 4 system,
to be embedded in 19{3:1}. . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.26 Foreground in the d = 11, gd = 4 system, to be embedded in 19{3:1}.1522.27 19{3:1} realization of the foreground from figure 2.26. . . . . . . . . 1532.28 Stufen for a descensive Q system with h = 4, d = 13, c = 24. . . . . . . . 1552.29 Normative voice leading in the 13-PC Q system. . . . . . . . . . . . . . . . 1562.30 Possible Ursätze for the quarter-tone Q system. . . . . . . . . . . . . . . . . 1562.31 Background structures derived from the second Ursatz. . . . . . . . . . . . 1572.32 Deep middleground with applied dominant. . . . . . . . . . . . . . . . . . . 1592.33 Deep middleground with function analysis. . . . . . . . . . . . . . . . . . . 1612.34 Development of bracketed passage in figure 2.33 to complete octave
ascent in bass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622.35 Middleground after development in figure 2.34, octave ascent complete. 1642.36 Development of bracketed passage in figure 2.35. . . . . . . . . . . . . . . 1662.37 Middleground after development in figure 2.36, linear counterpoint only.1672.38 Completed quarter-tone Q foreground. . . . . . . . . . . . . . . . . . . . . . 1692.39 Canonical templates for d = 7 and d = 9. . . . . . . . . . . . . . . . . . . . 1812.40 Computation of c values for contour-preserving 11-PC Q systems. . . . 1862.41 Table of calculations to find c for the target harmony. . . . . . . . . . . . . 1912.42 Diatonic templates derived from the calculations in figure 2.41. . . . . . 1922.43 Table of calculations to find c for the target harmony in tritave equivalence.193
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T presents a compositional theory of prolongation in equal, and some-
times microtonal temperaments. Because my audience is composers and scholars who
are interested in music theory I have tried to keep the text informal in comparison with
the body of music-theory literature which relies to any significant degree on mathemat-
ics. However, I have assumed the reader has some knowledge of Schenkerian theory
and other mainstream tonal theories, musical set-theory, and a passing knowledge of
music-theory approaches which generalize prior music theory, like the one found in
Lewin, Generalized Musical Intervals and Transformations, 1987.
Prolongation is a central concept in contemporary musical thought. Historically it
is in turn a theory of tonal masterworks, a theory of tonal syntax, a theory of perception,
a theory for analysis, and a compositional theory.1 Unsurprisingly there exists no single,
central sense in which one could construe such a term, and an attempt to generalize a
concept of such appreciable depth must be an exercise in interpretation, selection, and
in some cases suppression. I am primarily concerned with a model of prolongation that
contemporary composers might apply in their own work;2 thus my theory sits within
musical-creative traditions of prolongational theory. I have tried to avoid allowing my
thought to prescribe or proscribe anything specific about style or sound.
I seek to answer three groups of related questions in this essay. First, how might
prolongational systems behave in microtonal equal temperaments? What makes them1I am imagining work exemplified by Heinrich Schenker, Benjamin Boretz, Fred Lerdahl, Carl
Schachter, and Peter Westergaard, respectively.2is might be microtonal composition, but I believe the material discussed herein is sufficiently broad
that it might have application in traditional tunings and temperaments as well.
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prolongational? Second, are there necessary and/or sufficient conditions that must be
met in order for prolongation to be possible in a given temperament? And third, is there
any way to categorize the behavior of prolongational systems across different temper-
aments? I focus much of my discussion on construing prolongation as a composite of
interrelated syntactical systems that share various features, as opposed to something that
has a specific acoustic, philosophical, or cognitive essence.
Schenker’s theory is itself philosophically (if not logically) grounded in assertions
about the sonic nature of triads, but in order to reach a sufficiently general conception
of prolongation, I shall have much less to say about the theory’s sonic correlates. In fact,
it is the case that a contextualizing syntax has as much to do with how a sound is heard
and imagined as any features that might classify it as “sound-in-itself.”3 Whether one
need precede the other (conceptually or temporally) in the construction of a sound world
is a topic which I believe sheds as little light as other historically prominent musical
dichotomies such as the precedence of “form” versus “content” or that of “composition”
versus “theory.” Nevertheless, because I am generalizing only some of the syntactical
aspects of a large and relatively amorphous theoretical tradition, in this dissertation
prolongation syntax determines sound more often than acoustic concerns. In any event,
I hope nothing in my program can be construed to advocate for or against any specific
musical style.
I have restricted the discussion to equal temperament because of all tunings and
temperaments its structure is the simplest to model.4 ere is no reason this research
could not be extended to other types of tuning or temperament, but such would likely
rely on a conceptual, if not de facto equal temperament being in place. In common3Consider the difference in function of the familiar [0258] tetrachord in Beethoven’sHeiliger Dankge-
sang, Wagner’s Tristan prelude, Stravinsky’s Le Sacre, and Schoenberg’s Klavierstück Op. 33a.4I have settled on “equal temperament” as a familiar catch-all term for any equal division of an interval
of equivalence, which need not be an octave. So although “temperament” might imply adjustment ofotherwise just intervals, a just-tuning starting point should not be assumed.
3
practice tonality, diatonic spelling rather than frequency ratio determines intervallic la-
bel and function, and analogous extensions of this concept hold for the present study as
well. Benjamin Boretz’s concept of “pitch function,”5 which assigns function to pitch
intersubjectively and contextually allowing for a distinction, in Boretz’s words, between
“what can be heard” and “what is relevant to be heard” with regard to frequency dif-
ferences due to temperament, “playing out of tune,” vibrato, etc. might be the likeliest
means of relating the details of performance practice to the abstractions of Western
music notation.6
e body of the essay consists of two parts. In Part 1 (page 5) I discuss some prob-
lems with generalizing prolongation, and reflect on some assumptions about prolonga-
tion which permeate the rest of the essay. My discussion at first centers around the work
of Fred Lerdahl, John Rahn, and Joseph Straus.7 e ideas in these three papers provide
a framework for discussing other theories of harmony and/or voice leading generalized
from traditional theory, in particular work by David Lewin. I am also concerned with
showing how I believe prolongation can work as a compositional technology.
In Part 2 (page 99) I develop my theory of prolongation in equal temperament. e
discussion draws considerably on scale theory to showwhy some scale structures conduce
to prolongational voice leading. After a series of musical examples in four temperaments
that shows proof of concept, I end with an illustration of how my generalized diatonic
theory interacts with specific temperaments.
Before I begin in earnest a few words about my reasons for entertaining this research
are in order. From my perspective the most compelling compositional problem in the
history of Western music by far is the heterarchical interaction betweenmusical hierarchy5Boretz, Meta-Variations, 1994, pages 117-118; 130-134.6It may also be possible to do this more formally through a quantization function over one of the
quotient spaces in Callender, et al., “Generalized Voice-Leading Spaces,” 2008: see particularly the dis-cussion of tuning and temperament on page 348 of that work.
7Lerdahl, “Atonal Prolongational Structure,” 1989, Rahn, “Logic, Set eory, Music eory,” 1979,and Straus, “e Problem of Prolongation in Post-Tonal Music,” 1987.
I 4
and musical association.8,9 I have tried to accentuate this interaction between hierarchy
and association in my own tonal compositions; I have been especially drawn to canons
and other “learned counterpoint” forms in my tonal music, since almost no other tonal
genre is so definitively associative. After having explored various types of non-tonal
musical hierarchy, as well as microtonal serial structures, extending some features of
the tonal system to include nontraditional set-classes in the 12-tone universe and to
microtonal temperaments seemed a natural next step. I hope that my work in this essay
might be suggestive to composers as one way to think about new and traditional pitch
hierarchies without the usual attendant assumptions about style.
8See Hofstadter,Gödel, Escher, Bach, 1979 for what is probably the most popular exposition of “tangledhierarchies,” “strange loops,” “heterarchies,” and related concepts. I believe Hofstadter misses the markin his rather simplistic discussions of music, and it seems a shame that had he known about Schenke-rian theory (and felt he could write about it for a general audience) he could have given more pertinentexamples and made music much less of an afterthought in an otherwise lovely work.
9Cohn, “e Autonomy of Motives,” 1992 and Cohn & Dempster, “Hierarchical Unity, Plural Uni-ties,” 1992 treat these issues in considerable depth.
5
P 1Prolongational Approaches
Nothing complicated enough to be reallyinteresting could have an essence.
Daniel C. DennettDarwin’s Dangerous Idea, page 201
P 1 is a review of some relevant literature regarding generalization of prolongation,
and a reflection on the assumptions I rely upon as I develop my theory in Parts 2-4.
My theory is in many ways comparable to previous theories which generalize “tonal
function” and “voice leading” away from triad-based harmonic systems, and so this re-
view touches on some of that literature. Because my goal is generalization per se, I do
not review attempts to rationally reconstruct Schenker’s system, e.g. Brown, Explaining
Tonality, 2005, however useful those might be for locating the foundations of prolonga-
tion in tonal music. By the end of this part I hope the reader will come to appreciate my
understanding of what parts of traditional prolongation are appropriate for generalizing
and extending; in any case I believe my work is flexible enough that it can accommodate
viewpoints that differ with mine.
A N A L
It could be said that every theory of prolongation in some way involves mapping a con-
ceptual progression from background to foreground. Each well-formed step in such a
progression is called a “level” of the representation. e literature often uses a “high-
P A 6
low” metaphor to refer to “more background” vs. “more foreground” levels, but unfor-
tunately there is confusion about whether “higher” should mean “toward the surface” or
“toward the background.” is likely stems from borrowing metaphors from other do-
mains: for instance a “top-down” organization usually refers to an orientation in which
structure flows from the “big picture” (“background”) at the “top,” “down” to the details
(“foreground”). On the other hand, within a “globe” model, the “surface” refers to the
outermost or “top” layer, and one might progress “deeper” toward the “underlying core,”
(the phrases “underlying structure” and “underlying counterpoint” would comply with
this metaphor). ese metaphors are often mixed – one often reads of things like an
“underlying schema which controls the details from the top down.”1
For this dissertation, “higher level” will always mean “toward the background,” even
when an author I am reviewing uses the opposite metaphor. I believe this captures two
important themes: depending upon orientation, a Schenkerian Ursatz can be said to
control the surface and to be an emergent property of the surface – when focus is on
the former, the Ursatz is at the top of a top-down organization; when on the latter, it
supervenes on “lower” harmonic and contrapuntal processes of the foreground. In both
cases, the “higher level” is closer to the “background,” and I believes this helps cement
the intuition that the background can be both controlling and emergent.
1.1 T A G P
It is appropriate to begin with three papers, each of which could form the basis for a gen-
eralized theory of prolongation from a different perspective. ese are Lerdahl, “Atonal
Prolongational Structure,” 1989, Rahn, “Logic, Set eory, Music eory,” 1979, and1Another complicating factor is notational – Schenkerian level analyses tend to progress from the
Ursatz at the top of a page to the foreground at the bottom, whereas Lerdahl and Jackendoff ’s approach,which requires the top of the page for tree graphs pointing to events at the surface, progress in the oppositefashion. See below.
ree Authors on Generalized Prolongation 7
Straus, “e Problem of Prolongation in Post-Tonal Music,” 1987. Although all of
them are musically interesting and could deliver productive results, I shall ultimately
reject the first two of these approaches for my purposes.
F L: “A P S”
Fred Lerdahl’s approach in his 1989 paper is a distillation of some ideas that have
spanned his entire career as a theorist, represented most thoroughly in the monumen-
tal A Generative eory of Tonal Music (hereafter ), co-authored with linguist Ray
Jackendoff,2 and more recently in Tonal Pitch Space,3 which is a compendium of his
theoretic work on tonal music. Space and scope prevent any extensive review of these
books, and all of the features of Lerdahl’s theory that are relevant to the present study
are in the 1989 paper.
e title of the paper is to the point: Lerdahl purports to build a theory of prolon-
gation for atonal music. As always he builds his theory in terms grounded in cognitive
psychology. He argues that while musical set theory may have some explanatory power
for atonal music, its inability to segment the surface into cognitively significant chunks
and its ambiguity regarding equivalence and similarity among chord types already makes
it a questionable listening theory. Further, the cognitive status of the abstract relations
provided by the theory (e.g. Z, K, and Kh relations) for listeners is not borne out by ex-
periment. e most problematic features of atonal set theory from Lerdahl’s perspective
are its lack of inherent hierarchical structure, and the fact that it requires a different lis-
tening mechanism than that employed for tonal music – even multiple listening theories
for different pieces – which he believes is psychologically implausible. ese deficien-
cies, according to Lerdahl, are interrelated and reinforcing; for instance, the lack of
hierarchy is due to the difficulty involved in cognizing regular set relationships, and the2Lerdahl and Jackendoff, 1983.3Lerdahl, 2001.
P A 8
myriad “contradictory” listening mechanisms implied by set theory are due to the lack
of an integrating hierarchical theory.
Lerdahl sees the non-hierachical nature of atonal music as the most pathological
of the difficulties, and the problem is twofold. First, there is no a priori hierarchical
organization of pitch-space in atonal music, as there is in tonal music: atonal pitch-
space is “flat.” Second, because of this flatness, there is no way to create an unambiguous
grammar for atonal music that will reliably parse a composition’s surface into an “event
hierarchy,” which Lerdahl believes is essential for an adequate listening theory. To build
a listening theory for atonal music that complies with observable psychological principle,
he proposes an approach based on an updated version of the tonal theory developed at
length in .
In Lerdahl’s revised theory,4 an experienced listener is believed to divide a musical
surface into a nested hierarchy of time spans based on groupings in the meter. In a
tonal piece, a set of “stability conditions” will determine which event the listener will
hear as most stable within that time span. is leads to “time-span reduction,” and
its purpose is to tie tonal structure to rhythmic structure. Each higher level (i.e. away
from the surface) is represented in music notation such that increasingly larger dura-
tions become the shortest possible duration for that level (e.g. if the shortest duration
on the surface of a piece in common time is the quarter note, the next level away from
the surface in a time-span reduction might progress in half notes, and the next might
progress in whole notes). Exceptions to the rhythmic grouping rules are made to accom-
modate cadences, which are psychologically significant for marking phrase boundaries
and supersede the hierarchical nesting of rhythm. When two differing events within4is is by no means a comprehensive summary of the ideas in , nor is what follows a com-
prehensive critique. e best summary of the theory is the first chapter of Lerdahl, 2001, pages 3-40,“eoretical Foundations.” A thorough critique appears in Peel & Slawson, “Review of A Generativeeory of Tonal Music,” 1984, with a followup exchange in Lerdahl & Jackendoff, “A Reply to Peel andSlawson’s Review,” 1985 and Peel & Slawson, “Reply to a Reply,” 1985.
ree Authors on Generalized Prolongation 9
a time-span are equally stable according to the stability conditions, salience conditions
come into play – the more salient event is then chosen to represent that time span in
the next level away from the surface.
Lerdahl’s prolongational reductions are built on time-span reductions, and in con-
trast to Schenkerian methods the reduction process proceeds by psychological rules in-
ferred from experimentation involving tension and relaxation rather than matching to
a priori schemata. Two representations of a prolongation reduction are given: one is
the well-known “prolongation tree graph,” and the other is an equivalent representation
in music notation with slurs, in the Schenkerian spirit. A prolongation tree graph is
constructed by scanning successive levels of the time-span reduction, starting with the
highest level (i.e. furthest from the surface), and determining which events are more
stable within a prolongation span according to psychological theory. ose that are less
stable are either “tensing” events which lead away from a relatively stable event, or they
are “relaxing” events which lead toward a stable event; tensing events are represented
by a branch of a tree pointing rightward from a higher branch, and relaxing events are
represented by a tree branch pointing leftward. At each level a new set of branches is
added to the previous set until every event on the surface is connected to the tree by one
branch; tree construction follows strict well-formedness rules, such that no branch can
cross and each event on the surface is connected by one and only one branch.
e goal is not to produce an analysis that demonstrates the unification of the con-
straints of harmony and voice leading at different levels, but to chart how a given piece
“will be heard” in terms of the ebb and flow of tension and relaxation, as constrained
by rules of harmony and to a lesser extent voice leading; this distinction – which rep-
resents the main difference between Lerdahl’s approach and traditional prolongation
approaches – is crucial. I shall return to this below. Also important is the extent to
which information is preserved as the analysis progresses up from the surface. In broad
P A 10
terms, Schenkerian hierarchies are “inclusional” while hierarchies are “representa-
tional.”5 is precludes any attempt to follow rules of tonal voice leading in the notated
representation of deeper levels of such a reduction – the psychological integrity of the
whole prolongation tree is what is important, not the syntactic well-formedness of indi-
vidual levels. For notational and psychological clarity, the notated prolongation graphs
of different levels in a reduction omit “inner” or otherwise unimportant voices.
Figure 1.1 is an example of a -style hierarchical analysis of a Bach chorale.6 Note
the parallel fifths between soprano and bass at level d, bars 5-6; not all Schenkerian
approaches would permit this, and in those cases an analyst would need to transform
one or more of the chords at this level to comply with normative voice leading.
Lerdahl believes that all empirically supported listening theories for all musics will,
in the important details, look very much like his own; as he says, “[we] do not hear
Elektra and Erwartung in completely different ways.”7 Given this theory, then, atonal
music faces two problems. One involves a stylistic feature of much atonal music, viz.
that its rhythmic structure, having little to do with the concerns of tonal music, is not
always subsumable under a metric hierarchy.8 Because the very first step of analysis
in tonal theory is time-span grouping, applying this to music without a clear
metric hierarchy will be difficult from the outset. e second, more important problem,
is that atonal music lacks stability conditions: there is no syntactic way to determine
which events should appear at the next level up. Lerdahl suggests that in the absence
of syntactic stability conditions, an event’s relative salience will necessarily determine its
position in the perceptual hierarchy, as it does in tonal analysis when one is choosing5A representational hierarchy is one where lower parts of the hierarchy appear verbatim higher up in
the hierarchy as the “representative” of the lower part, while inclusional hierarchies allow some abstractionat higher levels. See Cohn & Dempster, “Hierarchical Unity, Plural Unities,” 1992 for a discussion ofthe distinction.
6Lerdahl, Tonal Pitch Space, 2001, pages 22-23.7Lerdahl, 1989, page 67.8One could argue that there is also much tonal music which also does not share these rhythmic con-
cerns.
ree Authors on Generalized Prolongation 11
Figure 1.1: A -style analysis of a Bach chorale.
P A 12
between events that meet the same stability conditions. He therefore proposes a set
of salience conditions for atonal analysis that could serve analogously to the stability
conditions of tonal analysis. After parsing the surface into levels according to relative
salience within time-spans, one connects events to a prolongation tree based on the
relative similarity of pitch, and based on whether or not a sequence of events intuitively
feels like a “departure” or “return.”
Lerdahl discusses some obvious problems with his approach, the most serious being
the question of whether salience in atonal music is in fact cognitively analogous to sta-
bility in tonal music. Indeed, “…if one event is more stable and the other is more salient,
there is a conflict in the rules [for determining cognitive dominance]. In tonal music
stability almost always overrides salience. One might say that the grammatical force of
tonal pitch structures can be gauged by their ability to override surface salience.”9 e
stability-salience analogy “…amounts to an acknowledgement that atonal music is not
very grammatical.”10 After some analysis of music by Schoenberg, he concludes by reit-
erating the abiding difficulty with a prolongational approach to atonal music, viz. its lack
of an abstract, a priori pitch/PC hierarchy. He implies that music organized around set-
classes or rows will only ever be cognitively valid to the extent that its surface is designed
specifically to project its relationships through such a salience-based event hierarchy.
Figure 1.3 is Lerdahl’s attempt at a prolongational reduction of the first nine bars
of Schoenberg’s Klavierstück, Op. 33a,11 shown in Figure 1.2.12 is is an interesting
piece to choose for prolongational analysis because its 12-tone structure is so well un-
derstood; if Lerdahl’s approach with this piece yields little analytical fruit, this will not
disrupt his claims about atonal music, as we shall see. Level f is the first step of the
time-span reduction from the surface, which normalizes the rhythm to quarter notes,9Lerdahl, 1989, page 73.
10Ibid., page 84.11Lerdahl, Tonal Pitch Space, 2001, pages 376-377.12Used by permission of Belmont Music Publishers, Los Angeles.
ree Authors on Generalized Prolongation 13
Figure 1.2: Arnold Schoenberg, Klavierstück, Op. 33a, bars 1-9.
and which removes pitches that are not salient according to the criteria. Most impor-
tantly, all “inner voices” are removed from each simultaneity occupying a given span, so
that no trace of the harmonies remains; at this level, the important tetrachordal associ-
ations in this piece are missing. In this way harmonies are treated very similarly both
to tonal harmonies in theory, and to motives in traditional Schenkerian theory.
Inner voices are no longer retained in deep levels of theory because at those levels
the harmonies are usually both psychologically more distant than the outer voices, and
are implied by the syntax. Similarly, there is no attempt to preserve foreground motivic
design in deep levels of traditional Schenkerian graphs if tonal syntax and motivic de-
sign disagree on their relevant joints, and given the associational nature of Schoenberg’s
tetrachords it may seem appropriate to discard these “motivic” features at early levels of
reduction. However, I believe this approach biases the analysis against the feasibility of
P A 14
Figure 1.3: Lerdahl’s prolongational reduction of bars 1-9 of Schoenberg’s Op. 33a.
ree Authors on Generalized Prolongation 15
psychological retention of simultaneities too early in the process, and thus forestalls any
possible “salience” that might result from interactions between pitches of inner voices.
ere is much more to critique in Lerdahl’s paper, even on its own terms. e mo-
tivation for the system he outlines is to avoid what he believes is the ad hoc nature of
musical set-theory analysis (and therefore the listening theory implied by such an analy-
sis). However, nearly every stage of analysis in this system succumbs to one or another of
his own criticisms of musical set theory. His complaint that there is no reliable method
of segmenting an atonal surface applies equally to the very first step in his atonal prolon-
gation method, since it relies on ad hoc and intuitive segmentation of the surface into
time spans, in the absence of metric hierarchies and cadential regularity. And while it
may be possible to apply the salience conditions consistently, the connections made in
the prolongation tree are also only ad hoc and intuitive. I do not think that Lerdahl
would disagree with these criticisms because much of his project is devoted to demon-
strating tonal music’s cognitive superiority over atonal music: if the only empirically
valid listening theory fails to deliver systematic results when applied to music that used
a systematic compositional theory, so much the worse for the latter theory’s cognitive
validity.
One might object (rightly, I believe) that Lerdahl’s approach to atonal music as-
sumes what it is trying to prove, viz. that his atonal prolongation theory, derived from
the tonal prolongation theory in , does not demonstrate anything syntactically
useful about atonal music because atonal music does not have the kind of grammar that
tonal music has. Of course, the problem is his disbelief that anything other than a hier-
archical prolongational model can possibly be the basis of a listening theory. He avoids
entirely begging the question by appealing to evidence from cognition experiments.13
e question of whether this evidence is convincing is complex; such a “cognitive con-13See, for instance, Lerdahl, “Cognitive Constraints on Compositional Systems,” 1988 for a review
of the evidence and a list of constraints Lerdahl believes a piece of music must satisfy in order to becognizable.
P A 16
straint” must be a constraint on what may be learned to be heard, not on what is heard
by an untrained listener. In my opinion this is too ambitious a claim to derive from the
available evidence, especially for music with an explicit grammar like Schoenberg’s or
Babbitt’s twelve-tone music.14
In any case, I shall set aside this question and consider some broader concerns about
Lerdahl’s paper: in what sense is it a generalization of prolongation, and what virtues
of the approach worth retaining? His method does have many features that resemble
those from more traditional systems of prolongation. In particular, the idea that there
are multiple levels of analysis, and that within each level a given event is present until it
is displaced by an event on the same level and controls or represents other events in lower
levels is indispensable for anything defined as “prolongation.” Moreover, it is likely that a
prolongation theory must include a grammar which constrains events; for this grammar
would limit the permissible harmonic tokens and would define an unambiguous syntax
over those tokens. Although the tonal prolongation theory shares all of these
features with Schenkerian theory, grammar is where Lerdahl’s extension of – by
his own design – begins to diverge from usual definitions of prolongation.
is divergence is useful for showing why the tonal model is a poor one to
generalize from. Lerdahl seems to envision a spectrum of music from tonal to atonal
in which stability conditions are more functional or less functional, but in which the
perceptual model is the same throughout.15 is preoccupation with perception makes14See Mead, “Twelve-Tone Organizational Strategies,” 1989 for an excellent foundation on which to
build a listening theory for twelve-tone music. Straus, Twelve-Tone Music in America, 2009, pages 214-218, “e Myth of Imperceptibility” is another cogent objection to the opinion that twelve-tone musicis aurally incomprehensible. Straus writes, “…every perceptual study of twelve-tone music of which I amaware has tested the wrong things. e abilities they are testing are not the ones that people need to makesense of this music.” I share this view and in general the attitude toward perceptibility that permeatesStraus’s book.
15In fact, he believes his approach will work the best for pieces from the 20th century which use bothtonal and atonal elements, and he justifies it with an observation about history: “…the historical develop-ment from tonality to atonality (and back) is richly continuous. eories of tonality and atonality shouldbe comparably linked.” (Lerdahl, 1989, page 67)
ree Authors on Generalized Prolongation 17
its relationship to compositional theory unclear, and although one could derive a strategy
for designing surfaces from Lerdahl’s approach, there is little basis on which to organize
higher levels. is is true even in the tonal theory because the higher levels of
reductions are not themselves grammatical.
ere are three related consequences for composition. First, all tonal surfaces (and
with Lerdahl’s extension, all musical surfaces) are supposedly explained by the theory,
so unlike with Schenkerian theory there is little in this theory that could separate com-
petent composition from inexpert composition. Any hope for creating a pedagogy that
trains composers to produce musical surfaces in relation to the theory is therefore for-
lorn. Second, because events at higher levels generally pass through as is to the surface
under theory, there is no good way to work from a syntactic background to a syn-
tactic foreground. ere are few well-defined transformational relationships between
levels that are needed for such a theory to work compositionally. Indeed, the back-
ground levels’ lack of syntax in analyses makes them in some sense uncognizable
as musical entities, and as such they can only be emergent products of surface details
and Lerdahl’s psychological constraints – background levels do not share the ability to
emerge from and control the surface simultaneously, which is such a rich feature of tra-
ditional prolongation. ird, because events find their place in the tree without change,
there is almost no way to define concepts like elision, where the next level up from the
surface might have more events than the surface itself.16 is third problem is not as
limiting as the first two, but it does have the effect of flattening the tonal space and the
tonal grammar by relativizing consonance and dissonance to a prolongation tree rather
than treating a dissonance as an event contrapuntally derived from higher-level con-
sonances. If a dissonant event takes its place verbatim as part of a prolongation tree
at all levels in which it participates, there can be no transformational relationship be-16ough, there are transformation rules that allow some limited deletion of pitches from events as
analysis progresses toward the background.
P A 18
tween consonance and dissonance, which further limits the theory’s utility as a
compositional theory.
e concerns outlined above amount to one main objection to the approach,
which is that it lacks well-defined transformational relationships from background to
foreground, making it difficult to define a corresponding compositional theory that al-
lows simultaneous deliberative work on multiple levels. us even if the theory
manages to be a good listening theory, it fails in one of its primary purposes, which is
to unite compositional theory with listening theory. Because the present study is about
composition, I will require a more rigorous understanding of “prolongation” than Ler-
dahl’s from which to generalize. However, there are two broad features of Lerdahl’s
work that I think are worth retaining.
One is the commitment to the relationship between prolongation and time. I do
not agree with the strict time-span reduction method – if, as he says, tonal grammar is
somewhat independent of salience and motivic relations, it can just as easily subvert any
common understanding of rhythm. Tonal syntax and rhythm work together, but the
relationship is not as easily defined as in . On the other hand, his intuition that
material can be prolonged backward in time from a stable anchor is an important and
somewhat misunderstood feature of Schenkerian syntax.17 In the Schenkerian litera-
ture such is called an auxiliary progression, and I would suggest that the ability to form
auxiliary progressions is a good test for the power of a prolongational system.
e second virtue of Lerdahl’s work, particularly in Lerdahl, Tonal Pitch Space, 2001,
is the intuition that the event hierarchy which constitutes a prolongational level analysis
itself depends to a large extent on abstract pitch or PC hierarchies which remain in play
over the course of a composition. e chief concern of my dissertation is connecting the
structure of equal-tempered systems with tonal hierarchies, so the further connection17In and Lerdahl’s subsequent work, this takes the form of a leftward-pointing branch from a
prolongation tree.
ree Authors on Generalized Prolongation 19
of tonal hierarchies with prolongational event hierarchies needs to be established. is
is also likely the best way to forge a strong relationship between neo-Riemannian the-
ory and Prolongation (see “Harmonic Function in Neo-Riemannian and Prolongational
eories”, page 92 below). I will argue for some sufficient conditions tonal spaces must
fulfill in order to give rise to prolongation in my review of Joseph Straus’s 1987 paper
on page 35 below. However, I must first consider John Rahn’s approach to generalizing
prolongation.
J R: “L, S T, M T”
John Rahn frames his 1979 paper as a pedagogical lesson in formalized theory build-
ing.18 Noting that Schenkerian analysis is too “broadly tailored” to comport with his
hearing of e.g. late Brahms, he believes it necessary sometimes to build more specified
theories for a given literature – or even for a given piece – which may share many fea-
tures with the broader theory.19 Because a theory must also be intelligible if it is to
be shared with others, a good way to build a theory is to define concepts in a formal
predicate logic.20 Any system based on logically well-formed definitions will have the
virtue of being unambiguous but flexible and extensible, but there are many potential18Rahn, “Logic, Set eory, Music eory,” 197919Rahn’s orientation is almost opposite to Lerdahl’s – Rahn wants different particular theories for
different pieces while Lerdahl wants one theory for all cognizable music. ese two approaches to gen-eralizing prolongation differ in the level at which the generalization occurs: Lerdahl’s strategy is to createa single generalized theory, while Rahn’s is more a framework for spawning a set of separate but relatedtheories. e ways in which the latter theories are related, and the extent to which they are so related,determines the strength of the intuition to call such an organizing framework a theory (if they are moreconnected) or a metatheory (if they are less so). Although my approach in this essay produces a set oftheories each of which only operates in its own limited domain, the connections among those theoriesjustify calling my framework “a theory”: it is a theory in much the same way the theory of combinatorialarrays in twelve-tone music may be considered a theory even if the ordering of a particular row producesa single, specifically tailored theory for a particular array and composition derived from that array.
20He suggests that discourse ought to be either formal or informal, and those authors who write infor-mally should not claim to define anything in case such a “definition” does not meet the criteria for formaldefinitions.
P A 20
obstacles. Even well-formed definitions may be vacuously instantiated,21 and others
may ultimately have very little cognitive, intuitive, or practical application.
As an example of such a project, Rahn puts forward a series of formal definitions
which are general enough to be used for building a prolongation theory that could struc-
ture many types of musical material. We need not be concerned with the first few
definitions, which point to some usual foundational concepts, such as “note,” “rest,”
“time-adjacency,” and “pitch-adjacency.” e more interesting definitions begin with
the “definitional schema” IVC, which states:22
IVC x and y are pitch-adjacent with respect to C IFF C is a cyclic orderingof pitch-classes and x and y are notes whose pitches are lessthan an octave apart and belong to pitch-classes that are ad-jacent in C.
is extends naturally to other definitions fitting this schema, for instance:
IVD x and y are chromatically adjacent IFF x and y are pitch-adjacent withrespect to the chromatic scale.
IVF x and y are extended diatonically adjacent IFF x and y are pitch-adjacent with respect to a major or harmonic minor ormelodic minor scale.
rough creative definition, “Pitch-Adjacency” is thereby generalized to any cyclical
ordering of PCs, away from the traditional notion that only major, minor, and – on
occasion – augmented seconds define pitch-adjacency.
From here, it is an easy step to define various neighbor relations, the most general of
which is:2321For example, suppose by definition X is an “anahedron” IFF X is a platonic solid in a three-
dimensional euclidean space and the faces of X are regular hexagons. e set of all such X is the nullset although the definition is well-formed.
22Rahn (1979), page 119.23Ibid., page 120.
ree Authors on Generalized Prolongation 21
VC x and y are neighbors with respect to C IFF x and y are time-adjacentand pitch-adjacent with respect to C (IVC)
And finally a corresponding notion of neighbor prolongation:24
VIC x and y NC-prolong z IFF x and y are neighbors with respect to Cand z is a note whose pitch equals the pitch of x or of y andwhose initiation (value of T1) is the earliest initiation of xor of y and whose release (value of T2) is the latest releaseof x or of y.
What this says in plainer English is that a pitch z may be prolonged by pair of pitches
x and y, which are adjacent with respect to cyclic ordering C and which are also time-
adjacent, so that either the first or last pitch in the pair is the same as z and the total
duration taken up by x and y together is the same as z’s duration. In Schenkerian voice
leading this would be considered an “incomplete neighbor,” and which pitch of the
two is considered “structural” would be determined by the context. By restricting the
definition to “incomplete neighbors,” Rahn allows it to work both for what would be
neighbor prolongation and passing-tone prolongation in Schenkerian theory, though in
principle under this definition the neighbor or passing tone would have to prolong either
the earlier or later pitch, with neither choice prescribed by the definition.25 Prolongation
by arpeggiation is defined similarly:26
VII A arp-prolongs B IFF A is a set of notes or rests and B is a setof notes and a pitch is in A IFF it is in B, and all initiations(T1) in B are equal to each other and equal to the earliestinitiation in A, and all releases (T2) in B are equal to eachother and equal to the latest release in A.
24Ibid., page 121.25ere is a similarity to theory here; because of the strict bifurcating tree structures in , a
neighboring or passing event may only attach to one node of the tree.26Ibid., page 121.
P A 22
is defines B as a simultaneity in which all the notes begin and end together. A set of
notes and rests A prolongs B by arpeggiation when all of its pitches (some of which may
be repeated) are found in B, and the total duration of A taken as a unit is shorter than
or equal to the total duration of B. B is uniquely determined by the pitches which occur
in A. Again, no attempt is made to ensure that all of the pitches in B create a “structural
harmony,” but such a system-refining rule could be defined.
Before I proceed further, I must stress that Rahn’s rudimentary model here is a sys-
tem of definitions of formal predicates rather than the more usual Schenkerian grammar
of harmonic types and transformations. In the latter case the types, transformations,
and structures such as levels all ought to be defined carefully, of course, but once they
are defined they take on different roles within the theory. e transformations become
procedural tools used to manipulate tokens of the harmonic types, in order to traverse
levels.27 By contrast, everything which counts in Rahn’s model is a definition. e
theory does not show how to get from one level to another by transforming musical
objects, but rather defines a relation (e.g. “next-background-to”) between two sets of
notes as a predicate on one of those sets so long as they meet the criteria specified by
the definition. Even “level analysis” is defined as a predicate on a set of sets of notes. In
such a system, operations on levels that generate other levels are implied if an analyst
or composer takes it as his or her task to produce the sets of notes that comply with the
definitions: all of the relevant operations will have been accomplished in virtue of hav-
ing found definition-compliant sets. So while the difference between a system designed
with predicate definitions and one designed with operations seems to be an intuitively
important difference with respect to the activity of the analyst or composer, both types
of systems are logically equivalent.27Near the end of the article, Rahn says his music theories have tended to be of this form: “…in-
dependent axiomatic or deductive systems, in which relations like neighbor-note prolongation becometransformation rules and things like Schenkerian Ursätze become axioms; each level of any analysis be-comes a theorem of such a theory.” (Rahn (1979), page 126)
ree Authors on Generalized Prolongation 23
Moving forward, Rahn gives three definitions involving levels and their relations:28
VIII A is a next-background to B IFF A and B are distinct sets andfor at least one set A1 and at least one set B1, A1 partitions Aand B1 partitions B and there is at least one one-to-one cor-respondence, X, from A1 to B1, such that for every memberof X, < a,b > ,b = a or b NC-prolongs a or b arp-prolongs a.
IX A is a level-analysis of B IFF B is a set of notes or rests and Ais a set of sets of notes or rests and B is an element of A andevery member of A except B is a next background to exactlyone member of A.
X A is a level IFF for some value of X and Y, A is an element ofX and X is a level-analysis of Y.
It may be counterintuitive to define level in terms of level analysis rather than the other
way around, and some may even find the notion circular – how can there be a level
analysis if we do not yet know what a level is? Any circularity, however, is apparent only
in the label affixed to the predicate; this is in fact a very efficient set of definitions. e
B of definition IX is most likely the surface of some composition or passage, but it need
not necessarily be – it could be a deep middle-ground structure, for instance. What
is important is the chain of next backgrounds, of which B is the “least background.”
Definition VIII is, as Rahn says, “slippery,” so let us recast it as an informal operation.
To obtain a next background to level B (call it A), first find a way to partition all of
B’s notes into a collection of note and rest sets (by whatever relevant means are defined
elsewhere), so that each note or rest appears in one and only one set in the collection. For
each member (i.e. set of notes/rests) of that collection, determine whether it could count
as an arp-prolongation or NC-prolongation of some other note/rest set. If so, place the28Rahn (1979), pages 122-123.
P A 24
latter note/rest set at the corresponding time point in A; if not, add the set from B
verbatim to A at the corresponding time point. A will then be a next background to B,
unless no prolongation was found, in which case A and B will be identical sets. On this
view, there could potentially be hundreds of levels for a given tonal composition.
Figure 1.4 is an example of a level analysis for the first four (and eventually eight) bars
of Mozart’s famous A-major sonata, K. 331.29 e level analysis complies with Rahn’s
definitions, but it is not the only level analysis which could. On the other hand it does
not comply with every Schenkerian approach – note the parallel fifths between bass and
soprano at level 6, for instance. Rahn’s analysis has features that are underdeterminied by
his definitions, however; the intuitive partitioning of each level into triads, for example,
is implied by none of the definitions given so far.
To illustrate why this is a problem, I have created the totally implausible level analysis
in Figure 1.5, which is also fully compliant with Rahn’s definitions. Mozart employs two
primary trichord types in the harmonization of the theme (viz. [037] and [025]); the
new level analysis takes [025] as structural. Normal slurs are used to indicate neighbor
prolongation into the next-background, while dashed slurs are used for arpeggiation
(the slurring in level 1 is left in its original form).
I deliberately included many similarities between the two level analyses. e second
level in my analysis is nearly identical in content to Rahn’s, save the E in the soprano
in bar 4. In level 3, Rahn and I both have the same [025] trichord in bar 3. In Rahn’s
seventh level he notes that the consequent phrase is necessary to connect the G# in
the second chord as a neighbor prolongation (while retaining the implicit triadic con-
straint); similarly in my fifth level a hypothetical consequent phrase reduction is needed
to connect the C# in the final chord of bar 4. In several places I have made serendipi-
tous use of the N*-prolongation definition, which (described informally) allows pitches
a perfect fourth or fifth away from another pitch to prolong that pitch as though it were29Ibid., page 125. Used with permission.
ree Authors on Generalized Prolongation 25
This content downloaded on Fri, 18 Jan 2013 17:57:50 PMAll use subject to JSTOR Terms and Conditions
Figure 1.4: Rahn’s K. 331 level analysis
P A 26
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Figure 1.5: Another K. 331 level analysis which complies with Rahn’s definitions.
ree Authors on Generalized Prolongation 27
a neighbor, provided both pitches appear in the bass voice. One difficulty appears in
level 6 in my analysis (marked with a ?): if the E in the soprano is a prolongation of
the following D, is this a neighbor prolongation or an arpeggiation of the [025] sonor-
ity? Given the definition and implicit constraints, this is an ambiguity that is difficult –
maybe impossible – to solve; I shall discuss ambiguities like this in the next section.30
My “[025]” level analysis is not meant to be a criticism of Rahn’s definitions. Ob-
viously, further definitions or amendments are required to make his level analysis com-
pliant and to rule out mine. Near the end of his article he does, in fact provide four
new clauses to definition IX (which defines “level analysis”), in order to further tailor
the theory to one in which his analysis would comply and mine would not:31
1. … and the most background level contains only one T1 and one T22. … and for any timespan within some level that contains only
pitches that belong to a certain diatonic collection, somemore background level must contain only pitches in thetonic triad of the diatonic collection within that timespan
3. … and the most background level of A must contain no pitchesother than those of a (root position) major or minor triad
4. … and some level of A contains only the triads I-V-I in that order,embedding a Schenkerian Ursatz.
I am not concerned about whether my “[025]” level analysis is really an example of pro-
longation – and thus by extension, whether Rahn’s definitions without the extra clauses
form a sufficient basis for a real prolongational system. However, it should be obvious
that Rahn’s definitions, without further specification, could lead to unsatisfactory, yet
compliant level analyses. For instance, a level analysis could comprise the surface of the
piece and only one other level: the chord which contains every pitch appearing in the30e same event occurs in the first bar of level 3, but since the E in the soprano is accompanied by the
C# in the bass, it seemed most appropriate to label it definitively as a neighbor prolongation. e samereasoning applies to the fourth bar of level 4 in the bass – the E in the second beat is accompanied by aC# in the soprano. In the latter case, the resulting chord (E-B-C#) is a “hidden” [025] sonority!
31Rahn (1979), page 126.
P A 28
surface; this chord would be an “arp-prolongation” of the surface. However interest-
ing such a simultaneity might be,32 a level analysis of this kind would not meet most
standards for an adequate model of prolongation.
From the tone and scope of the paper it is obvious that Rahn is presenting a theory-
building strategy, not a complete theory. A theorist using Rahn’s definitions as a point
of departure will likely wish to add “extra clauses” to definition IX in order to constrain
the system sufficiently. is means that two such theorists will most probably agree
in the spirit of their respective projects, but not on the particulars. It is difficult to
say just how much extra constraint is necessary to avoid underdetermined theories and
ambiguities like the one illustrated in the Mozart example above. In fact, a theorist
taking definitions IVC, VC, and VIC, which generalize pitch adjacency, the neighbor
relation, and neighbor prolongation, respectively, to a given cyclic ordering of PCs C, as
the only relevant pitch relations and without further constraint, would have a theory that
could produce a fully compliant level analysis for almost any musical surface in 12-tone
equal temperament.
I can think of no better way to show this than to derive a surface from a simple back-
ground, using only NC-prolongation and arp-prolongation. is will allow a proper
evaluation of Rahn’s approach as a compositional tool – see figures 1.6 and 1.7. e
cyclic ordering P is T2 of the first 12-tone row from Alban Berg’s Lyric Suite (i.e. the
transposition of the row starting on D). e cyclic ordering I is T0I of the same row
(i.e. the inversion starting on C). ese two forms of the row are related such that the
second and penultimate PCs are C#/Db and Gn; these two PCs shall form the most
background level, level 8. In addition, the PC content of the discrete hexachords in the
two rows differ from each other only by these two PCs. Let P-prolongation be “NC-32Especially for something like the first movement of Webern’s Symphony Op. 21, or the second
movement of his Piano Variations, Op. 27, each of which employs “frozen register.”
ree Authors on Generalized Prolongation 29
prolongation with respect to P,” and I-prolongation be “NC-prolongation with respect
to I.”
Because Berg’s Lyric Suite row is an all-interval row, P- and I- prolongation can
involve every interval-class; obviously this is quite different from traditional tonal pro-
longation, where each IC has a highly constrained range of syntactic functions. Here,
restricting types of NC-prolongation to two cyclic orderings serves a similar purpose:
each IC appears in only four PC adjacencies in the pair of cycles.33 However, one dif-
ficult problem is that a given PC adjacency in one cycle could only appear as an arpeg-
giation in the other; the immediate function of a given dyad can only be determined
contextually.
A brief description of the compositional strategy for this “level synthesis” follows. P-
prolongation is indicated with solid slurs, I-prolongation with dashed slurs, and promi-
nent arp-prolongation with brackets.34 Level 8 shows the source sonority mentioned
above, with an implied 5.5-measure phrase. Level 7 develops this with a few simple
arpeggiations, a simultaneous I-prolongation in the alto and bass of the last bar, and
a P-prolongation in bar 3. e voice-crossing between tenor and alto present a prob-
lem already, but we shall assume that there are contextual means other than register to
distinguish voice, e.g. instrumentation, or employing P- and I-prolongation more than
arp-prolongation, such that new pitches are more likely to derive from an adjacency in
one of the PC cycles. e rhythm is important here as well, as it was in Rahn’s Mozart
analysis: the attack points associated with the various prolongations tend to divide bars
into 3:1 or 1:3 duration ratios.33Without this restriction, any musical surface in 12-tone equal temperament could be derived from
just six forms of the Lyric Suite row (e.g. T0, T1, T2, T3, T4, and T5), using NC-prolongation only, sinceevery possible PC dyad would appear as an adjacency in the collection of rows.
34Note-value truncation is considered a type of arpeggiation in this system; I have not explicitly notatedtruncations.
P A 30
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Berg, Row Forms From Lyric SuiteT2 T0 I
Figure 1.6: “Level synthesis” using row forms from Berg’s Lyric Suite.
ree Authors on Generalized Prolongation 31
Level 6 develops the idea of simultaneous P- and I-prolongations and 3:1/1:3 dura-
tion ratios. Each of the four pitches from the source sonority participates in at least one
“double-neighbor” prolongation, revealing that conceptually each pitch from the source
sonority is doubled in levels 8 and 7. Further, each of the possible double-neighbors
using P- and I- prolongation appears at this level: both possible double-neighbors of
G in the soprano, both possible double-neighbors of C# in the bass, and one each in
the alto and tenor. Rhythmically, two of these prolongations (in bars 2 and 4) divide
half-notes into 3:1 and 1:3 durations ratios, implying a diminution of rhythm to at least
the 8th-note level. In bar 3, note that the double neighbor CnFn → C# occurs with an
anticipation of C# from the Cn.Levels 5 and 4 further employ further middle-ground development. Some motivic
features are beginning to develop. For instance, the I-prolongation Fn←C# in the alto
voice of bar 2, level 5, is repeated as C#→Fn in the baritone voice of bar 3, level 4. Some
regularity of simultaneity is beginning to come into focus as well: bars 4-6 in the bass
clef staff contains three trichordal simultaneities which appear elsewhere, and which also
seem to be in play linearly in the soprano, derived by other means. For convenience, level
4 is repeated at the top of figure 1.7.
In level 3, almost everything is in place. At this level, the most salient feature in
the first three bars is a division of the texture into a “melody” which begins with the
soprano Dn-C#-An “passing-tone” figure, and an “accompaniment” which manifests the
1:3 rhythmic idea as a quarter rest followed by three quarter notes. ere are two half-
notes in the bass – a Bn in the first measure and an Ab in the second – which require one
last prolongation to fit with the rest of the accompaniment texture. ese are completed
in level 2, as En←Bn in the first bar in order obtain the motivic Cn-En-Ab simultaneity in
beat 3, and Ab→Gn in the second bar to repeat the same prolongation which occurred
in the soprano between bars 3 and 4, and in the tenor in bar 1. Due to the rhythmic
P A 32
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Figure 1.7: “Level synthesis” using row forms from Berg’s Lyric Suite, continued.
ree Authors on Generalized Prolongation 33
constraints associated with the prolongation, the Gn in the alto voice of bar 3, which
was first generated at level 4, cannot last a full quarter note. us, in level 2 all of the
quarter notes in the “accompaniment” have been truncated to eighth notes. Level 1 is
an orchestration of the final texture for string quartet, assigning the “melody” to the first
violin, and the pitches of the “accompaniment” to the other three instruments simply
according to register. is is in fact the opening phrase of Schoenberg’s iconic Fourth
Quartet, op. 37.
Now obviously this is not a very plausible reading of Schoenberg’s composition, but
this is beside the point. e question at hand is whether a prolongation theory which
can produce a compliant analysis of Schoenberg’s Fourth Quartet using as the structural
sources two row forms from Berg’s Lyric Suite and a simultaneity (C#-Gn) which occurs
only twice on the surface – and there only incidentally – is recognizable as a prolongation
theory. For my purposes I must answer in the negative, even though it may be tempting
to say that this has a Wittgensteinian “family resemblance” to traditional prolongation
theory.
Chief among the many problems is the lack of any constraint on simultaneity at any
level, save for the seemingly accidental generation of the four discrete trichords from
Schoenberg’s row at the least background levels. Much of the richness of Schenkerian
theory comes from the self-similarity of the levels in the analysis, which follows from
employing the same contrapuntal and harmonic rules at all levels. Harmonic constraint
allows a systematic progression from foreground to background and vice-versa. is is
totally missing from the Berg-Schoenberg example here. While all the compositional
decisions made moving from background to foreground can be justified using NC- and
arp-prolongation, these decisions can scarcely be inferred analytically from anything
on the surface; there is no way to tell at any level which pitches are “more structural”
than others. And since simultaneity is not constrained, there can be no hierarchiza-
P A 34
tion among simultaneities. A second, related problem is that there is nothing as yet
analogous to a hierarchization of “regions” or “keys,” though one could be established
contextually. However, in traditional tonal theory, the hierarchical relations among re-
gions and keys is similar to the hierarchical relations among triads, which cannot be the
case in the Berg-Schoenberg example, again because simultaneity is not constrained.35
A basis for solving these problems emerges from the ideas taken up in the next sec-
tion. As with Lerdahl’s work, there are two features of Rahn’s approach which shall
prove useful. First, much like Lerdahl, Rahn’s schemata tie prolongation to rhythm,
and allow “backward prolongation.” Rahn’s treatment of rhythm is much more flexible
than Lerdahl’s, and has the potential to reveal “hidden motives” and rhythmic consis-
tencies across levels. However, by treating time spans literally – even at the highest
levels – elisions and Kirnbergerian “essential dissonance” will not always be derivable
from a more consonant next-background. Rahn’s second virtue is the idea that a system
of prolongation could be generalized to a cyclic ordering of PCs (a key insight is that
diatonic scales are both collections and orderings, and each perspective is important).
I have tried to show above that not just any cyclic PC ordering will create a coherent
system; much of my work is focused on what kinds of constraints on cyclic PC order-
ings are relevant for generating the kinds of highly regular and constrained pitch-spaces
which feature so prominently in Lerdahl’s recent work. Joseph Straus offers some very
helpful guidelines for what features a system must have in order for it to be a system of
prolongation; I take this up next.35Scotto, “A Hybrid Compositional System,” 2000, which uses Rahn’s framework as a foundation for
prolongation, would suffer greatly from these harmonic problems were it not for a deeper, strict organizingprinciple involving arrays derived from self-similar 12-tone rows. Unfortunately, Scotto’s fascinatingwork lies outside the scope of this dissertation.
ree Authors on Generalized Prolongation 35
J S: “T P P
P-T M”
In this important paper,36 Joseph Straus critiques what he believes are misapplications
of prolongation theory to post-tonal music such as Stravinsky and early Schoenberg.
Although he spends considerable space discussing specific problematic prolongation
analyses, these discussions are beyond the scope of this dissertation. Rather, it is the
simple and elegant (meta)theoretical foundation he opens the paper with which may be
considered the main point of departure for my own theory. According to Straus, any
music to which prolongation theory is applicable must meet four necessary conditions.37
Many musical grammars may meet one or two of these conditions, but the four taken
together are greatly constraining, while leaving enough flexibility to move beyond triads
and common-practice tonality, at least in principle.
Straus summarizes the four conditions thus:
First, there is the consonance-dissonance condition; we need a way based onpitch of distinguishing between structural and nonstructural tones. Second,there is the scale-degree condition; we need some kind of hierarchy amongthe consonant harmonies. ird, the embellishment condition; we need aconsistent model of voice leading that will enable us, for example, to tellan arpeggiation from a passing note. Fourth, there is the harmony/voiceleading condition; we need to be able to distinguish motions within a voicefrom motions between voices. Tonal music clearly meets all of these condi-tions; post-tonal music, in general, does not. As a result, post-tonal musicis not prolongational or, to put it another way, prolongation as an analyticaltool will not produce significant results.38
Conditions one, two, and four are best thought of as constraints on abstract tonal
pitch spaces, and condition three is a rather broad set of constraints on the kinds of re-36Straus, “e Problem of Prolongation in Post-Tonal Music,” 198737In this dissertation I shall often refer to them as “the Straus conditions.”38Straus (1987), page 7.
P A 36
lationships that can exist between syntactically well-formed levels. e reader may re-
member that in my critique of Lerdahl’s approach I indicated that nothing like the third
condition was present in his theory, while Rahn’s theory as it stood in a sense amounted
only to the third condition. Straus’s combination goes a long way toward harnessing
Rahn’s outline into a more meaningful generalization of prolongation. I shall now pro-
ceed with a discussion of each condition, saving condition three for last.
Straus’s first condition is the consonance-dissonance condition. In order for prolonga-
tion to apply, there must be a systematic way to tell between structural (stable, conso-
nant) sonorities and nonstructural (unstable, dissonant) sonorities. Further, this must
be a system of pitch relations, and while there is some room for contextuality, the im-
portant point here is that the consonant sonorities must be distinguishable from the
dissonances without very much extra context. In this case, “extra context” includes both
foundational, conceptual context and context within an actual musical composition. For
instance, the consonance of a sonority should ideally not depend upon its place in an
abstract scale or its place in an ordering of sonorities in a composition. In Schenkerian
terms, the consonance-dissonance distinction is the first and conceptually the most im-
portant way to tell which sonorities may be “composed out,” and thus which sonorities
may remain after a round of reduction moving away from the surface.
ere are two very important problems with this condition whichmust be addressed.
e first problem is that it is unclear whether “consonance” is a concept that applies to
intervals and combinations thereof, or to set classes; in other words, is a set class (e.g. the
major and minor triads in tonal music) admitted under the permissible consonances
in virtue of the consonant intervals it contains, or are consonant intervals consonant
only when they appear in or imply a given set class? In traditional theory the inter-
vallic approach would be best exemplified by Fuxian species counterpoint (and perhaps
the work of Zarlino), while the chordal understanding would be most compatible with
ree Authors on Generalized Prolongation 37
thoroughbass tradition (and tangentially, Rameau’s system). While this may seem like
a distinction without a difference in traditional tonal music, it has some fairly deep im-
plications with respect to generalizing the consonance-dissonance condition to other
equal temperaments. e second problem is a contextual one: might it be possible that
a dissonance is nevertheless contextually more structural than some of its surrounding
consonances? Another way to pose this question is to consider whether some “struc-
tural dissonances” may be composed out using ostensibly more consonant but contex-
tually less stable/structural sonorities. One approach might be to partition dissonances
into three categories: “essential dissonance” which may be composed out or which may
“stand for” a consonance (e.g. a dominant seventh chord which stands for a consonant
dominant triad), “permissible dissonant simultaneities” which may occur as the result
of voice leading but not be composed out (e.g. a suspension), and “impermissible disso-
nances” which may not occur in the syntax at all (e.g. a chromatic cluster of 6 pitches).
ese are tricky issues even in orthodox Schenkerian theory, so while I shall not
provide a universal solution to them, I will show why they are important to my gener-
alized theory of prolongation, and how I approach them in that context. I shall have
more to say about them presently, since Straus’s other conditions constrain the range of
possible solutions considerably. Let us call a set class containing the maximum number
of mutually consonant PCs in an equally tempered system a maximal consonance for that
system; more than one maximally consonant set-class may exist in a given system.
e scale-degree condition is the second of Straus’s conditions. According to Straus,
it is an extension of the first condition: prolongation requires not only a systematic dif-
ferentiation between consonant and dissonant simultaneities, but also a systematic way
to tell which of two or more consonant sonorities is most stable. Like the first condition,
this condition is meant to apply to abstract and a priori hierarchical relations among the
consonances rather than contextual relations among actual musical events. If the first
P A 38
condition is problematic, the second is even more so. Straus seems to suggest that there
should be a strong correspondence between tonal hierarchies and event hierarchies, to use
Lerdahl’s terms: if, say, a tonic triad is “more structural” than a dominant triad in a
tonal hierarchy, then a tonic triad should also be “more structural” than a dominant
triad when they occur adjacently in a musical passage (at the surface or at background
levels). is is to say that a sonority should never be used in the composing out of a
sonority that is lower in the tonal hierarchy.
As defined, Straus’s second condition is too strong even for common-practice tonal
music. However, since it is modeled on Schenker’s Stufen concept, which is an indis-
pensable part of Schenkerian theory, it will be necessary to salvage it. e best way to
do so is to split it into four less stringent sub-conditions. First, there must be a reference
scale type (a set class or a cyclic ordering thereof ) which embeds a subset of all the pos-
sible maximal consonances for the equal temperament under consideration.39 Each PC
of the scale must be contained in one or more of the consonances embedded in the scale.
Second, each PC of the scale must be distinguishable from others on the basis of the in-
tervallic relations between the PC and its complement set with respect to the scale. is
is often called “position finding” in diatonic scale theory. A consequence of this condi-
tion is that the set of consonances embedded in the scale will likewise be distinguishable
from each other on the same basis. ird, instead of an a priori hierarchy among con-
sonances, there need only be an a priori syntactic norm whereby each of the maximal
consonances in the equal temperament has a different, highly constrained set of func-
tions for each different transposition (or inversion) of a reference scale. is provides for
constraints on the function both of consonances embedded in a given instance of a scale
and those not so embedded. In most cases this third sub-condition requires the second
to be met as well. Fourth, and finally, the number of maximal consonances embedded39Normally any instance of such a scale includes only one of each of its constituent PCs, but in some
cases multisets and ordered multisets are possible.
ree Authors on Generalized Prolongation 39
in the scale must be nearly equal to the cardinality of the scale.40 is sub-condition
allows for an association between scale degrees and consonances built on them, as with
Stufen or roman numerals.
For an example of the revised scale-degree condition, let us examine how it functions
in common practice tonality. e major and minor triads are maximal consonances, and
the diatonic collection is a reference scale (for the purpose of the scale-degree condi-
tion, harmonic and melodic minor scales may be thought of as the result of mixture).
Together they satisfy all four sub-conditions just outlined. First, each transposition of
the diatonic collection contains three of the twelve major triads and three of the twelve
minor triads available in 12-tone equal temperament, and all PCs in a given diatonic
collection participate in one or more of these embedded triads. Second, no two scale
degrees have an identical intervallic relationship to the rest of a given diatonic scale, and
the same may be said of the embedded triads. ird, the function theories designed by
Riemann, Schoenberg, and neo-Riemannian theorists provide suitable syntactic norms
to constrain the way each triad may function within a key. Schenker’s own Stufen theory
is similarly suitable, from a different perspective. And fourth, a diatonic scale embeds
six consonant triads in its seven PCs.
As a counterexample, although the octatonic scale under triadic consonance fulfills
the first sub-condition, it does not fulfill the second and third. Each PC in a given
octatonic scale has intervallic surroundings identical to those of three other pitches in
the scale. Another way of saying this is that there are only two kinds of scale degrees
in an octatonic scale: the upper and lower PCs of chromatically adjacent dyads. Like-
wise, each of the four major triads in a given octatonic scale has the same intervallic
relationship to its octatonic complement set (and likewise for the four minor triads).
is means that they are not distinguishable in a way that would allow each to take a40For now this relation must be left vague, but see note 25 on page 115 for a more precise account.
P A 40
different set of functions in an established harmonic syntax.41 e octatonic scale under
triadic consonance does in fact fulfill the fourth sub-condition: it contains eight unique
major or minor triads (every PC is the third of a unique triad, rather than the root). An
example of a system which fulfills the first three but not the fourth is triadic consonance
in a hexachordal scale whose set-class is [023579]: there are only three embedded major
or minor triads among the six PCs, namely [037], [259], and [590].
I will postpone discussing the third of Straus’s conditions for now, and take up his
fourth, the harmony/voice leading condition, which – like the consonance/dissonance
and modified scale-degree conditions – depends largely upon relations among conso-
nances, reference scales, and the total “chromatic” under consideration. It consists in
two sub-conditions. First, a system of voice leading requires the ability to distinguish
individual “voices.” In a prolongational system, according to Straus, this is accom-
plished by an a priori distinction between harmonic intervals and intervals associated
with “stepwise motion.” In this sense, the fourth condition is also an extension of the
consonance/dissonance condition: all harmonic intervals are consonances and all voice-
leading (“stepwise”) intervals are dissonances, but not all dissonances are voice-leading
intervals. Second, voice-leading intervals must occur adjacently in a reference scale, and
harmonic intervals must not. Straus points out that in the diatonic scale, triads – the
“maximal consonances” – are also the simultaneities composed of the greatest number of
entirely mutually nonadjacent PCs with respect to the diatonic scale. Moreover, the tri-
ads can be generated from a single diatonic interval, the diatonic third. is all suggests
a second sense in which a maximal consonance is “maximal”: not only is it a set-class
with the greatest number of mutually consonant PCs, but it may also be desirable for
it to contain the greatest number of nonadjacent PCs in a reference scale which can be
generated from a single interval with respect to the scale.41A syntax could conceivably be established contextually, however, by means of phrase structure, or-
chestration, or other “secondary” (i.e. non-harmonic) norms.
ree Authors on Generalized Prolongation 41
In my discussion of my “[025]” level analysis of the opening phrase of Mozart’s
K. 331 (Figure 1.5, page 26), I noted an ambiguity in the harmony and voice-leading
constraints which create an analytical problem: “One difficulty appears in level 6 in my
analysis (marked with a ?): if the E in the soprano is a prolongation of the following
D, is this a neighbor prolongation or an arpeggiation of the [025] sonority? Given
the definition and implicit constraints, this is an ambiguity that is difficult – maybe
impossible – to solve.” At issue here is whether IC2 is a harmonic interval or voice-
leading interval for the purposes of Straus’s 4th condition, and whether it is therefore
possible in principle to determine what kind of embellishment is at play. Robert Morris
points out a possible rigorous solution to this kind of ambiguity: if voice leading rules
were such that voices were constrained to move only by a set of “voice-leading” intervals
from one consonant PC set to another and the PC set resulting from the union of the
two successive simultaneities were also members of a restricted set of permissible set-
classes, then any set-class can be a “consonance” in this context regardless of its interval
content.42 is works because any ICs which are both members of simultaneities and
voice-leading intervals are syntactically distinguishable with respect to their position
in the union set-class and its partition into the two simultaneities at play. Morris’s
voice-leading spaces have many other virtues, especially in their potential for creating
harmonic progression syntax which obey desired voice-leading rules, but my approach
differs from his in important respects.
Straus’s third condition is the embellishment condition. In order for a prolonga-
tional system to meet this condition, there must be a clear set of voice-leading rules
for how a set of notes may be transformed to create a new set of notes at a differ-
ent level.43 is statement is deliberately vague, because many authors have pointed42Morris, “Voice-Leading Spaces,” 199843In Lerdahl’s terminology, “transformation” has a different meaning from the one I have adopted here.
A “transformation” in Lerdahl’s work refers to an operation one may apply to a sonority at one level so thatit may appear in a different form in another. An important example is the deletion of pitches associated
P A 42
out different sets of such rules. For instance, at one extreme Rahn recognizes only
two kinds of transformations – “arp-prolongation” and “N-prolongation” (i.e. “arpeg-
giation” and “neighbor prolongation”); Straus recognizes these and “passing-tone pro-
longation” (for Rahn, passing-tone prolongation is a kind of N-prolongation). At the
other extreme, Schenker’s own set of transformations changes throughout his career,
but Matthew Brown points out no fewer than sixteen transformations.44
Following Brown’s classification, at the very minimum a traditional prolongation
theory requires four classes of transformation on triads and pitches:45
1. Arpeggiation of the pitches of a triad (repetition is included here, as is “respacing”
of a harmony at another level on octave-equivalence grounds).
2. Filling in the space between arpeggiated pitches with connectives (“passing tones,”
“neighbormotion,” and some kinds of “chromaticmixture”may be thought of as possible
connectives of the “filling-in” transformation).46
3. Support of a given pitch (which may have been a connective at a higher level) with
a locally stable triad (“chromatic mixture” also operates here).
4. Contrapuntally constrainedCombination of pitches from separate consonant struc-
tures, and/or Deletion or Elision of contrapuntal preparation and/or resolution of dis-
sonance. is last set of transformations is needed in order to explain “non-chord-
tones” as well as Kirnbergerian “essential dissonance,” so that in principle every 7thchord,
with “inner voices.” is transforms a low-level sonority so that only the two outer-voice pitches appearin higher levels. In contrast, my use of “transformation” includes Lerdahl’s as a proper subset, and addsto it a set of embellishment types.
44Brown, Explaining Tonality, 2005, pages 76-83.45What follows is an adaptation of Brown’s classification scheme for the sixteen Schenkerian transfor-
mations; a list of all sixteen would be too specific for the level of generalization under discussion.46“Filling in” is sometimes described instead as “displacement”: a “new” pitch comes to “displace” a
pitch that held sway earlier, sometimes forcing the other voices to respond. A pitch may be said to be“prolonged” when it is displaced by a given pitch (or succession of pitches) at one level, but not in another.For various reasons I believe that the “displacement” idea leads to unnecessary conceptual difficulties evenif it does comport with real-time experience of music. On the other hand, one advantage of “displace-ment” is its use in identifying “voices” and therefore in defining “polyphony.” See Boretz,Meta-Variations,1994, pages 208-215.
ree Authors on Generalized Prolongation 43
9thchord, suspension, appoggiatura, etc. is either a combination of notes from two con-
sonant chords (e.g. effected by “delay” of a pitch into the succeeding simultaneity as in
a suspension) or a contraction of some voice-leading motion in which the dissonance is
prepared and resolved.47 Matthew Brown notes that this fourth class of transformations
may usually not be applied recursively, because it renders some sonorities unstable and
therefore generally incapable of being composed out, except by arpeggiation.48
Usually avoidance of certain specific voice leadings (e.g. parallel fifths and octaves)
further constrains these transformations, but this depends greatly on the desired scope
of repertoire subsumed under the theory. An important feature of these transforma-
tions, as opposed to Neo-Riemannian transformations, is that these do not in general
have inverse operations, so prolongational structures are not group structures. is has
important implications for Schenker’s concept of levels.49
47A footnote from Boretz, Meta-Variations, 1994, pages 211-212 is worth quoting in full:
is “greater amount of inference” [required to account for a simultaneity which does notconform to a designated “consonant” chord type] arises from the condition that any suchsimultaneous presentation of succession-related pitches results from an “intensive” elab-oration of a preceding, “extensive” elaboration in which both simultaneities “implied” bythe presented event appear as a succession. us a “more background” level may have moreentities than a “more foreground” level generated out of it by intensive elaboration. So,for example, one might expect to find pre-foreground levels of Brahms’s Op. 116 No. 3(g minor) containing a considerably greater number of distinguishable (attack-specified)“events” than are represented in the “notational” foreground.
It is unclear whether Boretz’s “intensive elaboration” conceptually only involves combining pitches fromsuccessive consonant chords; there are some situations where deletion of an intervening consonance frombetween two dissonant simultaneities better fits the spirit of a musical passage.
In Blasius, Schenker’s Argument and the Claims of Music eory, 1996, the author calls this type of trans-formation “condensation” and discusses its connection to Schenker’s theory of counterpoint, auskom-ponieren, and his approach to musical psychology and epistemology. Note also that this type of trans-formation was one of the three conspicuously missing features of Lerdahl’s theory, which limited thecompositional utility of Lerdahl’s theory. See page 17 of this essay.
48Brown, Explaining Tonality, 2005.49See Harmonic Function in Neo-Riemannian and Prolongational eories on page 92 below for a
more detailed discussion of the relationship between prolongation and Neo-Riemannian eory.
P A 44
D S C
Straus presents his four conditions as necessary conditions for any system of prolongation;
is not clear whether he intends the four conditions, taken in conjunction, to be sufficient
conditions for prolongation as well. is would matter if there were some other essential
property of a prolongational system not captured by the Straus conditions; for instance,
Schenker purported to ground his system in the acoustic structure of harmonic sounds,
and for some readers this may be essential to prolongation. In the interest of casting
the widest possible net into the relevant design space, I will opt for taking the Straus
conditions as sufficient conditions for prolongation until any missing essential features
present themselves.50 e result is a set of systems which meet the Straus criteria but
which may be missing one or more features usually associated with tonal music.
It would be interesting to contemplate what musical systems would result from
meeting each of the fifteen combinations of the Straus conditions – for instance, what
kind of music could meet just the second and fourth conditions? Or just the fourth? But
since I am ultimately going to be interested in systems which meet all four conditions, it
will be most productive to discuss how the conditions interact with each other. e goal
of Part 2 is largely to enumerate which technical requirements of scales, simultaneities,
and voice-leading protocols will meet the four Straus conditions most generally; here I
reveal some of the relevant contours of the theory.
e most important interactions between the Straus conditions involve the con-
straints imposed upon the scales by the intervallic requirements of the first, second,
and fourth conditions (i.e. the consonance/dissonance condition, scale-degree condi-
tion, and harmony/voice leading conditions). Rahn’s NC-prolongation has no such
restrictions, so “any ordering will do,” but not any ordering fulfills the Straus condi-50I have included the epigraph at the beginning of Part 1 in order to make my views about essentialism
known without spending too much space discussing my methodology.
ree Authors on Generalized Prolongation 45
tions. For the remainder of this essay “scale” refers to a cyclic ordering of unique PCs,
not merely a PC set. For the set class associated with a scale ordering, I use “scale set”
or “scale collection” interchangeably.
e first and second Straus conditions together restrict the candidates for scale sets
to those which embed a large number of some set-class which meet the criteria for
“maximal consonance” within the temperament. e “position-finding” sub-condition
of condition two usually requires that scale collections not be transpositionally symmet-
rical.51 Because the second condition also requires a large number of embeddedmaximal
consonances relative to the cardinality of the scale, PCs must participate in more than
one maximal consonance. For this to be possible, the scale set must be generatable by
the consonances. is is best explained by illustration. Take the C-major scale as the
cyclic ordering <>; one may generate the PC contents of this scale using an
ordering of interlocking major and minor triads in “stacked thirds”: <>. All
of the imbricated trichords in this ordering are members of set-class [037]. Scale sets
with these properties are rare, and they are best discovered by generating them in this
fashion from predetermined consonances. David Lewin makes a formal theory of this
strategy in Lewin, “A Formal eory of Generalized Tonal Function,” 1982, which I
discuss below, beginning page 66.
Straus’s fourth condition (the harmony/voice leading condition) constrains the pos-
sible orderings of a scale cycle, and (less obviously) the cardinality of a scale set. e
fourth condition requires voice-leading intervals to occur between adjacent PCs in an
ordering and harmonic intervals to occur between non adjacencies. It also suggests (but
does not necessarily require) that maximal consonances be generatable by a single in-
terval with respect to the scale ordering.52 us a scale cycle must be ordered to ensure51Exceptions to this rule could arise when the transpositionally symmetrical scale collection under
consideration is made into a multiset by repeating one or more of its constituent PCs.52In this case, “intervals-with-respect-to-the-scale-ordering” must be understood as intervals in order-
position space rather than in PC space. If the ordering of the scale cycle is strictly ascending or descending
P A 46
regularity in the disposition of the maximal consonances. Moreover, the cardinality of
the scale set must also be a number which allows this kind of ordering to take place. To
illustrate, consider the scale set {}, with set-class [025] as the maximal conso-
nance. e scale contains five [025] trichords: {}, {}, {}, {}, and {},
so it contains the requisite number of consonances to pass the scale-degree condition.
However, since 6 (the number of PCs in the scale collection) is evenly divided by 3 (the
number of PCs in the maximal consonance), the scale can only be ordered so that two
of the five instances of [025] has mutually nonadjacent PCs. <CDFC> is an exam-
ple of such an ordering, with interlocking [025] trichords distinguished in the chain by
letter size.53
All of this leads to the conclusion that the structure of scales is a huge constraining
force on any systems that meet the four Straus conditions.54 As I shall show in detail
in Part 2 (page 99), the cardinality of a temperament, scale, and maximal consonance,
the intervallic structure of a scale, and the way scales can be realized in pitch space (as
opposed to PC space) are all the province of scale theory and all partially determine the
prolongational possibilities of any system. Before delving into the scale-theory particu-
lars, I shall need to discuss how traditional prolongation theory relates to other concepts
in tonal theory, with an eye toward the consequences of generalization. But first a mu-
sical interlude.through one modulus in PC space, then order-position and PC intervals are correlated; this has importantconsequences on the ability of a system to allow for the kinds of motivic association found in tonal music.
53is would fail to pass the other requirement of condition four, since ICs from [025] show up asadjacencies in this ordering. Note also that {} is not complementary to any of the other [025] trichordswith respect to the scale set, so it would not be usable as a maximal consonance in the first place.
54erefore I disagree with authors, e.g. Matthew Brown, who believe the structure of the diatonicscale has little ultimately to do with Schenker’s prolongation theory. See Brown, Explaining Tonality,2005, Chapter 4, “Schenker and e Myth of Scales,” pages 140-170 for his view on this matter.
Musical Interlude I: A Goldberg Canon 47
1.2 M I I: A G C
is interlude illustrates some of my approach in using prolongation as a compositional
tool, while highlighting many of the problems which shape Part 1.
F M O
Suppose I wished to reconstruct a compositional process for the first phrase of the canon
at the second from Bach’sGoldberg Variations, using prolongation to guide my decisions.
I am faced with several difficulties at the outset: since it is a canon at the second, and I
am aware of Bach’s concern with phrase length and harmony in these variations, I must
pay very close attention to the details of the surface. But since I am presumably going to
be working from the background to the foreground, I have to find a way to negotiate the
prolongation structure atmany levels at once: at the least I will need to find a background
which allows the canonic surface to emerge. Because this is a variation on a theme, I
have to decide which features of the theme I am going to preserve. It is not necessary
that it preserve prolongational background; in fact part of the variation process might
be to preserve features of the surface while varying the background or the interaction
between the background and foreground levels. Bach’s practice in the other variations
will be a good guide.
Figure 1.8 is the opening phrase from the Aria theme, which corresponds to the
variation I wish to compose. In general Bach preserves the function of the chords in the
progression but not necessarily the actual harmonies; in many of them he also preserves
the bass line, often embellished. I would like to do the same with this variation, but
the surface constraints may make this impossible. It is often a good idea to begin where
the constraints are the most severe, so I decide to find a plausible way to get a canon
at the second to fit with the chords, the bass line, or both, and refine the prolongation
P A 48
&?
#
#43
43
œ œ .œ! œŒ Œ œ.˙ ˙
œ rœ œ rœ ˙Œ Œ œ.˙ ˙
œ! .œ
"œ œ
Œ Œ œ#.˙ ˙œ œ œ œ œ œ jœ ˙Œ Œ œ˙ œ œ˙
œ œ .œ! œnŒ Œ œ.˙ .œ
&?
#
#
œ rœ œ rœ .œ JœT
Œ Œ œ˙ œ œ
œ .œ œ .œ œ .œ œ .œ jœ.œ œ
‰ jœ œ œ œ œ œ œ œœ œ ˙
œ .œ œ jœ ˙!
œ œ ˙.œ œ .œ! œ
œ
œ
? # œT
A œD
œP
œ œD
œT(
œP)
œD
œT
& # œœœB œœœœ œœ œœœ œœœ œœœ œœœ œœœ
& # ŒœC
œœ œœ œœ œ œ œœ œ œ œ œ? œœ
&?
##
œ œ œ œ œ œ œ œ œœœ œœœ œ œœ œœ œœ œœ œœ
˙ ˙ œ œ œ œ œ ˙
8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ*
( )( )
6 6
&?
##
##
##
œ œ œ œ œ œœœ œœ œœ œ œœ œœjœ
˙ œ œ œ œ ˙
*( ) &
?
œ œ œ
œ œ œ
OR œ œ œœ œ œ
Figure 1.8: Aria from Bach’s Goldberg Variations, Bars 1-8.
structure as needed. Figure 1.9 A is the bass line with the most usual chord functions (T
for Tonic, D for Dominant, P for Pre-dominant). e arrow from bars 5 to 6 indicates
that the tonic chord may be turned into a dominant-seventh harmony, tonicizing IV.
I will also need a point of imitation; I notice that bars two and four are both dominant
sonorities, so the canon at the second above will not work with a two-bar point of
imitation because none of the pitches in the second chord are a step higher than any
in the first. e same reasoning applies to a three-bar point of imitation and dominant
chords in bars 4 and 7. It is reasonable that comes must follow dux by one bar. I can
check this heuristically, as in Figure 1.9 B. In all cases two or more pitches in each chord
are a step higher than pitches in the one preceding.55
Canonic writing can benefit from making first-species outlines to help shape a strat-
egy.56 Because the bass line cantus firmus descends in stepwise fashion for the first four
bars, it will be impossible to run descending stepwise lines in the canonic voices in first
species without creating unwanted parallels. Ascending stepwise motion in first species55Another heuristic would be to check all root motions, to discover that only root motion by ascending
third does not meet the intervallic criteria, and then check to see that no ascending-third motions occurin the progression.
56Such an outline should not necessarily be construed as the tonal background to be composed out,but more as a guide to keeping the motivic design of the canon in line with the harmonic constraints.
Musical Interlude I: A Goldberg Canon 49
&?
#
#43
43
œ œ .œ! œŒ Œ œ.˙ ˙
œ rœ œ rœ ˙Œ Œ œ.˙ ˙
œ! .œ
"œ œ
Œ Œ œ#.˙ ˙œ œ œ œ œ œ jœ ˙Œ Œ œ˙ œ œ˙
œ œ .œ! œnŒ Œ œ.˙ .œ
&?
#
#
œ rœ œ rœ .œ JœT
Œ Œ œ˙ œ œ
œ .œ œ .œ œ .œ œ .œ jœ.œ œ
‰ jœ œ œ œ œ œ œ œœ œ ˙
œ .œ œ jœ ˙!
œ œ ˙.œ œ .œ! œ
œ
œ
? # œT
A œD
œP
œ œD
œT(
œP)
œD
œT
& # œœœB œœœœ œœ œœœ œœœ œœœ œœœ œœœ
& # ŒœC
œœ œœ œœ œ œ œœ œ œ œ œ? œœ
&?
##
œ œ œ œ œ œ œ œ œœœ œœœ œ œœ œœ œœ œœ œœ
˙ ˙ œ œ œ œ œ ˙
8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ*
( )( )
6 6
&?
##
##
##
œ œ œ œ œ œœœ œœ œœ œ œœ œœjœ
˙ œ œ œ œ ˙
*( ) &
?
œ œ œ
œ œ œ
OR œ œ œœ œ œ
Figure 1.9: A. e bass line with functions. B. Chords used most often in the variations,showing possible canonic relationships. C. An initial sketch for the canon.
is also out of the question because the two voices would simply double each other. us,
if I am to retain the constraint to use this bass line, I am going to have to make the
canonic voices leap about, and then embellish the leaps. Bach prizes extreme motivic
consistency in these variations, however, and leaping about with no guarantee of inter-
vallic consistency is not a good way to meet that constraint. One last avenue presents
itself: I could use the cantus firmus in the upper voices. A quick, literal check in Fig-
ure 1.9 C, running the cantus firmus in canon with itself in first species, shows some
promise for the first four bars. e process derails in the last four, so if it is to work, the
cantus firmus must be shared between the two voices. is approach has the additional
virtue that it retains the G-A motive from the beginning of the Aria’s soprano line, but
here as the initial imitation between voices.
At last I am ready to begin sketching the prolongational structure, keeping the fore-
going in mind. My tactic is to start in the middleground in Figure 1.10, confident that
if it is syntactically well-formed it will connect to a well-formed background. After
confirming a background, I will be free to focus on building the canon. I regard the
middleground structure as a hypothesis which will be confirmed or disconfirmed by its
P A 50
&?
#
#43
43
œ œ .œ! œŒ Œ œ.˙ ˙
œ rœ œ rœ ˙Œ Œ œ.˙ ˙
œ! .œ
"œ œ
Œ Œ œ#.˙ ˙œ œ œ œ œ œ jœ ˙Œ Œ œ˙ œ œ˙
œ œ .œ! œnŒ Œ œ.˙ .œ
&?
#
#
œ rœ œ rœ .œ JœT
Œ Œ œ˙ œ œ
œ .œ œ .œ œ .œ œ .œ jœ.œ œ
‰ jœ œ œ œ œ œ œ œœ œ ˙
œ .œ œ jœ ˙!
œ œ ˙.œ œ .œ! œ
œ
œ
? # œT
A œD
œP
œ œD
œT(
œP)
œD
œT
& # œœœB œœœœ œœ œœœ œœœ œœœ œœœ œœœ
& # ŒœC
œœ œœ œœ œ œ œœ œ œ œ œ? œœ
&?
##
œ œ œ œ œ œ œ œ œœœ œœœ œ œœ œœ œœ œœ œœ
˙ ˙ œ œ œ œ œ ˙
8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ*
( )( )
6 6
&?
##
##
##
œ œ œ œ œ œœœ œœ œœ œ œœ œœjœ
˙ œ œ œ œ ˙
*( ) &
?
œ œ œ
œ œ œ
OR œ œ œœ œ œ
Figure 1.10: Middleground hypothesis for the Goldberg canon.
ability produce the desired result. If it does not, I can modify it based on what I learned
from the first attempt.
is middleground establishes the 8-chord progression from the Goldberg phrase
with an 8-line. Since the cantus firmus bass line from the Aria descends from b8 to b3,it is easily instantiated in an 8-line. To complete it I have harmonized the ascending
portion of the cantus firmus with two chords in first inversion, following the established
Goldberg progression. e A in parentheses under b7 will likely be the beginning of the
second canonic voice; the parentheses under b6 indicate a decision to be made later: will
this chord be VI or IV6? e previous variations make use of both solutions, so I am
not constrained to make a decision here yet.
e phrase features what is either a long dominant prolongation from b7 to b2, or a
dominant prolongation from b7 to b5−b4, and a final tonic prolongation from b3 to b1. I have
left these unaccounted for parts unattached in the graph for now; the decision about
which of the two solutions is “correct” is immaterial at this point, but will depend both
on background structures and how the foreground develops. For a Schenkerian, what
I am calling a middleground might actually be an Ursatz with Leerlauf on b4, but I can
find ways to move yet further into the background. For instance, I have bracketed the
span from b7 to b5 of figure 1.10 with an asterisk, indicating a voice-exchange between
Musical Interlude I: A Goldberg Canon 51
&?
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#43
43
œ œ .œ! œŒ Œ œ.˙ ˙
œ rœ œ rœ ˙Œ Œ œ.˙ ˙
œ! .œ
"œ œ
Œ Œ œ#.˙ ˙œ œ œ œ œ œ jœ ˙Œ Œ œ˙ œ œ˙
œ œ .œ! œnŒ Œ œ.˙ .œ
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#
œ rœ œ rœ .œ JœT
Œ Œ œ˙ œ œ
œ .œ œ .œ œ .œ œ .œ jœ.œ œ
‰ jœ œ œ œ œ œ œ œœ œ ˙
œ .œ œ jœ ˙!
œ œ ˙.œ œ .œ! œ
œ
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A œD
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œP)
œD
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& # œœœB œœœœ œœ œœœ œœœ œœœ œœœ œœœ
& # ŒœC
œœ œœ œœ œ œ œœ œ œ œ œ? œœ
&?
##
œ œ œ œ œ œ œ œ œœœ œœœ œ œœ œœ œœ œœ œœ
˙ ˙ œ œ œ œ œ ˙
8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ*
( )( )
6 6
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œ œ œ œ œ œœœ œœ œœ œ œœ œœjœ
˙ œ œ œ œ ˙
*( ) &
?
œ œ œ
œ œ œ
OR œ œ œœ œ œ
Figure 1.11: Goldberg canon background structures.
soprano and bass which could be reduced. is is especially well-motivated because b4is not supported.
Figure 1.11 explores this reduction, replacing the voice-exchange with a dominant
first-inversion triad (indicated with the bracket and asterisk). is changes the counter-
point by moving the b5−b4−b3 span from the middleground into a lower voice, forcing the
leading tone under the asterisk to resolve locally into the implied tonic in the follow-
ing chord. is local implied resolution suggests that the tonic chord may be a passing
sonority within a larger dominant prolongation, and the subsequent movement from
the alto back to the soprano suggests another voice-exchange. is reduces further to
one of the tonic-dominant-tonic progressions on the right side of figure 1.11, which
are voice-inversions of each other. e choice depends on whether the root-position
dominant triad or the first-inversion dominant triad is the more structural chord in the
voice-exchange, and this may depend further upon motivic details on the surface. In any
case, these further background structures are now part of the middleground hypothesis,
and they too can be revised if I am forced to do so by foreground constraints. Confident
that my middleground connects to a well-formed background, I can begin moving from
middleground to foreground.
e first step toward the foreground is the transformation in figure 1.12. In order to
further explore the nested voice-exchange motives from the background, and to main-
tain Bach’s characteristic motivic consistency, I have decided to prolong the chords of
P A 52
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œ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
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jœ œ œ œ œ.œœ œ œ œ œ œ
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.œ .œ
.œ
&
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.œ
œ œ œ œ œ œ
.œœ œ œ œ œ
œ œ œ œ œ œ
œ œ œ œ œ.œ
œ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
2
Figure 1.12: First transformation of the Goldberg canon middleground.
the progression with voice-exchanges; this also gives me more pitch material to work
with in inner voices. is new level-hypothesis has some interesting features: first, it
hides the descending 8-line structure by producing an ascending 10th sequence activated
by the voice-exchanges. Second, the brackets show a motive characterized by a descend-
ing 2nd and 3rd, which could be a motive around which the canon could be created. In
fact, the first two instances of it are in exactly the correct position for the canon, and
I could have surmised this ahead of time by noting that the ascending line in the bass,
combined with sequential voice-exchanges, would be guaranteed to produce figures in
the upper voices related by a 2nd above, as desired.
e motivic situation looks promising, so I can try to permeate the entire passage
with these motives in strict canonic form. Figure 1.13 is the result of this attempt.
An extension of the descending 2nd-3rd motive is obtained by chaining the motive in
each canonic voice, creating a descending 2nd-3rd-3rd. is canon resolves the harmonic
ambiguity in measure three (I did not know whether it was VI or IV6) by forcing a
VI chord. In measure four I have had to move b5 to the previous measure in order to
maintain the rhythmic profile (marked with an asterisk). It now functions as a passing
tone, and so I will have to put the same passing tone in the other instances of the motive.
is causes a problem, however, because it makes a parallel 5th between the bass and
the upper voice; since the bass line is somewhat more flexible than the canonic lines, I
Musical Interlude I: A Goldberg Canon 53
&?
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œ
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œ œœ
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œ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œœ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
&
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#
.œ
œ œ œ œ œ œ
.œœ œ œ œ œ
œ œ œ œ œ œ
œ œ œ œ œ.œ
œ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
2
Figure 1.13: First attempt at a canon based on the transformed middleground.
will need to modify the bass line. A more serious problem is that I cannot extend the
chained motives all the way through the phrase, since the last three measures lose their
harmonic meaning entirely.57 I try to remedy these problems in the next level down,
shown in figure 1.14.
is kind of parallel-5th problem is often remedied by arpeggiating to a different
voice in the harmony, so in order to do that and to retain the upper-line passing tone
in the third measure, I have turned this into a third-species exercise and arpeggiated
the bass. ird species also suggests including passing tones within the voice exchanges
as well, creating descending scale patterns in the upper voices and a characteristic se-
quential motive in the bass. I put the same bass motive in the first measure to ensure
consistency, even though it does not participate in a voice exchange. e dotted slurs in
this example indicate a passing connection between pitches that do not result from an
arpeggiation within a harmony, but rather from skips between harmonies. is gives
me a most fortuitous solution to the last three measures: the descending four-note scale
pattern in the upper voices can take different harmonic meaning depending on which
part of the harmony it outlines; in the opening sequence it outlines descending 3rd pat-57Prior harmonic and contrapuntal constraints also suppress the temptation to view the ascending G-
major scale spelled by the half notes in the canonic voices as a structural line, as such a line would formparallel fifths with the bass, requiring recomposition of the entire excerpt. e appearance of a motivicascending 8-line in the context of a structural descending 8-line is interesting, and shows how richly tonalmusic can exploit the partial independence of prolongational structure and motivic design.
P A 54
&?
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œ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œœ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
&
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#
.œ
œ œ œ œ œ œ
.œœ œ œ œ œ
œ œ œ œ œ œ
œ œ œ œ œ.œ
œ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
2
Figure 1.14: ird species canon based on the previous attempt.
terns with passing tones, but in measures six and seven it can be made to outline a
descending 4th between the root and 5thof the triad. I want to retain the E-F#-G line
in the bass from these last three measures in higher levels, since that provided the best
counterpoint to the middleground 8-line and cantus firmus.
Unfortunately this creates yet new motivic problems. For the sake of variety I would
like not to include the same bass motive in every measure, but the bass motives should
all be related if possible. Bar seven includes a diatonic retrograde inversion of the bass
motive in order to retain the F#, but I am not able to do the same in bar six because
to leap down to E from the G the bar before would create an awkward parallel with
the soprano. Repeating the G is a possibility, but I would not be able to maintain the
third-species texture and hit the E required at the end of the bar. e only option left is
to leap up to C from bar five to bar six, and leap down a 6th to the E, and hope to find
a solution in the next level down.
It would appear I have two options to fill in the descending sixth in the bass in bar
six. e first would be C-G-F#-E, which is serviceable, but I am starting to notice the
five consecutive repetitions of the bass motive, which is uncharacteristic of Bach. A
triple meter would give me the extra rhythmic leeway I need to vary the bass motive and
it would allow me to fill in the entire descending 6th with a scale. With all this in mind,
I may proceed to the hypothetical foreground of figure 1.15.
Musical Interlude I: A Goldberg Canon 55
&?
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œ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œœ œ œ œ œ œ
jœ œ œ œ œ.œœ œ œ œ œ œ
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.œ .œ
.œ
&
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œ œ œ œ œ œ
.œœ œ œ œ œ
œ œ œ œ œ œ
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œ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
.œ
2
Figure 1.15: A hypothetical foreground for the Goldberg canon.
e bass of measure six now has the descending 6th scale fragment. e leap of a 4th
between bars five and six in the bass is extended back to the other measure transitions
as far as possible (marked with solid brackets), which gives each measure a turn figure
in addition to the third-species motive from the next level up. ere are many possible
solutions for turning the four-note descending motives from the upper parts in the pre-
vious level into triple-meter figures, but including suspensions clearly differentiates the
two canonic voices in what is a single-timbre texture, while avoiding awkward octaves
between the upper voices and the new pitches in the turn figure of the bass. Bar seven’s
bass figure is one of many solutions which retains the retrograde-inversion portion of the
passage’s bass motive, and measure eight is provided with a transition from this phrase
to the next phrase which will start in bar nine. ere is one last problem to address,
which is the consecutive repetitions of the bass motive. Since the bass of the first mea-
sure is less constrained than in the other measures I have removed the bass motive from
that bar in order to create something different but related. I also notice that I have a
potential parallel octave between the bass and an upper line between bars one and two as
it stands, and I still have four repetitions of the same bass motive from bars two through
five (marked with a dotted bracket), which is still one too many repetitions for Bach’s
usual practice. I am close enough to the surface that these remaining problems may be
addressed there, in figure 1.16
e bass of measure two borrows the lower neighbor from the solution to measure
seven, which also creates a retrograde-inversion of the four-note turn figure in bars three
P A 56
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jœ œ œ œ œ.œœ œ œ œ œ œ
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jœ œ œ œ œ.œœ œ œ œ œ œ
.œJœ œ œ œ œ œœ œ œ œ œ œ
.œ .œ
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&
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œ œ œ œ œ œ
.œœ œ œ œ œ
œ œ œ œ œ œ
œ œ œ œ œ.œ
œ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
.œœ œ œ œ œœ œ œ œ œ œ
œ œ œ œ œ.œœ œ œ œ œ œ
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2
Figure 1.16: e surface of the Goldberg canon.
through five. is motive is easily copied into the first measure, giving me two repe-
titions of this motive (marked with a solid bracket), followed by three repetitions of
the other (marked with a dotted bracket). Incidentally, the direct motivic association
between the first and second bars suggests that the D in the bass of measure two is
at the same level as the G in the first measure. is would favor the first of the two
background structures from figure 1.11 (page 51), in which the bass proceeds through
a G-D-G arpeggiation. It is thus interesting to note that different background struc-
tures may generate identical foregrounds, but decisions made at the surface can resolve
such ambiguities at background levels. is is one of many reasons I like to think of
the various levels as hypotheses rather than reified entities, especially in using prolon-
gation as a compositional technique; decisions made at any level on the basis of another
hypothetical level can create changes or resolve ambiguities in the latter.
T M5/M7 T: [] [] T
One of the most striking passages in seventeenth-century repertoire occurs in the in-
troduction to Monteverdi’s Lamento della ninfa from his eighth book of madrigals (see
figure 1.17). e poetry translates to “On her pallid face her grief could be read”; my fo-
cus is on the four simultaneities Monteverdi uses to set il duo dolor: “her grief ” (marked
with a bracket in figure 1.17). is is a prolongation of a dominant triad in A minor
by means of an upper-neighbor in the bass and suspensions in the two tenor voices. In
seventeenth-century music this contrapuntal combination sometimes results in a cross
Musical Interlude I: A Goldberg Canon 57
BB?
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ccc
c
c
‰ jœ jœ jœ# .œ Jœsul pal li det to
‰ Jœ Jœ Jœ .œ Jœsul pal li det to
‰ jœ jœ Jœ .œ Jœsul pal li det to
‰ œ œ œ .œ jœœ œ œ# .œ Jœ‰ œ œ œ .œ Jœ
œ jœ jœ œ jœ jœvol to scor gea si il
œ Jœ Jœ œ Jœ jœ#vol to scor gea si ilœ Jœ Jœ œ Jœ Jœvol to scor gea si il
˙ œ œ œœ œ œ œ œ œ#œ œ œ œ œ œ
.˙ œsuo do
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.˙ œ.˙# œ#
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wlor
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ww#w
- - - -- -
- - - - - -
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-
-
-
BB?
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c
jœiljœ#il
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.˙ œsuo do
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.˙ œsuo do
.˙ œ.˙# œ#
.˙ œ
wlor
w#lor
wlor
ww#w
-
-
-
Score
Figure 1.17: A passage from the introduction to Lamento della ninfa by Monteverdi.
relation: the bass ascends from b5 to n b6 while the leading tone is suspended. In order
to avoid resolving this suspension by means of an augmented second down to n b6, it is
instead “resolved” to # b6, which creates an augmented unison or octave between the bass
and this voice. What makes this passage particularly interesting is the combination of
this cross relation with the other suspended voice. e full suspension simultaneity (in
this case [FnG#En]) and its resolution ([FnF#Dn]) are [014] trichords related by inversion.
Neither of these is a locus of stability in tonal music, so neither can be composed out;
but since [014] and [037] are related by the “circle of fifths” transforms, M5 and M7, it
is interesting to imagine what inversionally related [037] triads could be formed by ap-
plying one of these operations to Monteverdi’s sonorities. e most parsimonious route
is to apply M7, which preserves the PCs of one whole-tone collection and transposes
the others by a tritone; since the dominant triad on either side of the [014] trichords
is actually represented by a major 3rd, the M7 operation can preserve it and the sus-
pended pitches, while transposing the Fn in the bass down by a tritone to Bn. e result,
P A 58
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‰ jœ jœ jœ# .œ Jœsul pal li det to
‰ Jœ Jœ Jœ .œ Jœsul pal li det to
‰ jœ jœ Jœ .œ Jœsul pal li det to
‰ œ œ œ .œ jœœ œ œ# .œ Jœ‰ œ œ œ .œ Jœ
œ jœ jœ œ jœ jœvol to scor gea si il
œ Jœ Jœ œ Jœ jœ#vol to scor gea si ilœ Jœ Jœ œ Jœ Jœvol to scor gea si il
˙ œ œ œœ œ œ œ œ œ#œ œ œ œ œ œ
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wlor
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-
-
-
Score
Figure 1.18: An M7 transform of the dominant prolongation from Lamento della nifna.
shown in figure 1.18, is a 6-54-3 suspension over b2 in the bass: not entirely idiomatic for
Monteverdi, but not out of the question.
In Rahn’s terms, the bass line of the original and its M7 transform are conceptually
related: the original bass line is an N-prolongation, while the transformed bass line is an
N*-prolongation. erefore it seems that while much of the expressive quality of the two
[014] trichords has been removed with the transformation to [037] triads, not much has
actually changed in the counterpoint. Both the original and the result of the operation
contain a bass voice which moves away from and back to a stable pitch, and the upper
voices undergo no change at all, neither in their pitch content nor their contrapuntal
function. In fact, the transformed progression wouldmaintain a very conventional [037]
harmonic norm throughout the passage, and if it were what Monteverdi had actually
composed, this madrigal would no longer be the curious harmonic oddity it is. e
obvious next question is what happens if M7 is applied to the entire passage? Figure
1.19 is one possible rendering. Now [014] is the harmonic norm, while the two [037]
trichords logically serve the same anomalous harmonic function the [014] trichords did
Musical Interlude I: A Goldberg Canon 59
BB?
&?
‰ Jœ# Jœ# jœ# .œ# Jœ#sul pal li det to
‰ Jœn Jœn jœn .œn jœ#sul pal li det to
‰ jœn jœn Jœn .œn jœnsul pal li det to
‰ œn œn œn .œn jœ#œn œn œ# .œ# Jœn‰ œn œn œn .œn jœn
œ# Jœ# Jœ# œn Jœn jœnvol to scor gea si il
œn jœn Jœn œ# jœ# jœ#vol to scor gea si il
œ# jœ# Jœn œn jœn Jœnvol to scor gea si il
˙# œn œn œnœn œn œn œ# œ# œ#œ# œ# œn œn œn œn
.˙n œnsuo do
.˙# œ#suo do
.˙n œnsuo do
.˙n œn.˙# œ#
.˙n œn
wnlor
w#lor
wnlor
wn w#wn
- - - - - - -
- - - - - - -
- - - - - - -
BB?
&
?
����
�
2
Figure 1.19: An M7 transform of the entire Lamento della ninfa passage.
in the original passage. But do the contrapuntal and prolongational structures persist
after applying the transform? ere are good reasons to answer in the affirmative as well
as the negative; the rationales for each case will be made clear by applying M7 or M5 to
the Goldberg canon from the foregoing.
T G C U M5/M7 T
By applying M5/M7 to a tonal passage, one thereby applies it conceptually to the en-
tire tonal system. For my purposes this raises an important question: to what extent
does the transformed system retain its prolongational capacities? is depends entirely
on the assumed framework. In Rahn’s, for instance, such a transformation does not
change anything essential: imagine applying M5 to the entire Berg-Schoenberg deriva-
tion above;58 nothing in the logical structure of the level analysis would change even
with the differences in pitch, pitch-class, set-class, etc. In all relevant ways a level anal-
ysis and its M5 dual are isomorphic. In contrast, Lerdahl’s approach could not readily
accommodate the transformed system. It will take some work to determine whether it58See figure 1.6 on page 30 and figure 1.7 on page 32.
P A 60
&&
T8 M5
œ œ œ œ œ œ œ# œ
œn œn œb œ# œ# œn œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&
T6 M7
œ œ œ œ œ œ œ# œ
œn œn œn œb œb œb œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&?
œ œ œ
œ œ# œ
1 7 1
1 5 1ˆ ˆ ˆ
œ œ# œœ œ œ
1 5 1
1 7 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
œ œ œ
œ œb œ
1 7
1 5 1
1ˆ ˆ ˆ
ˆ ˆ ˆœ œb œ
œ œ œ1 1
1
7
5 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
.œ
œ œ œ œ œ œ
.œn œ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œn œ.œbœn œ œ œ œ œ
.œb œ œb œn œn œnœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ .œ
.œ
&?
.œ
œ œ œ œ œ œ
.œœ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œ œ.œbœn œ œ œ œ œ
.œb œ œbœ œ œœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ
.œ .œ
3
Figure 1.20: M5/M7 transforms of the G-major scale.
meets the four modified Straus conditions, however, and it is with this in mind that I
continue with the investigation by applying M5/M7 to the Goldberg passage.
I shall want to keep track of the PCs associated with each scale degree; this book-
keeping is shown in figure 1.20: M5 and M7 transforms of the G-major scale, retaining
Gn as b1. Since M5 and M7 are related by inversion (formally M11), the contours of the
resulting pitch cycles are also inversionally related. e M5/M7 transforms preserve all
ICs but 1 and 5; M5 preserves PC intervals 3, 6, and 9, and inverts PC intervals 2, 4,
and 8, while M7 preserves 2, 4, and 8, and inverts 3, 6, and 9. erefore, as in the major
scale, the two transformed cycles have two whole-tone segments – inverted in M5 and
transposed in M7 – and each is separated from the other by a single IC (IC1 in a major
scale, and IC5 in the transformed cycles).
An M7 transform of a scale-based passage like the Goldberg canon will thus preserve
more of the contour of the original than M5, and if a major-key passage is transformed,
M7 preserves the b1 and b3 PCs. On the other hand, under M5 the b5 and b7 PCs are
exchanged, so the “dominant triad” will be similar to the major key’s. is has a sub-
tle effect for the Goldberg passage, as seen in figure 1.21, which contains M5 and M7
transforms of the deep backgrounds from figure 1.11 (page 51): the M5 background has
identical PC content to the figure 1.11 backgrounds.
Another consequence of the M5/M7 transforms of the G-major scale (figure 1.20) is
that since the diatonic collection may be generated by a cycle of fifths, an M5/M7 of the
Musical Interlude I: A Goldberg Canon 61
&&
T8 M5
œ œ œ œ œ œ œ# œ
œn œn œb œ# œ# œn œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&
T6 M7
œ œ œ œ œ œ œ# œ
œn œn œn œb œb œb œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&?
œ œ œ
œ œ# œ
1 7 1
1 5 1ˆ ˆ ˆ
T8 M5
œ œ# œœ œ œ
1 5 1
1 7 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
œ œ œ
œ œb œ
1 7
1 5 1
1ˆ ˆ ˆ
ˆ ˆ ˆ
T6 M7
œ œb œ
œ œ œ1 1
1
7
5 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
.œ
œ œ œ œ œ œ
.œn œ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œn œ.œbœn œ œ œ œ œ
.œb œ œb œn œn œnœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ .œ
.œ
&?
.œ
œ œ œ œ œ œ
.œœ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œ œ.œbœn œ œ œ œ œ
.œb œ œbœ œ œœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ
.œ .œ
3
Figure 1.21: M5/M7 transforms of the deep backgrounds from Figure 1.11
&?
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œ
8
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œb œ
7
10
V
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6
10
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5 4ˆ ˆ
10
V 65
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3
10
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IV 6
œn œbœ
2
6
V 6
œœœ
1
I
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8
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œb œ
7
10
V
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œb œ
6
10
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5 4ˆ ˆ
10
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œ œn
3
10
I
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œn œbœ
2
6
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œn œœ
1
I
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œ œ œ œ œb œ
.œn œ œ œ œ# œ#
œ# œ# œ# œn œ œ#
œ œn œ œn œ#.œbœn œ œ œ œ œ
.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 VI 864 V 86 I
&?
˙
œ œ œ œ œ œ# œ œ
˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
œ œnœ œ œ œ
4
Figure 1.22: T8M5 of the Goldberg canon passage.
diatonic collection results in an ordered chromatic collection. In order to employ the
smallest pitch intervals, the PC cycles have to “wrap around” and arrive at the pitch on
which they started; in other words, a complete scale cycle with the most parsimonious
voice leading does not traverse an octave the way a diatonic scale does. e result of this
is that voices hewing to the smallest pitch intervals will tend to stay in the same register,
causing a “stratification” of voices. e M5 transform of the Goldberg surface in figure
1.22 displays this stratification: neither of the two canonic voices leaves its octave. is
makes Schenkerian procedures that rely strongly on register, e.g. “motion into an inner
voice,” far more difficult to achieve.
is scale cycle wrapping and stratification can become pathological when two voices
occupy the same register: see figure 1.23 – T6M7 of the Goldberg canon passage – which
as expected preserves a bit more of the original’s contour. Here it is impossible to desig-
nate an “upper” and a “lower” voice; stem direction in the two canonic voices is notated
by fiat rather than by the logic imposed by register in the diatonic system. Indeed, were
P A 62
&&
T8 M5
œ œ œ œ œ œ œ# œ
œn œn œb œ# œ# œn œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&
T6 M7
œ œ œ œ œ œ œ# œ
œn œn œn œb œb œb œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&?
œ œ œ
œ œ# œ
1 7 1
1 5 1ˆ ˆ ˆ
T8 M5
œ œ# œœ œ œ
1 5 1
1 7 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
œ œ œ
œ œb œ
1 7
1 5 1
1ˆ ˆ ˆ
ˆ ˆ ˆ
T6 M7
œ œb œ
œ œ œ1 1
1
7
5 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
.œ
œ œ œ œ œ œ
.œn œ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œn œ.œbœn œ œ œ œ œ
.œb œ œb œn œn œnœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ .œ
.œ
&?
.œ
œ œ œ œ œ œ
.œœ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œ œ.œbœn œ œ œ œ œ
.œb œ œbœ œ œœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ
.œ .œ
3
Figure 1.23: T6M7 of the Goldberg canon passage.
&&
T8 M5
œ œ œ œ œ œ œ# œ
œn œn œb œ# œ# œn œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&
T6 M7
œ œ œ œ œ œ œ# œ
œn œn œn œb œb œb œn œn1 2 3 4 5 6 7 8ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&?
œ œ œ
œ œ# œ
1 7 1
1 5 1ˆ ˆ ˆ
T8 M5
œ œ# œœ œ œ
1 5 1
1 7 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
œ œ œ
œ œb œ
1 7
1 5 1
1ˆ ˆ ˆ
ˆ ˆ ˆ
T6 M7
œ œb œ
œ œ œ1 1
1
7
5 1ˆ ˆ ˆ
ˆ ˆ ˆ
&?
.œ
œ œ œ œ œ œ
.œn œ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œn œ.œbœn œ œ œ œ œ
.œb œ œb œn œn œnœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ .œ
.œ
&?
.œ
œ œ œ œ œ œ
.œœ œ œb œb œbœb œb œb œb œ œb
œ œ œ œb œb.œnœ œ œb œ œ œb
.œb œ œn œ œ œbœn œ œ œ œ œ
œ œn œn œ œ.œbœn œ œ œ œ œ
.œb œ œbœ œ œœb œn œ œn œ œb
œ œb œb œn œn.œœ œb œ œb œb œ
.œœ œ œ œ œ œœ œb œb œb œn œn
.œ
.œ .œ
3
Figure 1.24: T6M7 of the Goldberg canon passage, with clearer voice separation.
it not for the “suspensions,” long notes, and unbroken “stepwise motion” through the
scale cycle, parsing this surface into three voices could be difficult by means of pitch
alone.
Recalling that voice partitioning is the purpose of the fourth Straus condition, this
M5/M7 system may not sustain prolongation without very careful treatment of voices.
In cases where “stepwise motion” prevails, contrapuntal voices may be inferred even with
considerable voice overlap, and since stepwise motion and small leaps are normative in
Schenkerian counterpoint, perhaps this may be salvaged as a prolongational system after
all. A second solution depends on the idea that using the smallest pitch intervals is not
a requirement. Figure 1.24 is a recomposition of figure 1.23, with some perfect-5th
“steps” in the canonic voices, effecting better voice separation. Loosening the voice-
leading constraints can make the overall contrapuntal picture more clear.
Assuming for the moment that it is always reasonably possible to resolve contrapun-
tal voices in the M7 system employed in figures 1.23 and 1.24, what would a middle-
ground reduction of the canon passage look like? Two possibilities derived from the two
M7 surfaces are given in figure 1.25; each preserves the registration of the surface from
which it was derived, the top from figure 1.23 and the bottom from figure 1.24. ese
middleground reductions correspond to the same level found in the analysis of the orig-
Musical Interlude I: A Goldberg Canon 63
&?
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7
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6
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2
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7
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.œn œ œ œ œ# œ#
œ# œ# œ# œn œ œ#
œ œn œ œn œ#.œbœn œ œ œ œ œ
.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 VI 864 V 86 I
&?
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˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
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4
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.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 VI 864 V 86 I
&?
˙
œ œ œ œ œ œ# œ œ
˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
œ œnœ œ œ œ
4
Figure 1.25: Middleground reductions of figures 1.23 and 1.24.
inal canon passage from figure 1.12 (page 52), using the same kinds of voice-exchanges
in the same places. Other operations such as arpeggiation and filling in likewise function
in analogous ways.
However, the reader may recall that one motivation for the middleground structure
in figure 1.12 was the motivic parallelism between the opening figures of the two voices,
effected by the voice-exchanges. Canon is after all themost concentrated kind ofmotivic
design, requiring literal preservation of motivic material between two or more voices.
I have bracketed the equivalent pitches in the bottom of figure 1.25; to what extent
can these bracketed pitch successions be considered equivalent motives? Items against
their equivalence include significant differences in contour and pitch-interval size. In
support of their equivalence is identical intervallic size with respect to the scale cycle.
is question is one of many about the ability of the M5/M7 system to support motivic
P A 64
association. I will not render judgment at this point; full treatment of motive theory is
far beyond the scope of this essay, but the following points may be made.
First, while motivic association is an obviously important part of the tonal repertoire,
it is not difficult to imagine a fully prolongational music which intentionally subverts
motivic association; a system need not support motives to support prolongation, though
such a system may lack much of the richness of tonal music.
Second, the motive concept is not well enough defined, and takes on so many roles
in compositional theory and practice, that it makes some sense to imagine association
as a general kind of musical organization. For instance, motivic repetition can designate
similarity or identity in pitch-contour and/or rhythm; similarity or identity in succes-
sion of ordered pitch or PC intervals to within a diatonic or chromatic collection; or, as
in Schoenberg, something far more abstract, sometimes approaching a definition syn-
onymous with set class. Association comes in many flavors and its form depends greatly
on musical context. One strategy might be to take traditional tonality as a paradig-
matic system that definitely supports motivic composition in each of its senses above,
and explore the ability of other prolongational systems to support a subset of motivic
organization types (or analogues thereof ).
ird and finally, there is a sense in which the classes of prolongational transforma-
tion themselves count as sources of associative interconnection, as they occur in different
contexts within the prolongational hierarchy. is was Schenker’s own conception of
motive in his late theory.59 In this sense of “motive,” any system which satisfies the third
Straus condition can support motivic association. Defining motives in this manner has
some advantages, the principal of which is the ability to create high-level motives that
are systematically reflected in different levels of a level analysis. Establishing connec-
tions between high-level motives and surfacemotives is often a difficult prospect because59SeeCohn, “eAutonomy ofMotives,” 1992 for a helpful discussion of the progression of Schenker’s
thought on motive.
Prolongation and Other eories of Tonality 65
two instances of a given motive need not have the same function and structure in a pro-
longation syntax, and indeed some kinds of motivic association can only happen on the
surface (e.g. the systematic use of [025] trichords in the Mozart example from figure
1.4). Multidimensional association is an especially fertile artistic technique which pro-
vides the motivation for 12-tone array composition, “infinity series,” fractal structures,
and various literary conceits. Association of this kind is determined in large part by the
constraints at play, whether those constraints are designed or externally imposed. For
example, the first sentence of second paragraph of this dissertation, on page 1, also reads
the same down the left column of that paragraph. is required foreknowledge of page
formatting and typesetting, which count as externally imposed constraints that may not
hold were this essay to appear in another publication, where I may not have any control
over formatting minutiae. In tonal music, the relationship between prolongation and
association is so difficult to systematize – it is so heterarchic – that multidimensional
repetition can potentially work differently in every piece.
is ends the musical interlude. e following section touches on the relationships
between prolongation theory and other theories as they relate to the Straus conditions,
in preparation for the discussion of scale theory in part 2.
1.3 P O T T
e interaction of prolongation with other music theories has a fascinating history in
the literature, but alas, there is a great deal more that is worthy of consideration than
scope allows.60 e issues relevant to the rest of this essay will be found in an important60e status of prolongation in highly chromatic tonal music and music whose harmonic norm centers
more around 7th and 9th chords is especially problematic, and seemingly implied by the designation “othertheories of tonality.” But although it deserves attention in this context, it does not figure in my theoryas laid out in part 2, and its discussion in this essay would muddy already less-than-transparent waters.e chief reason for this is that my purpose is to discover systems in other temperaments which supportprolongation as unambiguously as possible, and these edge cases in the tonal repertoire tend to illustrate
P A 66
paper by David Lewin. en for a wider view, I include a brief survey of the concept of
“harmonic function” as it applies in neo-Riemannian and prolongational theories.
L’ G T S
David Lewin’s 1982 paper “A Formal eory of Generalized Tonal Functions” is an
obvious starting point from which to connect generalized prolongation to harmonic
function theory. It will provide a preliminary strategy for forming scale cycles that
comply with Straus conditions 1, 2, and 4. He begins with a structure borrowed from
Moritz Hauptmann, called the “canonical listing” for the diatonic set (see figure 1.26).61
is is an ordering of the C-major collection in stacked thirds, which clearly shows the
tonic triad and the subdominant and dominant triads on either side. e arrows signify
intervals from b1, b4, and b5; the lowercase d and m denote “dominant” and “mediant”
intervals, respectively. All of the d intervals are the same size (perfect 5th), as are all
of the m intervals (major 3rd). is ordering of the diatonic is prominent in many
19th-century “function” theorists, including Hauptmann, Oettingen, and Riemann. In
function theory, scales are usually not system primitives, but are instead generated by
the kinds of intervallic relationships given in figure 1.26, and so this ordering of the
diatonic has much more to say about the structures in a function theory than a scale
written in ascending order does.62
prolongation ambiguously at best. e generalized prolongation theory I arrive at is complex enoughwithout further complicating things by poking too hard at boundaries that will not actually be challenged.
61Lewin, “A Formal eory of Generalized Tonal Function,” 1982, page 24. Used with permissionfrom June Lewin.
62See Klumpenhouwer, “Dualist Tonal Space and Transformation,” 2002 for a detailed exposition ofthis “triads without scales” phenomenon in function theory.
Prolongation and Other eories of Tonality 67
d
C e G
d
F/, F a C
d
G b D
F a C e G b D
Figure 1
24
Figure 1.26: e “canonical listing” for C-major.
Lewin’s project is to generalize this by allowing the size of d and m to vary, and
then to explore the resulting group properties of the systems that result.63 He names
the derived systems Riemann Systems, and provides the following definition:64
D. By a Riemann System (RS) we shall understand an orderedtriple (T , d , m), where T is a pitch class and d and m are intervals, subjectto the restrictions that d = 0, m = 0, and m = d .
e canonical listing of such a system, together with its primary “tonic,” “dominant,”
and “subdominant” triads is isomorphic to the C major system. Lewin’s graph of the
generalized canonical listing structure is given in figure 1.27.65 Every entry in the canon-
ical listing is a PC, so the intervals are taken modulo the octave; this means that when
instantiated in pitch, m can be a larger interval than d . Lewin’s model applies to the
scale cycles from the previous section. For the G-major scale, d = 7, m = 4, and the63Although conceptually any tuning system may be used, Lewin concerns himself only with 12-tone
equal temperament.64Lewin, “A Formal eory of Generalized Tonal Function,” 1982, page 26.65Ibid., page 28. Used with permission from June Lewin.
P A 68
d
T (T+m) (T+d)
d d
m~~~~~~ (1N~
(T-d) (T-d+m) T (T+d) (T+d+m) (T+2d)
(T-d) (T-d+m) T (T+m) (T+d) (T+d+m) (T+2d)
Figure 2
Table 1
canonical listing
Fbbe#D F#aCebG,bbbb Dbb FgCdGaDJ
BbdCeDf#E A tbebCgEbG# FebCbb GfD Bb eCf4Dg#E Ab bbCdEf#G#
diatonic set
(0, 4, 8) (0, 3,6, 9) (0, 2, 5, 7, 9) (10, 0, 2, 4, 6) (1.1,0, 3, 4,7, 8) (10, 0, 2, 3, 5, 7) (0,2, 4, 6, 8, 10) (0, 2,4, 6, 8, 10)
RS
(C, 8,4) (C, 6,3) (C, 7,2) (C, 2, 4) (C, 4, 7) (C, 7, 1 0) (C, 2,6) (C, 4,2)
28
Figure 1.27: e generalized canonical listing for a Riemann system.
canonical listing is <C G D # A>; for the associated M7 system, d = 1, m = 4, and
the canonical listing is <Gb b Gn n Ab n An>. So each of those is a Riemann System.
A canonical listing always has seven entries, and for clarity the PCs are notated
with alternating large and small letters. e structure may also be seen as two inter-
leaved d-cycles separated by m. e intervallic complement of m with respect to d (the
“submediant” interval in a major scale) is labeled m′. e “mediant” and “submediant”
triads occur in a canonical listing as entries 4-6 and 2-4, respectively (with entries in-
dexed from 1). Lewin calls these the “secondary triads,” and it is worth noting that
there is neither a “supertonic” nor a “leading tone” triad; the structure is a series, not a
cycle, so it is missing the wraparound necessary for building those triads. e secondary
triads are always PC inversions of the primary triads.
Here are two more definitions from the paper:66
D. e diatonic set of [the Riemann System] (T , d , m) is theunordered set-theoretic union of the primary triads, comprising the variouspitch classes T − d ,T − d +m,T ,T +m,T + d ,T + d +m, and T +2d .
D. A Riemann System will be called redundant if its diatonic sethas fewer than seven members, irredundant if its diatonic set has exactlyseven members.
66Ibid., pages 26 and 27.
Prolongation and Other eories of Tonality 69
e latter definition means that although the canonical listing for a Riemann system
contains seven entries, some PCs may be repeated. An example would be to switch d
and m from the G-major Riemann System, so that (T , d , m) is (G, 4, 7), yielding the
canonical listing <Eb b G B # D#>; Eb/D# is repeated, and so the generalized diatonic
set for this Riemann system has only six PCs, namely [2367AB], a hexatonic collection.
erefore the Riemann system type (d , m) = (4,7) is redundant. I am exclusively con-
cerned with irredundant systems in this essay.
Finally, in order to explore the group structure of Riemann systems, he needs to
define the group operations:67
D. e conjugate system of the RS (T , d , m) is the RiemannSystem (T , d , d−m). e operation that transforms a givenRS into its con-jugate will be called “C ONJ ”. We shall write, symbolically, C ONJ (T , d , m) =(T , d , d −m).
D. e TD-inversion [i.e. “tonic-dominant inversion”] of theRS (T , d , m) is the Riemann System (T +d ,−d ,−m). e transformationwhich takes any RS into its TD-inversion will be called “T DI NV ”. Weshall write, symbolically, T DI NV (T , d , m) = (T + d ,−d ,−m).
D. e retrograde of the RS (T , d , m) is the Riemann System(T +d ,−d , m− d ). e transformation which takes any RS into its retro-grade will be called “RET ”. We shall write, symbolically, RET (T , d , m) =(T + d ,−d , m− d ).
D. By the identity operation on the family of Riemann Sys-tems, we mean the operation “I DENT ” which, when applied to the speci-men RS (T , d , m), yields (T , d , m) itself as a result. We write symbolicallyI DENT (T , d , m) = (T , d , m).
When applied to triads, the T DI NV operation is identical to the Riemannian “P” op-
eration, which maps the triad D-F#-A built up from the root D to the triad D-F-A built
down from the “root” A, a structure which might better be written A-F-D. is nicely67Ibid., pages 31-36.
P A 70
reflects a precedent from Riemannian function theory, which for methodological rea-
sons builds minor triads down from their “roots” in this manner. e C ONJ operation
maps the D-F#-A triad with root D to a D-F-A triad with root D by retaining T and d
while swapping the order of m and m′. us T DI NV and C ONJ of a triad each yield
a triad with the same PCs, but while C ONJ retains the root PC, T DI NV swaps the
root and fifth PCs. Finally, RET yields the same PCs as its input, but like T DI NV
swaps the root and fifth PCs. e group structure of these operations together with
I DENT is isomorphic to the classical twelve-tone operations P, I , R, RI , and formally,
I DENT is a P operation, T DI NV is an I operation, RET is an R operation, and
C ONJ is an RI operation.
As the definitions note, the operations are more properly applied to entire Riemann
systems than individual triads, using the tonic triad as the focus of the operation. Figure
1.28 shows the systems resulting from applying the group operations to the Riemann
System (G n, 5, 4). All of the entries in each canonical listing is written in “ascending”
order to make the PC identities clear, but conceptually the RET and T DI NV trans-
forms should be read from right to left in the direction of the arrows, making C the
tonic of those two systems.
Another obvious way to understand a canonical listing is to look at it simply as an
ordered series of PCs with two alternating intervals (m and m′) as one moves “up” from
the subdominant. A series with two alternating intervals is an example of an “RI -chain,”
which Lewin describes in .68 An RI -chain is derived by repeating an ordered set
of 2 or more PCs using the last interval in that set to begin an RI transform of that set,
and then using the last interval in the RI transform to begin a transposed repetition of
the first ordered set. is can be repeated indefinitely. In the I DENT line from figure
1.28, the ordered set <2, 6, 7> is RI -chained with <6, 7, 11>, and the pattern is repeated
until the three primary triads have been built.68Lewin, Generalized Musical Intervals and Transformations, 1987, pages 180-189.
Prolongation and Other eories of Tonality 71
F ♮e ♮C ♮b ♮G ♮f ♯D ♮IDENT: (G n, 5, 4)m′
d
m
F ♮d ♭C ♮a ♭G ♮e ♭D ♮CONJ: (G n, 5, 1)
F ♮e ♮C ♮b ♮G ♮f ♯D ♮RET: (C n, 7, 11)
F ♮d ♭C ♮a ♭G ♮e ♭D ♮TDINV: (C n, 7, 8)
Figure 1.28: e Riemann System (G n, 5, 4) and its transformations.
Figure 1.29 shows examples of this RI -chain type taken from the second movement
of Beethoven’s Op. 132 quartet, and the third movement of Webern’s Op. 5 pieces for
string quartet. In the Beethoven example, because the PC content of these four bars
is repeated immediately in the next four, it is possible to imagine beginning the RI -
chain in third measure of the excerpt and wrapping back to the first two measures; then
the chain lasts for an astounding nine PCs.69 e Webern example, from a movement
that is a trichordal study, is more straightforward; the violin and cello play inversions
of the same RI -chain class, completing the aggregate with the last two notes of the
first violin in the previous measure. One could reimagine at least the Beethoven excerpt
as an arpeggiation through the harmonic space of a Riemann system, taking the [015]
trichord as the consonant norm with which to continue.69It is remarkable how much of the motivic saturation in this movement is mod 12 rather than mod 7.
A very lovely feature of this movement is Beethoven’s use of both the P and RI forms of the three-notemotive (+1,+4, and +4,+1, respectively), both in phrasing and rhythmic placement. In fact he managesto slip the +4,+1 version into the excerpt in figure 1.29: the F#-A#-B succession in the last note of bar 2and the first two notes of bar 3. Subsequent appearances of this RI version occur using the same beats.
P A 72
Figure 1.29: RI -chains in Beethoven, Op. 132 and Webern, Op. 5.
In just-tunings, a two-interval RI -chain can conceptually run infinitely in both di-
rections because of the various commas between almost-equal PCs. In equal tempera-
ment, however, every interval has a periodicity with which it returns to a PC it began
on, and so Riemann system RI -chains become RI -cycles with the same periodicity as
a Riemann system’s interval d . e Riemann system (B b, 2, 5), for instance, has the
canonical listing <Ab b Bb b Cn n Dn>. Extending this RI -chain in both directions
yields <En n F# n | Ab b Bb b Cn n Dn | n En n F#>; the canonical listing is enclosed
in | | bars. It is easy to see that because the RI -chain begins and ends with En n F#, theRI chain will repeat. ere are six distinct <+5,-3> trichords in the chain, as expected
with d = 2; the large-typeface notes together comprise a whole-tone scale, as do the
small-type notes.
e RI -chain operation, which Lewin calls RI C H , is itself a group operation, and
extending the RI -chain of a Riemann system through its entire orbit (i.e. exhausting
the trichord content possible in a given chain) yields a space of regions for the Rie-
mann system.70 ere are as many orbits for RI C H as there are for the d interval’s
transposition cycle. For example, since there are three T3 cycles – <0369>, <148A>,
and <259B> – there are three distinct RI -cycles when d = 3 (the octatonic scale in
ascending order is an example). e SH I F T operation on a Riemann system moves70Because the just-intonation space is infinite, a RI C H orbit is likewise potentially infinite. I will be
assuming equal temperament for the rest of the discussion.
Prolongation and Other eories of Tonality 73
E ♮c ♯A ♮f ♯D ♮b ♮G ♮e ♮C ♮a ♮F ♮d ♮B ♭g ♮E ♭c ♮A ♭
E ♮
c ♯
A ♮
f ♯
D ♮
b ♮
G ♮
e ♮
C ♮
a ♮
F ♮
d ♮
B ♭
g ♮
E ♭
c ♮
A ♭
m′m
d
Figure 1.30: Creating a Tonnetz fragment from the (C , 7, 4) RI -cycle.
its 7-PC canonical listing boundaries one space to the left or right along its associated
RI -cycle. SH I F T (<Ab b Bb b Cn n Dn> , left) (i.e. on the previous example, the (B b, 2, 5) system) yields <En n F# | n Ab b Bb b Cn n | Dn n En n F#>. e new canonical
listing is a transposition of the C ONJ form of the source. Two shifts in one direction
will yield a transposition of the source; this is analogous to “modulating” to a different
key or “region” in tonal music by means of maximal common tone preservation. If there
is more than one orbit for an RI -cycle class, then SH I F T modulation cannot exhaust
the total available number of regions, so another operation like C ONJ is necessary to
modulate to regions that exist on other orbits.
Given his theory’s reliance upon Riemann et al., Lewin perhaps misses an opportu-
nity to relate the resulting spaces to the various dualist theory Tonnetze. e canonical
listing graphs from above would be isographic to portions of Tonnetz graphs to within
the directional connections of canonical listings. Creating a Tonnetz from an RI -cycle
is just a matter of moving the small letters up and reconnecting the nodes to make trian-
gles. is is illustrated in figure 1.30, using the (C , 7, 4) Riemann system’s RI -cycle to
create a classic Riemmanian Tonnetz fragment, which can be extended in all directions.
Nodes from the source Riemann system are placed in circles.
Implied in this reasoning is the idea that Tonnetze can be built from any Riemann
system, complete with generalized Riemannian transformations. Richard Cohn’s Ab-
P A 74
stract Tonnetz, shown in figure 1.31, is an example of this.71 Although Cohn prefers
squares to equilateral triangles in his Tonnetz representations, the structures are identi-
cal. Cohn’s interval terms map to Lewin’s thus: {x = m}, {y = m′}, and {x + y = d};.e bold right triangle depicts a source triad Q, and the others represent a generalized
neo-Riemannian P LR-family of operations on Q; reflection across the 0, (x + y) di-
agonal is a P operation, reflection across the x, (x + y) vertical is an L operation, and
reflection across the 0, x horizontal is an R operation. In Lewin’s terms, P is a C ONJ
operation and L and R are SH I F T operations, all on triads. Since RI -cycles may have
more than one orbit, a given Riemann system class may generate multiple Tonnetze.
Morris, “Voice-Leading Spaces,” 1998 also includes examples of Tonnetze related by
“Stravinskian rotation,” which maps rows and/or columns to diagonals and vice-versa
(thus preserving the Tn cycles from each dimension by mapping them into another).
A family of related Tonnetze can be derived from the Riemannian Tonnetz using these
rotation operations, with an interesting consequence that to get the same result by op-
erating on PCs, one would need a nonstandard PC operator.72
D L’ G T S
Lewin’s theory has an interesting relationship with the modified Straus conditions, and
also a few deficiencies which will need to be patched in order to use it as a basis of a
generalized system of prolongation. e first, second, and fourth Straus conditions all
work to constrain the structure of a scale as it relates to the properties of the chromatic
world it inhabits and the harmonic norms it embeds. To review, the first condition is71Cohn, “Neo-Riemannian Operations, Parsimonious Trichords, and their Tonnetz Representations,”
1997, page 10. Reproduced here with permission of the author.72More recently, Dmitri Tymoczko has been advocating for further geometric generalization of pitch
space, including Tonnetz models for PC sets with more than three members (building on work by Gollinand Cohn), but while these models are relevant to the immediate topic and fascinating in their ownright, they do not bear directly on my prolongation theory. See Tymoczko, A Geometry of Music, 2011and Tymoczko, “e Generalized Tonnetz,” 2012.
Prolongation and Other eories of Tonality 75
-2x + 2y -x + 2y 2y x + 2y 2x + 2y
-2x + y -x +y y- x + y - 2x + y
I/1/ -2x -x 0O x 2x
-2x-y -x-y -y x-y 2x-y
Tx-2y (Q)2x - 2y -2x - 2y -x - 2y -2y x - 2y-2x - 2y
Figure 6: The Abstract Tonnetz
the paper, which uses PLR-family operations to navigate the Tonnetz, in its various versions at various degrees along the abstraction/specification continuum.
2.2. The Generic Tonnetz
Our investigation of the Tonnetz of harmonic theory initially situates it as a member of an infinite class of two-dimensional matrices whose generic form is presented in Figure 6. The primary axes of Figure 6 are generated by the generic intervals x and y, in the sense that each row increments from left to right by the value of x, and each column incre- ments from bottom to top by the value of y. The figure should be inter- preted as projecting its structure beyond its boundaries. The elements of Figure 6 represent real numbers as they increment to infinity, and should not be interpreted in the context of the closed modular systems that were the focus of our previous work. Figure 6 is neither more nor less than the Cartesian coordinate plane of analytic geometry.
In terms of Def. (2), the primary axes of Figure 6 are interpreted as the smallest two step-intervals of a prime form trichord Q = { 0, x, x+y } with step-intervals <x, y, -(x+y)>. The remaining step-interval is the inverse of the sum of the two smaller step-intervals, and hence generates the diag-
10
Figure 1.31: Cohn’s abstract Tonnetz.
the “consonance/dissonance” condition: a way to distinguish consonant intervals from
dissonant ones (or alternatively consonant simultaneities from dissonant ones) is re-
quired. e second (which I have modified) is the “scale-degree” condition: the scale
cycle itself must be sufficiently structured so that a maximum number of the maximal
consonances are embedded in the collection, that each embedded consonance be dis-
tinguishable from the others on the basis of the intervallic structure of the scale (the
“position finding” sub-condition). And finally, the fourth Straus condition, the “har-
mony/voice leading” condition, requires distinction between “stepwise” intervals and
“harmonic” intervals. Ideally, no chromatic interval which appears in a maximal conso-
nance is also used as a stepwise interval; when this is not the case, the distinction must be
maintained contextually. In addition, the consonances of the scale must be embedded
so that their PCs are mutually nonadjacent and generated from a single interval with
respect to the scale (e.g. a diatonic third – as opposed to a major or minor third – in a
major scale).
P A 76
? œ œn œ œ œ œ# œ œˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 2 3 4 5 6 7 8
? œœœ œœœn# œœœ œœœ# œœœn œœœ# œœœn œœœI IIIII? IV V VI VII? I
[045] [019] [015] [045] [045] [015] [01A] [045]
? œ œ œ œ# œn œ œ œ# œn œ1 2 3 4 5 6 7 8 9 10ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
-1+1
? œœœn œœœ# œœœn œœœ## œœœnn œœœ# œœœn œœœ# œœœ#n œœœn[045] [045] [015] [015] [045] [045] [015] [015] [048] [045]
? œœœœn œœœœ## œœœœnn œœœœ## œœœœnn œœœœ# œœœœ#n œœœœn# œœœœ#n œœœœn[0459] [0459] [0156] [0156] [0459] [0459] [0159] [0156] [0489] [0459]
I II III IV V VI VII VIII IX I
? œn œs œf œn œs œf œn œs œn œs œf œn œs œf œn œs œn œs œf0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
? œn œn œn œs œn œn œn œs œn œn1 = 0 2 = 3 3 = 6 4 = 9 5 = 8 6 = 11 7 = 14 8 = 17 9 = 16 1 = 0
-1+2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
? œœœœn œœœœss œœœœnn œœœœss œœœœnnœœœœs œœœœsn œœœœns œœœœsn
œœœœn[0 6 8 14] [0 6 8 14] [0 2 8 10] [0 2 8 10] [0 6 8 14] [0 6 8 14] [0 2 8 14] [0 2 8 10] [0 6 12 14] [0 6 8 14]
I II III IV V VI VII VIII IX I
Scale:
Trichords:
Scale:
Trichords:
Tetrachords:
Scale:
Chromatic:
Tetrachords:
Figure 1.32: Scale and trichord implementation of Riemann system (G n, 5, 4).
Returning to Lewin’s paper, he himself recognizes some of the problems with his
approach as a generalized model of tonality:
It should hardly be necessary to point out that there are important tradi-tions in tonal theory which the work of this paper does not generalize. emost significant such tradition involves the study of voice leading and coun-terpoint in relation to tonal functionality; our generalized theory, with itscanonical listings, must perforce remain mute on such matters in gener-ality, though one could of course work out protocols for voice leading andcounterpoint in connection with specific individual Riemann Systems otherthan the tonal ones.73
Lewin makes “working out voice-leading protocols” sound far easier than it is. e
M5/M7 Riemann system from the Goldberg interlude above is theoretically corrigible
for the reasons mentioned in that section. Let us return to (G n, 5, 4), the Riemann
system introduced in figure 1.28 to test other possibilities. e canonical listing for this
is <Dn # Gn n Cn n Fn>. If one were to arrange it in scale order, by analogy to a diatonic
scale – that is, maintaining maximally even spacing within the scale ordering for each
of the consonant trichords, per Straus condition four – it would look like figure 1.32.
e set types written above the trichords are referenced with respect to the root of each
chord.73Lewin, “A Formal eory of Generalized Tonal Function,” 1982, page 57.
Prolongation and Other eories of Tonality 77
How does this system measure up to the Straus conditions? While the scale cycle
itself has some intervallic consistency, the inclusion of three ICs in an unpredictable
order makes it harder to use. ICs 1, 4, and 5 are consonant, and 2, 3, and 6 are dissonant,
and the dissonant intervals are all used in stepwise motion, so it appears the first and
part of the fourth of Straus’s conditions are met: there is a strong distinction between
consonant and dissonant intervals, as well as between stepwise and harmonic intervals.
Looking at the chords, the I, III, IV, V, and VI chords are each one of the inversionally
related [015] or [045] trichords as designed in the Riemann system specification, but II
and VII are members of two different set classes ([014] and [013], respectively). ere
are thus only five usable consonances of the same set class embedded in the scale, but the
second Straus condition requires at least six; two of the three dissonant ICs appear in II
and VII, which is an unwelcome ambiguity that can only be resolved contextually, for
instance by avoiding one or both of these chords except in passing or cadential situations.
VII is usually not prolonged in major keys, but is the lack of a prolongable supertonic
chord in this Riemann system that big a problem? After all, the ii◦ chord is dissonant in
traditional minor keys, and therefore cannot be prolonged much deeper than the surface
without using mixture to transform it into a minor triad. If mixture is analogous to
“borrowing from the C ONJ system” in Lewin’s terms, perhaps it is possible to turn the
II chord into [045] or [015] via such a borrowing. e C ONJ system here is (G n, 5, 1),
with canonical listing <Dn b Gn b Cn b Fn>, and supertonic triad Fn-Dn-b. As in
the major-minor system, b2 and b4 are the same in each, but here a change in b6 does no
good to make the II chord a usable consonance. Fn-F#-b would be another possibility
– a b is available via a SH I F T left in the C ONJ system, which is also where bII lives
in the minor system, so there could be some “neapolitan” connection. It will be most
fruitful, though, to find out what it is about the major Riemann system that provides it
a consonant supertonic triad.
P A 78
A ♯D ♯G ♯C ♯F ♯B ♮E ♮
f ♯b ♮e ♮a ♮d ♮g ♮
A ♮D ♮G ♮C ♮F ♮B ♭E ♭
f ♮b ♭e ♭a ♭d ♭g ♭
Figure 1.33: “Wolf closure” on a Tonnetz derived from Riemann system (C n, 7, 4).
e answer is given in figure 1.33. Formally, only the Dn on the right (in the primary
chain of d intervals) is part of the Riemann system and included in its canonical listing.
In principle, however, the dn on the left (in the secondary chain of d intervals) could be
included to close the system. Under 5-limit just intonation, these two Dn ’s are in fact
different PCs, differing by a syntonic comma (frequency ratio 81:80, or about 21.51¢),
and the interval between the Dn on the primary chain and the an on the secondary is also
not a true fifth, but rather an unusable “wolf fifth.” Dispatching this discrepancy is the
motivation for meantone temperament, and equal temperament does it automatically.
Henceforth I will call closure of a Riemann system by means of PC identity occurring
between primary and secondary d chains “wolf closure.” Wolf closure in the major-
scale system is what allows the supertonic triad: locating the same PC at both ends of
an RI -chain allows it to serve two functions, primarily as the d of the dominant triad,
and secondarily as the root of SH I F T (subdominant), which is the II Stufe.
e Riemann system class (1,4) has wolf closure, being the M7 transform of (7,4),
but Riemann system class (5,4) does not (see figure 1.34). erefore (5,4) does not
have a proper supertonic triad, as discovered in figure 1.32. Examining the (5,4) graph
at the bottom of figure 1.34, it is apparent that the RI -chain can be extended to the
left two PCs and to the right one, to achieve wolf closure on Dn. is would necessi-
Prolongation and Other eories of Tonality 79
tate a 9-PC scale, which is not provided for under Lewin’s framework. Nevertheless
this scale is worth investigating; it is an example of what I will call a “wolf-extended”
Riemann system, meaning the 7-PC set of a Riemann system has been extended left
or right or both until each side reaches the same pitch. ere are also “wolf-truncated”
Riemann systems, which are arrived at by removing redundant pitches from the 7-PC
set in order to get an ordering of PCs all of which occur once in the ordering except
the PC at the edges, which occurs twice. If a given Riemann definition requires ex-
tension to achieve wolf-closure, “WE” (for “wolf-extended”) may be appended to the
definition, e.g. (F #, 5, 1)-WE. “WT” may be appended for wolf-truncation. Not all
closed Riemann systems can be wolf-closed. An instance of this is the RS (B b, 4, 5),
with canonical listing <Gb b Bb b Dn n F#>. e Gb-F# convergence in this hexatonic
system occurs along the primary IC4 chain (with large note names). Its transpositional
symmetry precludes PC duplication between the primary and secondary IC4 chains.
is shift in perspective necessitates “wolf-closed canonical listings,” which are the
Riemann system RI -chains extended or contracted until they are wolf-closed. It is
appropriate to make the tonic PC bold since it cannot necessarily be inferred from the
structure of the listing. e “wolf-closed” canonical listing for the Riemann system
(F n, 7, 3) is <Bb b F n b Cn b Gn b>. Notice here that it is the subdominant PC rather
than supertonic PC that is repeated at both ends; the intervallic structure of the RI -
chain determines which PC appears at the border. At this point an explanation of why
this is the case would be out of place, but I will come back to it a number of times. For
now, note the inversional relationship between this Riemann system and one of the class
(7,4) – in the major system, the primary pitch on the right was discovered on the left of
the secondary d chain, and in the minor system the reverse is true.
With these formal matters out of the way, I can return to the wolf-extended (5,4).
If I again choose Gn as tonic, my wolf-extended canonical listing for this system might
P A 80
(Cn,1,4)
A ♯A ♮G ♯G ♮F ♯F ♮E ♮
f ♯f ♮e ♮d ♯d ♮c ♯
E ♭D ♮D ♭C ♮B ♮B ♭A ♮
b ♮b ♭a ♮a ♭g ♮g ♭
(Cn,5,4)
F ♯C ♯G ♯D ♯A ♯F ♮C ♮
d ♮a ♮e ♮b ♮f ♯c ♯
E ♭B ♭F ♮C ♮G ♮D ♮A ♮
b ♮g ♭d ♭a ♭e ♭b ♭
Figure 1.34: Wolf closure in (C n, 1, 4) and (C n, 5, 4) Tonnetze.
Prolongation and Other eories of Tonality 81
be <An # Dn # G n n Cn n Fn n>. Another possibility is to extend in the opposite
direction, thus: <Dn # G n n Cn n Fn n Bb n>. e former is preferable for a number
of reasons, but the most important is that the “dominant” side of the RI -chain has
a more important structural relationship to the tonic in prolongational systems than
does the “subdominant” side. Indeed, in Schenkerian literature, one encounters the
terms “tonic,” “dominant,” and “pre-dominant,” the last being a catch-all term for many
different chords, diatonic and chromatic; in my experimentation I have found it much
better to keep the dominant chord alone at the right end and pile all of the “extra” PCs
next to the subdominant where they take on various pre-dominant roles. In either case,
it is easy to see that each of the imbricated trichords in the wolf-extended RI -chain is
a member of SC [015] or its inversion [045], as desired.
Now the 9 PCs need to be ordered in “ascending” scalewise fashion. If I employ the
first wolf-extended canonical listing from above using Gn as tonic, I know the following:
{ b1 = Gn}, { b3 = n}, { b5 = Cn}, { b7 = n}, and { b9 = Fn}, exhausting the odd scale degrees by
climbing “up” the RI -chain. If I continue “up” and wrap around at the wolf-PC, I will
fill out the even scale degrees: { b2 = n / An}, { b4 = #}, { b6 = Dn}, and { b8 = #}, which
returns to the tonic Gn. Concatenating them yields the scale cycle listed on the first line
of figure 1.35. e second line gives all of the embedded [045] and [015] trichords with
only the chord built on Fn forming another SC. Another question is pertinent here: in
the 7-PC diatonic system, one rationale for the fourth Straus condition’s constraining
maximal consonances to triads was that they are the largest set of nonadjacent PCs in
the diatonic ordering. is is not true for the trichords in the present 9-PC system, but
tetrachords do adhere to this constraint. ey are provided on the third line of figure
1.35, consisting of [0459] and [0156] sets on all scale degrees except for VII and IX,
which are [0159] and its inversion [0489] (all set type labels are referenced from the
roots of the chords).
P A 82
? œ œn œ œ œ œ# œ œˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 2 3 4 5 6 7 8
? œœœ œœœn# œœœ œœœ# œœœn œœœ# œœœn œœœI IIIII? IV V VI VII? I
[045] [019] [015] [045] [045] [015] [01A] [045]
? œ œ œ œ# œn œ œ œ# œn œ1 2 3 4 5 6 7 8 9 10ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
-1+1
? œœœn œœœ# œœœn œœœ## œœœnn œœœ# œœœn œœœ# œœœ#n œœœn[045] [045] [015] [015] [045] [045] [015] [015] [048] [045]
? œœœœn œœœœ## œœœœnn œœœœ## œœœœnn œœœœ# œœœœ#n œœœœn# œœœœ#n œœœœn[0459] [0459] [0156] [0156] [0459] [0459] [0159] [0156] [0489] [0459]
I II III IV V VI VII VIII IX I
? œn œs œf œn œs œf œn œs œn œs œf œn œs œf œn œs œn œs œf0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
? œn œn œn œs œn œn œn œs œn œn1 = 0 2 = 3 3 = 6 4 = 9 5 = 8 6 = 11 7 = 14 8 = 17 9 = 16 1 = 0
-1+2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
? œœœœn œœœœss œœœœnn œœœœss œœœœnnœœœœs œœœœsn œœœœns œœœœsn
œœœœn[0 6 8 14] [0 6 8 14] [0 2 8 10] [0 2 8 10] [0 6 8 14] [0 6 8 14] [0 2 8 14] [0 2 8 10] [0 6 12 14] [0 6 8 14]
I II III IV V VI VII VIII IX I
Scale:
Trichords:
Scale:
Trichords:
Tetrachords:
Scale:
Chromatic:
Tetrachords:
Figure 1.35: “Wolf-extended” Riemann system (G n, 5, 4)-WE.
ere are a number of points to unpack here. First, we want to think of the 9-PC
scale ordering as an instance of a generalized diatonic scale of some sort; let us further
assume that it is analogous to “major.” ere are some similarities between this and the
traditional major scale: it has only two stepwise intervals (which happen to be 1 and 2
semitones in this case), with relatively long chains of one interval punctuated by roughly
equally-spaced instances of the other. In comparison with the scale from figure 1.32,
this extended version is superior in its intervallic consistency. Dissimilarities with the
traditional diatonic aside from the obvious difference in cardinality and the two changes
in direction in the scale ordering include a “leading-tone” (the Fn) which is not in an
obviously unique place in the scale – the structure of the scale ordering does not make
it an obvious “tendency tone” – and IC 1 is represented both as a step and as a “third,”
descending and ascending respectively (indicated by brackets in the figure).
From a harmonic standpoint, the Fn turns out to be the best choice for the leading
tone because the trichord built on it is a different SC from all of the others by analogy
to the traditional diatonic. I will show in Part 2 that the leading tone’s lack of obvious
tendency here emerges from some number-theoretic properties of the integer 9 (the
cardinality of the scale). e “diatonic” contours of this system differ greatly depending
Prolongation and Other eories of Tonality 83
on whether the harmonic norm is trichordal or tetrachordal. If it is trichordal, the
consonant ICs are 1, 4, and 5, the dissonant ones are 2, 3, and 6, and the stepwise
1 and 2. Neither IC 3 nor 6 appears as a stepwise interval or as an interval in the
leading-tone triad, so their harmonic function is not clear, and IC 1 is both a consonant
and a stepwise interval. is means that harmonic and contrapuntal clarity requires
IC 1 to be consonant only when it is part of an [015] or [045] trichord, or when the
surrounding context makes it clear that it forms a scale third.74 A secondary concern is
that in the diatonic scale, all root-motions within the scale preserve PCs except stepwise
root-motion, while here neither stepwise root motion nor motion by a “fourth” preserves
PCs. is is again because the trichords are not the largest mutually non-adjacent PC
sets.
Tetrachords are, however, and they are likely preferable as harmonic norms for this
scale than trichords. Not only do they preserve PCs in all root motions but stepwise,
they preserve a role for ICs 3 and 6. In the tetrachordal system, consonant ICs are
1, 3, 4, 5, and 6, the dissonant IC is 2, and the stepwise ICs are 1 and 2. Instead of
two inversional forms of one consonant SC, here there are two consonant SCs, each
of which is inversionally symmetrical. And like the major scale, the two exceptional
tetrachords are inversionally related (the V7 and vii∅7 chords in the major scale, and the
VII and IX chords here).75 e Straus-condition status of this tetrachordal system is
difficult to evaluate. e consonance/condition is barely met; almost all of the intervals
are consonant, but this is balanced by the dissonant IC 2 which is also the predomi-
nant stepwise interval. Intervals 3 and 6 are further constrained by the SCs they can
appear in as consonances, and in none of them do they appear together. e scale-
degree condition is clearly met within the diatonic, but because it has nearly saturated74On the other hand, like many ambiguities in the tonal system, this could be exploited for interesting
harmonic purposes, e.g. to get from one region to another quickly.75Note, though, that here the tetrachords are purportedly a harmonic norm, while in the major scale
they are not.
P A 84
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7
10
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6
10
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5 4ˆ ˆ
10
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3
10
I
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IV 6
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2
6
V 6
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1
I
&?
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8
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7
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5 4ˆ ˆ
10
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3
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IV 6
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2
6
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1
I
&?
.œ
œ œ œ œ œb œ
.œn œ œ œ œ# œ#
œ# œ# œ# œn œ œ#
œ œn œ œn œ#.œbœn œ œ œ œ œ
.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 II 864 V 86 I
&?
˙
œ œ œ œ œ œ# œ œ
˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
œ œnœ œ œ œ
4
Figure 1.36: (G n, 5, 4)-WE middleground (cf. Goldberg middleground in figure 1.10).
the chromatic, regions have far more overlap in PC-content; the smallest overlap is 6
PCs. e harmony/voice-leading condition is still threatened by the dual role of IC1 as
consonance and stepwise interval.
I have so far said very little about the third Straus condition – the “embellishment”
condition – which requires a set of prolongational transformations. I save full treatment
of this condition to Part 2, but I offer figures 1.36 and 1.37 as proof of concept. e
first is a middleground “10-line” structure analogous to the Goldberg middleground in
figure 1.10 (page 50). A “Roman numeral” analysis appears on the bottom for reference.
I would like to draw attention to two features of this passage. First, the chord progres-
sions display a remarkable fluidity: common tones abound, as does stepwise motion
in all of the voices; because these are tetrachords, five-voice harmony is necessary for
enough doubling to ensure this fluidity. Second, the larger cardinality of this scale en-
tails a longer phrase than the analogous G-major structure. Figure 1.37 uses the first
seven chords of the middleground as a skeleton for a canon at the second, analogous
to the Goldberg canon. Because of the scale’s larger cardinality, a 24 time signature is re-
quired to fit the scale fragments that comprise the canonic voices, which in turn forces
a slight change in the left-hand accompaniment strategy. is is a very clear instance of
foundational assumptions and structures functioning as “compositional determinants.”
Prolongation and Other eories of Tonality 85
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7
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6
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5 4ˆ ˆ
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1
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3
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2
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1
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.œœ œ œ œ# œ#
œ# œ# œ# œn œ œ#
œ œn œ œn œ#.œbœn œ œ œ œ œ
.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 VI 864 V 86 I
&?
˙
œ œ œ œ œ œ# œ œ
˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
œ œnœ œ œ œ
4
Figure 1.37: Hypothetical canonic surface derived from the first 7 chords of figure 1.36.
Many of the problems with the 9-PC scale can be mitigated if it is situated in a
larger chromatic collection; to do this we need a microtonal scale. e best of these for
this illustration is 19, because many of its intervals are very close to the same size
as intervals in 12, which means that the two systems can function as “cognates” –
two systems which sound nearly equivalent. In 19, IC3 sounds nearly as a whole-
step does in 12. IC5 sounds as a “minor third,” IC6 as a “major third,” IC8 as a
“perfect fourth,” and IC11 as a “perfect fifth.” IC1 and IC2 sound as two different
sizes of “minor second.” And because of the larger chromatic cardinality, there are more
intervals to work with, and chances for a much better distinction between consonant
and dissonant intervals.
Microtonal systems are difficult to notate, but one rule of thumb is to try as hard as
possible to reflect the size of intervals in ways which map well to musicians training in
reading music. Intervals which sound similar or identical to a minor third (say) from
12 should look like a third in the notation if at all possible. e top line of figure
1.38 is one of many ways to notate 19. e s and f symbols are accidentals devised
by composer and theorist Easley Blackwood for microtonal notation.76 e advantage
of using Blackwood’s accidentals over # and b is that the latter symbols are already as-
sociated with 12 intervals; Blackwood’s symbols clearly indicate something other
than 12. e way I have rendered the notation preserves as much as possible from
12 intuitions: the natural notes would make a cognate major scale, steps between76I have deviated somewhat from Blackwood’s use of these accidentals. See Keislar, “Six American
Composers on Nonstandard Tunings,” 1991, pages 189-190.
P A 86
? œ œn œ œ œ œ# œ œˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 2 3 4 5 6 7 8
? œœœ œœœn# œœœ œœœ# œœœn œœœ# œœœn œœœI IIIII? IV V VI VII? I
[045] [019] [015] [045] [045] [015] [01A] [045]
? œ œ œ œ# œn œ œ œ# œn œ1 2 3 4 5 6 7 8 9 10ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
-1+1
? œœœn œœœ# œœœn œœœ## œœœnn œœœ# œœœn œœœ# œœœ#n œœœn[045] [045] [015] [015] [045] [045] [015] [015] [048] [045]
? œœœœn œœœœ## œœœœnn œœœœ## œœœœnn œœœœ# œœœœ#n œœœœn# œœœœ#n œœœœn[0459] [0459] [0156] [0156] [0459] [0459] [0159] [0156] [0489] [0459]
I II III IV V VI VII VIII IX I
? œn œs œf œn œs œf œn œs œn œs œf œn œs œf œn œs œn œs œf0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
? œn œn œn œs œn œn œn œs œn œn1 = 0 2 = 3 3 = 6 4 = 9 5 = 8 6 = 11 7 = 14 8 = 17 9 = 16 1 = 0
-1+2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
? œœœœn œœœœss œœœœnn œœœœss œœœœnnœœœœs œœœœsn œœœœns œœœœsn
œœœœn[0 6 8 14] [0 6 8 14] [0 2 8 10] [0 2 8 10] [0 6 8 14] [0 6 8 14] [0 2 8 14] [0 2 8 10] [0 6 12 14] [0 6 8 14]
I II III IV V VI VII VIII IX I
Scale:
Trichords:
Scale:
Trichords:
Tetrachords:
Scale:
Chromatic:
Tetrachords:
Figure 1.38: e 19 “cognate” of the extended Riemann system from figure 1.35.
two s notes or two f notes sound like a whole step, the interval of 6 between Bn and
D s sounds like a major third, and so forth. Not all equal-tempered microtonal systems
map as cleanly to 12, so a few other notation strategies will be necessary when they
come up. When our context is microtonal, Riemann system labels also need the N
value, e.g. 19 : (B f, 11,5) (cognate to Bb minor in 12).
e cognate 9-PC scale is listed on the second line of figure 1.38, with scale degrees
matched to PCs from the aggregate on the first line. e larger chromatic cardinality
already pays intervallic dividends: the two roles for IC1 in 12 are now handled by
ICs 1 and 2, eliminating that previous ambiguity. ere are a total of 9 ICs in 19;
the consonant ICs are 2, 5, 6, 8, and 9, the dissonant ones are 1, 3, 4, and 7, and the
stepwise are 1 and 3. IC7 is, like IC6 in 12, a dissonant interval that can still be
used in harmonies. IC4 is not present between any two PCs of the scale so one could
make a case that it is not just a dissonant interval, but one that is requires great care to
handle. is system meets the Straus conditions far more comfortably than its 12
cognate.
Finally, as preparation for many of the themes in part 2, an analysis of the ordering
of the 9-note scale in 19 : (G n, 8, 6)-WE is in order. ere is one very important feature
Prolongation and Other eories of Tonality 87
of wolf closure I have neglected to mention, but which the Tonnetz structures show
very clearly: wolf closure guarantees that the PC-content of a scale will be generated
by the interval d . Because the primary and secondary d-chains in a Tonnetz are joined
at the wolf identity, these d-chains may be concatenated to form one long d-chain
containing all of the PCs of the scale. It also guarantees that the scale set cardinality is
odd, because the primary and secondary d-chains must be the same length and have one
PC repetition, so the cardinality of the scale set is always 2n− 1 where n is the length
of the two d-chains. is brings the study directly into the realm of scale theory, where
musical objects like generated scales have been analyzed thoroughly. Figure 1.39 gives
a taste both of the underlying theory and of the common heuristics for scale design.
e purpose of figure 1.39 is to show how the orderings of a scale, its RI -chain,
and the underlying generator IC cycle interrelate, but before I discuss it I must lay out
some principles it relies upon. If I take any odd number n (5, say) I can write down
the integers mod n in order, and imagine them as a cycle (0 follows n − 1) : <01234>
for n=5. Sometimes cycles need to be explicitly distinguished from rows, in which case
they can be notated with -k at the end, where k is the zeroth term, like so: <01234-
0>; this is not necessary when the context is always cyclical. Cyclical ordering also
guarantees that a rotation of the notated cycle is congruent to that cycle: <01234> is
the same as <23401>. Because n is odd here, I can reorder the cycle by taking every
other number (and wrapping around) like so: <02413>. If I continue the process, I
get <04321>→ <03142>→ <01234>, the last of which is the original cycle. ese are
called multiplicative permutations because each of these orderings can be obtained from
the original by multiplying order number positions. If k is a value and o is its order
position number (indexed from 0), the original cycle may be rewritten as an unordered
set of pairs (k , o): [(0,0)(1,1)(2,2)(3,3)(4,4)]. To take every other number, one has
to multiply each of the order numbers by the multiplicative inverse of 2, mod n. If
P A 88
multiplication by 1 is the identity operation, then the multiplicative inverse of 2, call it
x, satisfies the formula 2x = 1. For n = 5, x = 3. Multiplying the o term in each (k , o)
pair from the original cycle by 3, mod 5, yields [(0,0)(1,3)(2,1)(3,4)(4,2)]. In cyclic
notation this is <02413>, which is the result of taking every other number from the
original cycle, as desired. Although order number multiplication is formally the correct
principle at play, it is usually easier to just think of a “skip factor” s for multiplicative
permutation. Moving through the cycle in order is indicated with s = 1. Taking every
second number means s = 2, s = 3 means taking every third number, and so forth. e
result of each possible permutation operation on the original cycle is a multiplicative
ordering, and together they comprise a multiplicative ordering class.77
While the RI -chain of a Riemann system is foundational for Lewin’s approach, it is
better thought of as emergent in wolf-closed systems, where instead the generator inter-
val is foundational. Whether or not an interval can generate a wolf-closed scale depends
on the numeric interaction between that interval, the cardinality of the generalized di-
atonic scale, and the cardinality of the chromatic collection. However, heuristically it
sometimes makes sense to begin with the RI -chain, e.g. when a given consonant sonor-
ity is desired.78 Perhaps surprisingly, it is almost never easy to start by trying to create
the scale ordering itself; it is also almost always emergent.
e generator chain, scale ordering, and RI -chain are all instances of the same mul-
tiplicative ordering class. Figure 1.39 demonstrates this for the G major scale and for
the 9-PC 19 scale from figure 1.38. e top line of each example shows the PC
content of the scale generated by the generator interval. Notice that because this is a
PC-cycle, it wraps back to the beginning, which introduces one more interval into the
cycle. e intervals for each step in the cycle are listed above each line of the example;77In group theoretic terms, they comprise an orbit of the order number multiplication operator.78In some more advanced cases, the ordering which provides the harmonic norm for the system is not
actually an RI -chain; this need not concern us until later.
Prolongation and Other eories of Tonality 89
?œn œn œn œn œn œn œ# œn
&7 7 7 7 7 7 6
& œn œn œn œn œn œn œ# œn2 2 1 2 2 2 1
? œn œn œn œn œn œn œ# œn3 4 3 4 3 4 3
?œs œs œn œn œn œn œn œn œn œs
&8 8 8 8 8 8 8 8 12
& œn œn œn œs œn œn œn œs œn œn3 3 3 18 3 3 3 18 3
? œn œs œn œs œn œn œn œn œn œn6 2 6 2 6 2 6 2 6
2
Generator:
Scale:
RI-Chain:
Generator:
Scale:
RI-Chain:
12:(G,7,4)
19:(G,8,6)-WE
Figure 1.39: Multiplicative orderings of 12 : (G n, 7, 4) and 19 : (G n, 8, 6)-WE.
P A 90
for G-major it is a cycle-of-fifths followed by a tritone, and for the 19 system it is a
cycle of interval 8 (sounding as a perfect fourth) followed by interval 12.79
e second line of each example shows how the scale ordering is derived by applying
a multiplicative permutation on the generator cycle, with s = 2 (s is the “skip factor”).
Rather than keep the same first PC in each line, I have kept the last PC the same in
each line. is is often a good heuristic for finding the “leading tone” of the system that
is analogous to “major,” and by inference the tonic for that system. e RI -chain of
the third line results from a second application of the multiplicative permutation, and a
third application would lead back to the generator cycle. A crucial point is that for each
line, there are exactly two intervals, and the two intervals differ by the same amount in
each (viz. 1 in the G-major system, and 4 in the 19 system). is is called Myhill ’s
property, and will feature in much of the discussion below.
Why are the arrows from generator-cycle to scale-ordering crossed in the 19
system but not in the G-major system? is derives from the properties of the numbers
7 and 9, the cardinalities of each collection. In the G-major system the permutation
operation is applied twice with s = 2, and so the move from generator to RI -chain is
itself a multiplicative permutation with s = 4. In fact the s = 2 operation generates a
group of multiplicative permutations, which is the cycle 2 j mod 7 for each successive
operation, beginning with j = 0. is cycle is <1,2,4-1>, giving the three s values for
the multiplicative orderings for the system. By contrast, if this is done mod 9, the cycle
is <1,2,4,8,7,5-1>, which gives six different multiplicative orderings. Note, however,
that if the generator cycle results when s = 1, when s = 8 the result is just the retro-
grade of that cycle. is is true for any complementary s pair. Retrogression does not
change anything essential about the ordering, though – it is analogous to the difference
between ascending and descending versions of the same scale. So in the 19 case, the79e top example of figure1.39 is very similar in spirit to the foundational insight from jazz composer
and theorist George Russell’s lydian chromatic system. See the example on page 2 of Russell, LydianChromatic Concept of Tonal Organization, 2001.
Prolongation and Other eories of Tonality 91
ascending scale-order from the second line derives from taking every other PC from the
descending generator cycle. is happens quite often in constructing systems for pro-
longation, but so long as the prime and retrograde forms of each cycle are treated as
interchangeable it does not pose a problem.
Lewin’s work is indispensable for any theorist who seeks to generalize. His Riemann
system idea is exactly the kind of simple, flexible, and suggestive framework needed to
arrange PC-space so that it can grow many varieties of prolongational fruit. Given
that Lewin’s work here and as a whole is most at home among other transformation
and function theories of the past, a good transition to part 2 ought to take a broader
look at the historic compatibility (or lack thereof ) between prolongation theories and
transformation/tonal function theories, the chief of which is neo-Riemannian theory.
P A 92
H F
N-R P T
As I have shown above, the transformational approach underlying Lewin’s Riemann
system framework and neo-Riemannian theory works very well to establish uniform
PC spaces which, when combined with scale theories, form a “tonal substrate” upon
which additional prolongational transformations may be built. I do believe, however,
that neo-Riemannian theory, as sophisticated as it is, is simplistic when it is used to
analyze actual tonal events. Why should this be the case? Let us look at some of what
these theories assume.
Onemay be tempted to look at any theory of function derived from theHauptmann-
Oettingen-Riemann tradition as a dualist theory which sets up and explores the impli-
cations of tonal “oppositions” or “dualisms,” for example “major-minor,” “dominant-
subdominant,” “tonic-phonic,” “overtones-undertones,” and so forth.80 But it seems
more useful conveniently to forget the dualist history of function theories and their
questionable methodological baggage,81 and reformulate them as theories which to a
first approximation deal more with PC than with pitch, and which admit PC inversion
as a foundational transformation between triads.82 In comparison to the main theoretic
traditions from the eighteenth century, function theories’ emphasis on PC over pitch
ties them closely to Rameau’s earlier theories and divides them from species counter-
point and thoroughbass theories.83 ey are also theories which are more likely to em-
phasize the particularities of progressions from chord to chord on the surface of a piece,80I will assume that the reader has some familiarity with this material; excellent introductions are
offered in Harrison, Harmonic Function in Chromatic Music, 1994, part 2 (chapters 5-7), and Klumpen-houwer, “Dualist Tonal Space and Transformation,” 2002.
81e status of “the undertone series” in Oettingen and Riemann is especially problematic.82PC inversion is not to be confused with “chord inversions” in traditional tonal theory.83As noted in Klumpenhouwer, 2002, p. 456, Schenker thought of Riemann’s theoretic heritage as
“little more than warmed-over Rameau,” while he based his own theory on the legacy of species andthoroughbass.
Prolongation and Other eories of Tonality 93
which give some weight to voice leading in PC space, and also to particular movements
within a space built from the assumed foundational transformations, as instances of
those transformations.
In other words, if this type of theory is designed to reveal the “underlying structure”
of a tonal piece, it does so by mapping a wide terrain of potentialities through which any
tonal piece may progress and then by describing events in the piece at hand as instances
of the available transformations producing movements in this terrain.84 Further, the
transformations involved may be collected into categories of similarity, and each musi-
cal event may be assigned a function based on the types of transformations surrounding
it in its local context. e collection of transformations in such a system typically forms
a generalized interval system or ,85 and the ability to form a stems from the fact
that these systems operate within PC-space. is does not mean that the transforma-
tions operate on PCs, but rather that the operations map PC-space triads onto other
PC-space triads, which limits their ability to account for register, doubling, spacing,
etc. when these matter.
A path through such a space (a “chord progression”) need not be goal-oriented, but
each entity (a “chord”) may have a putative “goal” based on the function assigned to it,
which may differ depending on the scope of the context. If goal-orientation is desired,
a structure allows chains of transformations to be collected into musical phrases,
such that the opening chord of the phrase may be associated with the last chord in the
phrase by means of a transformation within the group, which is itself the composition
of all the transformations from event to event in the phrase.86 It should be emphasized
that the space in question need not have literal geometric representation: the intervals
in a are enough to ascribe distance algebraically. e approach in (Callender et al.,84For instance the so-called L, P, and R transformations, which are inversions around each of the three
dyads of a triad, are sufficient both to establish and move about a Tonnetz.85Lewin, Generalized Musical Intervals and Transformations, 1987.86See Hook, Uniform Triadic Transformations, 2002 for a comprehensive description of such a .
P A 94
2008)87 and (Tymoczko, 2011)88 provide topological descriptions of continuous quo-
tient spaces formed under equivalences among “musical objects” (e.g. “chords”), and
which then treat chord progressions as a set of discrete motions through one of those
quotient spaces. is could be a promising reinterpretation along geometric lines with
obvious implications for microtonal music.
While diatonic scales do function in these theories, it is important to note that they
are usually the consequence of the transformations among triads rather than a construct
from which triads are derived. In many theories, including what I have outlined in the
discussion of Lewin’s work above, scale construction is in fact a specific goal of triad (or
other maximal consonance) transformation. Certainly in the historical function theo-
ries, derivation of the major and minor modes was a constraining force on the mapping
of the triadic terrain. Scales may figure heavily in cases where deviations from the un-
derlying tonal space require explanation; such deviations can occur by creating special
combinations of, or analogies among, the set of available transformations with an ear
toward a specific scale or mode. Daniel Harrison’s functional discharge idea,89 which
constructs chord function on the basis of the combination of scale degrees, is one such
theory. In this system scale degrees derive their normative functions from prior tri-
adic transformational constructs, but they can then function in analogous non-triadic
or chromatic contexts which display similar voice leading among scale degrees. is al-
lows a partial hierarchization of the tonal space by conferring function to the correlates of
triadic transformations rather than to the triads and their transformations themselves;
the resulting terrain is thus less “flat” than those in traditional Tonnetze or “charts of
regions.”
In Harrison’s theory the various non-triadic chords which arise in chromatic tonal
music are thus explained by locating one or more core scalar voice-leading motion(s)87Callender, et al., “Generalized Voice-Leading Spaces,” 2008.88Tymoczko, A Geometry of Music, 2011.89Harrison, Harmonic Function in Chromatic Music, 1994, chapter 2 and pages 90-126.
Prolongation and Other eories of Tonality 95
amongst the pitches of a chord pair (say # b7−b1), which may be the result of a conceptu-
ally prior triadic transformation (a Leittonwechsel in the example) – Harrison calls the
first member of such a pair a “functional agent” and the voice leading itself “functional
discharge.” One then locates parallel or contrary motion elsewhere in the voice leading
and ascribes it an accompanying role in the harmonic motion. e results are categories
of tonal motion, e.g. those involving # b7−b1 (“Dominant-Tonic”) or b b6−b5 (“Subdominant-
Tonic” or “Subdominant-Dominant”), which need not necessarily involve the tonic,
dominant, or subdominant triads at all. Perhaps ironically, the flexibility in chord con-
struction which accompanies this focus on voice-leading categories necessarily brings
the theory more into the pitch world than PC world, and requires scale as a foundation
almost equal in priority to triadic transformation; both of these make it more compatible
with prolongation theories.90
How this all relates to prolongation theory is difficult to disentangle, in part be-
cause in Schenkerian literature the assumptions appear more modest (or simply more
traditional) than those behind function theory, and in part because there is disagreement
among Schenkerian theorists about what the assumptions actually are. e triad plays as
strong a role in prolongation theory as in function theory, but it is a quite different role:
in prolongation theory, diatonic and chromatic scale areas are derived by filling in the
space between pitches of the root-position tonic triad rather than by group operations
on triads. However, filling-in, like its triadic-transformation counterparts in function
theory, is both an a priori tonal-space-oriented procedure and a literal operation which
generates musical events via passing/neighbor motion between arpeggiated members of
a triad: in prolongation theory a triad is a locus of stability and is therefore an originator90William Benjamin’s “Pitch-Class Counterpoint” is another model which constructs voice leading
from normative harmonic progressions – involving descending fifths – and which could be used in con-junction with a system of prolongation. It is probably reformulable in Neo-Riemannian terms, but thedetails of this theory may have some application in Schenkerian accounts of invertible counterpoint. SeeBenjamin, “Pitch Class Counterpoint in Tonal Music,” 1981 and Renwick, Analyzing Fugue: a Schenke-rian Approach, 1995, pages 82-86.
P A 96
of tonal motion.91 As a result, this motion is to be understood more contrapuntally than
harmonically, but it is not fair to say that it is purely contrapuntal because the harmonic
content resulting from such motion is not by any means unconstrained. e major and
minor scales are assumed to be the only properly tonal diatonic modes, and the rest of
the chromatic derives from modal mixture.92 While Schenker may have regarded the
fact that diatonic scales embed several triads – each in a different intervallic relation with
the others – as a simple fact of nature, the foregoing discussion shows that orderings of
this kind are rare and must be constructed. Without this property of diatonic scales,
the archetypal Schenkerian concepts “composing out of triads,” Stufen, and therefore
Schichten, would all be impossible.
In most constructions of Schenkerian prolongation, octave equivalence is syntacti-
cally contextual rather than absolute, so that pitch-class is not quite as robust a concept
in prolongation theory as in function theory. Traditionally these systems are “bass-
oriented,” so that two triads having identical PC content but a different PC in the bass
will have different syntactical meanings.93 is is not to say that a Db is sometimes not a
Db, but just that two or more instances may have very different functions in the syntax,
even within the same chord.
e most obvious overlapping concept between prolongation and function theory is
the Schenkerian Stufe – “scale degree” – which has both abstract and concrete manifes-
tations. In its simplest, most conceptual form it is just a triad built on one of the scale91See Schachter, Unfoldings, 1999, “e Triad as Place and Action” (chapter 6, pages 161-183). is
concept also has a Platonic flavor, as in this passage from Phaedo: “Let us examine whether those that havean opposite must necessarily come to be from their opposite and from nowhere else, as for example whensomething comes to be larger it must necessarily become larger from having been smaller before…” Onecan imagine Socrates saying “and motion necessarily comes from rest” – an idea explored most famouslyin Aristotle’s discussion of the Unmoved Mover.
92With the possible exception of #b4/bb5, which may arise only from a tonicization or other secondaryrelationship: see Brown, et al., “e #IV(bV) Hypothesis,” 1997.
93Bass-oriented harmony may be a canonical trait of tonal practice, but there is no logical reason forthis to be the case – soprano orientation is logically just as possible, as is no orientation to a particularvoice at all (in which case voice leading would conceptually be among PCs rather than pitches).
Prolongation and Other eories of Tonality 97
degrees of a major or minor scale. In the applied theory it is a term usually reserved for
describing a triad that has been or has the potential to be prolonged by one or more of
the transformations at a lower level. A given Stufe has no a priori harmonic function
in prolongation theory in the sense it would in function theory. Instead, a set of voice-
leading paradigms which prolong the tonic (Schenker’s “Ursätze” and “deep middle-
ground structures,” for instance) endow each Stufe with a contrapuntal function within
that schema. Harmonic function is thus mostly paradigmatic – triads which have the
same contrapuntal function in most or all of the voice-leading paradigms thereby attain
harmonic function as a shorthand for the typical collective movement of all the voices.
e Stufe idea also provides constraint for harmony in strict counterpoint. Aside from
a few prescriptions concerning initial sonorities and cadences, the harmonic system in
traditional Fuxian counterpoint remains purely intervallic. However, if each consonant
simultaneity in strict counterpoint is thought to represent a triad, then a progression
syntax can be joined with the contrapuntal scheme, resulting in something more akin
to thoroughbass.
I have tried to show that in order to generalize prolongation – in order to meet the
Straus criteria – one needs a very well-regulated PC space. So far I have tied together
some ways of constructing these PC spaces. But much of what makes neo-Riemannian
theory simplistic as a theory of musical events is its inhabitance of PC space alone, with
very little actual pitch-space constraint. From my perspective, a of PC-space triads
(or other n-chord) is not a model of tonal syntax, nor does a syntax emerge necessarily
from it; rather, such a is a description of a regularity imposed on a PC space in
preparation for the creation of a syntax by other means. An analogy may help make the
point: a neo-Riemannian is like a chessboard. A chessboard’s grid structure makes it
possible to measure distances in different ways. e rules of neither chess nor checkers
predictably emerge from the grid structure itself, but both depend on the grid to get
P A 98
going, and the structure of their rules and principles of competent play are constrained
by the structure of the grid. A system of prolongation is one of many “games” that can
be played using a neo-Riemannian (or similar) as a necessary measuring constraint.
is all leads to a curiously tangled layering of structure and design. We tend to think
of transformation and function theories, with their network structures, as associational
rather than hierarchical systems. Paths along aTonnetz or other PC space can occur with
various degrees of similarity; many of Wagner’s leitmotivs are such paths in PC space,
and in those cases the association is not between melodies or contours but between har-
monic progressions. Prolongational systems built on these PC spaces necessarily impose
some hierarchy both on the abstract PC spaces and on musical events; the same path
in PC space can be instantiated in very different prolongational paradigms, depending
on how objects in pitch space are deployed. In my conception of prolongation theory,
motives can act as yet another layer of association built on top of hierarchically designed
voice-leading structures. None of these layers of structure and design in tonal music can
function on its own, and the potential ways in which each one can constrain the other
two is what makes tonal music so rich with possibility and expression. My hope is to
extend this expressive potential to areas of composition not usually considered “tonal.”
99
P 2Prolongation in Equal Temperament
He imposes orders as he thinks of them,As the fox and snake do. It is a brave affair.Next he builds capitols and in their corridors,
Whiter than wax, sonorous, fame as it is,He establishes statues of reasonable men,Who surpassed the most literate owl, the most
erudite
Of elephants. But to impose is notTo discover. To discover an order as ofA season, to discover summer and know it,
To discover winter and know it well, to find,Not to impose, not to have reasoned at all,Out of nothing to have come on major weather,
It is possible, possible, possible. It mustBe possible. It must be that in timee real will from its crude compoundings come,
Seeming, at first, a beast disgorged, unlike,Warmed by a desperate milk. To find the real,To be stripped of every fiction except one,
e fiction of an absolute—Angel,Be silent in your luminous cloud and heare luminous melody of proper sound.
Wallace StevensNotes Toward A Supreme Fiction:
It Must Give Pleasure, VII
H the hypothesis that prolongation depends upon the structure of
scales (scale cycles) and tonal spaces in part 1, I develop these ideas further in part
2. is part functions as the full presentation of my theory of prolongation in equal
temperaments.
P E T 100
2.1 C G D
S C F P
Because the structure of scales plays a large role in determining the kinds of prolon-
gation that can be built on them, I must draw some concepts and terminology from
the literature on scale theory. What follows is therefore not by any means meant to
be exhaustive. Scale theory tends to be a bit more mathematically involved than other
tonal theory. Although I endeavor to keep the presentation as informal as possible, I
assume some knowledge of simple group theory.1 e best place to start is the structure
of scale cycles, without placing too much emphasis on a given scale’s relationship to the
equal temperament it is embedded in. e reason for this is that scale cycles of the
same cardinality but different temperaments are more similar to each other than differ-
ent scale cardinalities in the same temperament. In a later section I explore embedding
scale cycles in equal temperaments.
M G n M P
I introduced multiplicative permutations in Part 1 (page 87); the following is a more
complete explanation. A scale cycle of cardinality d may be represented as an ordered set
of all the integers mod d , mapping the tonic scale degree to 0, b2 to 1, b3 to 2, and so forth.
A 7-PC scale cycle would thus be represented as < 0,1,2,3,4,5,6− 0>. A chromatic
aggregate of cardinality c may likewise be represented as an ordered set of all integers
mod c (in common practice c = 12, but c is different in other equal temperaments),
mapping 0 to a given PC (Cn, say), and each successive integer to successively higher1efirst chapter ofMorris,ClassNotes for AdvancedAtonalMusiceory, 2001 (pages 1-19) introduces
all of the terms I use here. e first three chapters of comprise an alternative approach. See Lewin,Generalized Musical Intervals and Transformations, 1987, pages 1-59.
Constructing Generalized Diatonic Scale Cycles For Prolongation 101
PCs in PC space. In this section I show what cardinalities of scale cycle are useful for
meeting the Straus conditions, and under what circumstances.
A mod n system is denotedZ/nZ. Z is the set of all integers from−∞ to+∞, and
nZ is the set of all integer multiples of the integer n. e “division” notation in Z/nZsays that we are dividing the set of all integers into sets of size n ranging from a multiple
of n to the next multiple of n, minus 1. If each of these sets is taken as equivalent,
the result is the set of possible remainders from dividing any integer by n, yielding
the common understanding of modular systems. In a given modular system, all of its
integers collectively form a group under addition.2 is is not true for multiplication
mod n: for instance, 0 cannot be a member of any modular multiplication group because
it maps every integer to 0 and thus has no inverse.
Amultiplication groupmod n, denoted (Z/nZ)× (the× symbol indicates the group’s
operation type), consists of all integers mod n which are coprime to n.3 is set of
integers coprime to n is sometimes called a reduced residue system mod n. Euler’s to-
tient function ϕ(n) is a catalog of the cardinality of the reduced residue system mod n
for every positive integer n. For mod 10, the reduced residue system is {1,3,7,9}, so
ϕ(10) = 4. It is easy to show that all modular multiplication groups contain an even
number of members if n > 2. For any mod n system, if n is even, then n2 is not rel-
atively prime to n, and for any mod n system even or odd, any m < n2 coprime to n
has a complement with respect n which is also relatively prime to n. An integer m’s
complement mod n is is congruent to m · (n − 1) mod n or equivalently n − m mod
n. is is analogous to multiplying by -1 or subtracting from 0 in the regular integers.2is means every mod n addition operation results in a unique member of the set of integers mod
n (the operation is one-to-one and onto), every operation has a unique inverse which “undoes” thatoperation, and there is one identity operation which maps each integer onto itself. In a mod n system,adding an integer a to any member of the group can be undone by subtracting a from the result. Adding0 to any integer returns that integer, so 0 is the identity operation for addition.
3at is, they share no common factor. Relatively prime has identical meaning.
P E T 102
a× 1 5 7 11
1 1 5 7 11
5 5 1 11 7
7 7 11 1 5
11 11 7 5 1
b× 0 1 2 3 4 5 6 7 8 9 10 11
1 0 1 2 3 4 5 6 7 8 9 10 11
5 0 5 10 3 8 1 6 11 4 9 2 7
7 0 7 2 9 4 11 6 1 8 3 10 5
11 0 11 10 9 8 7 6 5 4 3 2 1
Figure 2.1: a) e table of operations for (Z/12Z)×. b) e multiplicative operationsacting as automorphisms on all integers mod 12.
Since the integers in a multiplication group form complementary pairs when n > 2, the
ϕ(n) function always returns an even number.
e mod 12 multiplication group is common in musical set theory. ϕ(12) = 4, and
the integers coprime to 12 are {1,5,7,11}.4 Figure 2.1a shows the multiplication table
for the mod 12 reduced residue system. Because this is a group, each member of the
group appears once in each row and each column.5 Each member of a multiplication
group is also an automorphism for the entire set of integers mod n.6 e action of each
multiplication operator on the integers mod 12 is listed in figure 2.1b. If these integers4I am deliberately using the base-10 integer 11 instead of the base-12 integer B (or e) as in standard
musical set theory. e reason for this is that in this essay I am dealing with larger modular systemswhere arithmetic using letters for integers larger than 9 becomes cumbersome. What is I+L mod 24?e answer is much easier to find when it is rewritten 18+21 mod 24.
5is group is isomorphic to the Klein four-group. e group formed by the traditional serial opera-tions P, I, R, and RI is also isomorphic to the Klein four-group.
6An automorphism is an operation that maps a set onto itself.
Constructing Generalized Diatonic Scale Cycles For Prolongation 103
represent PCs, then the four operations map the ascending chromatic to an ascending
chromatic, a circle of fourths, a circle of fifths, and a descending chromatic, respec-
tively. e M11 operation is an inversion operation, which maps integers onto their
complements. When it is applied to order numbers rather than PCs, the result is order
inversion, which is a retrogression.7 is will become relevant shortly. For traditional
scales we tend to think of the ascending and descending forms of the scale – which are
related by order inversion – as just two manifestations of the same scale. I adopt this
stance for all scale orderings in this essay: one scale ordering always has two possible
directions of traversal analogous to “ascending” and “descending” forms of traditional
scales.
For my purposes here I turn to a common representation of multiplication groups
mod n: the cycle graph. A multiplication cycle is much like an interval cycle in musical
set theory, but using multiplication operators rather than addition operators. To create
a multiplication cycle, choose an integer σ from the multiplication group to start the
cycle, and an integer m to use as a base of multiplication mod n (σ and m can be equal).
Begin the cycle on σ , and generate new members using successive multiplication by m:
< σ , σ ·m, σ ·m2, σ ·m3, σ ·m4 . . . σ ·m r > mod n where m r ≡ 1 mod n (this just
means that the cycle ends when the process returns to σ).8 For instance, if n = 20, the
multiplication group is {1,3,7,9,11,13,17,19}. If σ = 7 and m = 3, then the resulting
cycle is< 7,1,3,9−7>. ere are usually many such available cycles in a multiplication
group. e idea behind a cycle graph is to create a single connected graph using the
group’s longest available cycles. Such a graph uses σ = 1 for all of its cycles. One can7Since 0 persists under all M operations, the zeroth order position remains. For a traditional retro-
gression, one must apply a rotation (order transposition) of one place backward after M11 on order. isis analogous to the T11I operation in PC space, which maps 0 to 11 and 11 to 0, or in the case of order,moves the last to the zeroth and the zeroth to the last.
8e ≡ symbol means “is equivalent to” or “is congruent to.”
P E T 104
make an account of all the available σ = 1 multiplication cycles for a given group to see
how they might fit together to make a single connected graph.
(Z/9Z)× {1,2,4,5,7,8} (σ = 1) (Z/16Z)× {1,3,5,7,9,11,13,15} (σ = 1)
Length m Cycle
2 8 < 1,8− 1>
3 4 < 1,4,7− 1>
6 2 < 1,2,4,8,7,5− 1>
Length m Cycle
7 < 1,7− 1>
2 9 < 1,9− 1>
15 < 1,15− 1>
43
5
< 1,3,9,11− 1>
< 1,5,9,13− 1>
12
48
7
5
(Z/9Z)×
1
3
9
11 513
157
(Z/16Z)×
Figure 2.2: Multiplication cycles and graphs for (Z/9Z)× and (Z/16Z)× where σ = 1.
In figure 2.2 I have listed all of the σ = 1 multiplication cycles from groups (Z/9Z)×
and (Z/16Z)×, organizing the lists by length of cycle. For the mod 9 system, we see
that ϕ(9) = 6, and there is a cycle of length 6; therefore it can be graphed as one single
cycle, displayed below the table in figure 2.2. Examining the shorter cycles, it is easy to
notice that they form subgroups of the mod 9 multiplication group. e cycle of length
3, < 1,4,7− 1>, results from skipping every other member of the main cycle, and the
cycle of length 2 results from skipping 2 members of the main cycle. e mod 16 system
is a bit more complicated: ϕ16= 8, but the longest cycle has length 4. Notice first that
the operators 7 and 15 each appear in only one cycle, each of which is of length 2, so
those cycles must be drawn independently. Also notice that operator 9 occurs in the
Constructing Generalized Diatonic Scale Cycles For Prolongation 105
same place in the two cycles of length 4, so those cycles can be drawn concentrically,
with shared nodes at 1 and 9. As may be expected, the identity operator 1 serves as
a manner of nexus for the graph.9 ose graphs which can be expressed as a single
complete cycle have two or more generators or primitive roots, which are multiplication
operators that can generate the entire group. Generally, primitive roots exist for mod
n multiplication groups where n = 2, n = 4, n = pk , and n = 2 pk , where p is an odd
prime and k is a positive integer. e existence of a primitive root for a given mod n
group does not figure greatly into my theory, but groups with primitive roots can be a
bit more flexible than others.10
C S C M
P O
Recall from part 1 that order-position multiplication is what maps a scale ordering to a
(potentially wolf-extended) canonical listing, andmaps the canonical listing to a chain of
a generating interval. erefore the multiplication group associated with a given scale’s
cardinality matters a great deal, and in fact not all scale sizes are usable. Also recall
that I take orderings related by order inversion (retrogression) to be two different forms
of the same ordering. is means that multiplicative orderings whose skip factors are
complementary integers are equivalent; therefore the number of essential multiplicative
orderings for a scale of cardinality d is ϕ(d )2 . What I have called an essential ordering may
also be called a multiplicative permutation class; I use both terms interchangeably.9Every cycle graph for a modulo multiplication group is isomorphic to a cyclic group or direct sum of
cyclic groups. e reasons for this are beyond the scope of this essay, but it is a well-known result fromgroup theory and number theory.
10at odd primes p all have primitive roots is also what makes Mallalieu-type rows possible, due toan isomorphism between the multiplication group of p and the addition group of p − 1. Every equaltemperament of cardinality p − 1 has a Mallalieu-type row.
P E T 106
We need to know two things to proceed: first, what properties of the multiplication
group associated with the cardinality of a scale are required for that scale to be viable for
prolongation? Second, what cardinalities provide these properties? e latter question
depends in part on the structure of cycle graphs. To that end I have drawn cycle graphs
for every mod n system from n = 3 to n = 24 in figure 2.3, and will be referring to them
throughout this discussion.11 e goal is to find multiplicative orderings representing
1) the scale order, 2) the canonical listing, and 3) the generator chain, such that the skip
factors which map 1) to 2), 2) to 3), and 3) back to 1) (or its order inversion) form a
coherent whole. Occasionally we may find that all three transformations apply the same
skip factor, and in many cases we may find that there are more than three multiplicative
orderings. e former case applies to the traditional diatonic scale of cardinality 7, and
provides the greatest flexibility. at “two steps make a third, two thirds make a fifth,
and two fifths make a step” is responsible in part for the great number of harmonic
sequences in tonal music, for instance.12 In the latter case, there are extra orderings
that may have no consequence at all, or we may find that an ordering that is not an
alternation of two intervals may serve better as the canonical listing. e theory is not
developed enough at this point to decide on these points, so for now I will examine a
loose categorization of possible generalized diatonic systems based on the structure of
the scale cardinality’s multiplication cycle graph.
To begin, some cardinalities may be eliminated outright: any whose multiplication
group does not contain at least 3 distinct members with which to represent the scale
ordering, canonical listing, and generator chain. For example, cardinality 10 has only
two distinct multiplicative orderings, viz. the ordering represented by skip factors 1 and
9, and the one represented by skip factors 3 and 7 (1 and 9 are related by order inversion,11A scale of cardinality 24 is certainly on the threshold of unusability, since it would need to be em-
bedded in an even larger chromatic. It could work in an equal-tempered version of Partch’s 43-tone scale,but this is itself very close to the limits of cognition as a chromatic construct.
12See Clough, “Diatonic Interval Sets and Transformational Structures,” 1979.
Constructing Generalized Diatonic Scale Cycles For Prolongation 107
(Z/3Z)× (Z/4Z)× (Z/5Z)× (Z/6Z)× (Z/7Z)× (Z/8Z)×1
2
1
3
1
2
4
3
1
5
13
26
4
5
157
3
(Z/9Z)× (Z/10Z)× (Z/11Z)× (Z/12Z)×1
2
48
7
51
3
9
7
1 2
4
8
5109
7
3
6
1711
5
(Z/13Z)× (Z/14Z)× (Z/15Z)× (Z/16Z)×1 2
4
8
361211
9
5
107 1
3
913
11
5 1
2
4
8 137
1411
1
3
9
11 513
157
(Z/17Z)× (Z/18Z)× (Z/19Z)× (Z/20Z)×1 3
910
13
515
1116148
7
4
122
6 15
717
13
111 2
48
16
13
714
9181715
11
3
6
125
10
1
3
9
7 1713
1911
(Z/21Z)× (Z/22Z)× (Z/23Z)× (Z/24Z)×
12
48
16
11
20
13
519
1710 1 7
5
13
32115
17
9
191 5
210
4
20
8
1716
1192218
2113
19
3
15
67
1214
1
57
11
13
23
19
17
Figure 2.3: Multiplication groups for integers 3-24.
P E T 108
as are 3 and 7, each being a complementary pair mod 10). e relation by order inver-
sion for the two complementary pairs is made clear in the full listing of multiplicative
orderings for cardinality 10 yielded by successive applications of skip factor 3:
M1: < 0,1,2,3,4,5,6,7,8,9− 0>
M3: < 0,3,6,9,2,5,8,1,4,7− 0>
M9: < 0,9,8,7,6,5,4,3,2,1− 0>
M7: < 0,7,4,1,8,5,2,9,6,3− 0>
M1: < 0,1,2,3,4,5,6,7,8,9− 0>
10 is not an appropriate cardinality for prolongational structures because one of the
required essential multiplicative orderings is missing. By the same reasoning, the cardi-
nalities 3, 4, 5, 6, 8, 10, and 12 can all be eliminated.13
We can further eliminate systems that fail to provide for tertian or quartal harmony,
which correlate with skip factors of 2 and 3, respectively. I restrict the investigation to
tertian and quartal harmony only because it is useful to voice leading for the scale to
be exhausted or nearly exhausted by a maximal consonance and all of its neighboring
PCs with respect to the scale ordering. So while a quintal-harmony system is logically
possible, its voice leading may be prohibitively difficult. Because a skip factor must be
relatively prime to the cardinality, we can eliminate those which are relatively prime to
neither 2 nor 3, which are all multiples of 6. us 18 and 24 are also not appropriate for
my prolongation framework. Henceforth I call tertian systems T systems and quartal
systems Q systems. is leaves the following 13 scale cardinalities which can potentially
(depending on further constraints I develop presently) be used for prolongation: 7, 9,
11, 13, 14, 15, 16, 17, 19, 20, 21, 22, and 23.
Starting with some nomenclature, I label the scale ordering S , the canonical listing
H (for “harmony”), and the generator chain G; I also use d (for “diatonic”) for the scale13Actually, cardinality 5 is small enough that the generator can also serve as a dyadic maximal conso-
nance. is is an exceptional circumstance with interesting possibilities.
Constructing Generalized Diatonic Scale Cycles For Prolongation 109
cardinality, h for the cardinality of the maximal consonance, and c for the cardinality of
the chromatic. For T and Q systems, the S→H skip factor is constrained to be either
2 or 3. Assuming that the chromatic generator interval (labeled gc ) is a consonance in
a system, it must occur at least once in all of the system’s maximal consonances. e
H →G operation determines which chord member of a system’s maximal consonance
is gc higher than the root. is skip factor must be strictly less than h and greater than
1; if (H → G) ≥ h, then the generator could not occur between PCs in a maximal
consonance. So it is always the case that 1 < (H → G) < h. e composition of
these two operations together determine the diatonic interval associated with the gen-
erator (labeled gd ), which imposes some constraints on harmonic progression, primary
arpeggiation in the bass voice of an Ursatz, and modulation to other regions.
To find h for a given system, first recall that the fourth Straus condition (the har-
mony/ voice-leading condition) requires that “stepwise” intervals be strictly distinguished
from “harmonic” intervals, both within a scale ordering and the chromatic in which it
is embedded. In the next section I show that in Q systems, stepwise intervals consist
in both diatonic seconds and thirds; this expands the notion of non-adjacency and also
expands the number of voice-leading chromatic intervals to four (rather than the usual
two).14 In my discussion of wolf-extended Riemann systems in Part 1, I suggested that
for a 9-PC scale, tetrachords rather than triads made for more consistent voice leading
(see figure 1.35, page 82). Maximal evenness, a concept from scale theory, proves useful
here and warrants a brief explanatory excursion.
A maximally even set is “a set whose elements are distributed as evenly as possible
around” the cycle of a given superset.15 For sets where intervals between adjacencies
are each one of two possible values – the traditional diatonic scale, e.g., with its whole
steps and half steps – the scarcest intervals must be distributed as far as possible from14See “Tonicization” on page 153.15Clough and Douthett, “Maximally Even Sets,” 1991, page 96
P E T 110
one another to satisfy the criteria for maximal evenness. us the ascending melodic
minor set is not maximally even because its two semitones are not as far apart as they
are in the major mode. Clough and Douthett point out that triads and seventh-chords
are maximally even sets with respect to the traditional diatonic. is evenness accounts
for the parsimonious voice leading available to these collections.16 However, seventh
chords do not comply with the fourth Straus condition as a harmonic norm because the
seventh is a stepwise interval.
With the foregoing in mind, it is clear that h must be large enough to enable par-
simonious voice leading but not so large that it contains a stepwise interval: we are
looking for the largest T or Q harmonies which do not contain stepwise intervals. T
systems employ a skip factor of 2 and are therefore only usable in odd scale cardinali-
ties; this means that all T harmonies for prolongation consist of a stack of thirds plus a
fourth. For T systems, h = [ d2 ], where h is the cardinality of the harmony, d is the scale
cardinality, and the brackets indicate truncation.17 e d value for Q systems must be
relatively prime to 3, so it will always be 1 or 2 greater than a multiple of 3. Since dia-
tonic seconds and thirds count as stepwise intervals, Q harmony for prolongation must
be a stack of fourths plus a fifth, or a stack of fourths plus a sixth. So for Q systems,
h = [ d3 ]. In general, h = [ d
S→H ]. With these formulas for h, the 1 < (H → G) < h
inequality may be solved in terms of the other known values.
e G→ S operation is either the member of the mod d multiplication group that
maps the composition of S → H and H → G to 1 mod d or the one which maps the
composition to (d −1)mod d , whichever such multiplier is smaller. When the result is
1 mod d , the skip operation maps the generator chain back to the scale ordering. When16Ibid., pages 169-170. ey call the diatonic and its pentatonic complement first-order maximally
even sets with respect to the totally even 12-tone equal tempered chromatic, and triads and seventh chordssecond-order maximally even sets because they are only maximally even with respect to the diatonic, notto the chromatic.
17[x] yields the largest integer not greater than x.
Constructing Generalized Diatonic Scale Cycles For Prolongation 111
the result is (d−1)mod d , however, the skip operation maps the generator chain to the
retrograde of the scale ordering. In this latter case, I put a bar over both the S and the
integer in the operation (G → S = 3, e.g.) to indicate that the result is the retrograde
of the scale ordering. I call systems with a G→ S operator ascensive systems, and those
with a G→ S operator descensive; in contexts where either operator might apply, I use
G → (S⊻S).18 e reason the smaller multiplier of the two is used rather than simply
always using the one which maps the composition back to 1 mod d , has to do with
the fact that harmony is built from and referenced to the bass. A pair of examples will
suffice to illustrate the concept. Let d = 13, and S → H = 3, making it a Q system;
then h = [ d3 ] = [
133 ] = 4. erefore the H →G skip factor may be either 3 or 2 in order
to satisfy 1 < (H → G) < h. In a bass-oriented system, intervals are measured from
low to high, “root position” indicates that the root is in the bass and that the other chord
members are measured above it, and so forth. So if H → G = 3, then the generator
spans the diatonic interval between the root and the fourth member of the chord above
the root. If it is 2, then then the generator spans the interval between the root and the
third member.
Figure 2.4makes this clear by expressing ascending intervals as clockwisemovements
around a graphical cycle of 13 elements, numbering ascending scale steps as consecu-
tive integers indexed from zero. In the top half of the figure, H → G = 3, and since
it is a Q system, gd (the diatonic interval spanned by the generator) is 3× 3 = 9. In
the bottom half, H → G = 2, so gd = 3× 2 = 6.19 Let us assume that the generator
is also the interval between tonic and dominant, understood in the Schenkerian sense
where the dominant attains its function from its participation in a primary arpeggiation
within the tonic chord. For the traditional 7-PC diatonic system, it takes moving two
fifths up to move one PC to an adjacent PC in the scale order. e progression ii-V-I18⊻ is the “exclusive or” symbol: (S⊻S) means “either S or S , but not both.”19Recall that a diatonic interval of 6 is called a 7th in traditional music theory.
P E T 112
0 1
2
3
4
567
8
9
10
11
12
S→H = 3H →G = 3
0 1
2
3
4
567
8
9
10
11
12
G→ S = 3
0 1
2
3
4
567
8
9
10
11
12
S→H = 3H →G = 2
0 1
2
3
4
567
8
9
10
11
12
G→ S = 3
Figure 2.4: Two possible Q systems for d = 13.
makes this relationship explicit in the PCs which are shared between adjacent chords
in the progression: ii and V share b2, and V and I share b5, which is what makes the
primary bass arpeggiation between b1 and b5 possible. In the top half of figure 2.4, then,
it takes moving three generators up (that is, 3 clockwise motions by 9 steps) to reach
a PC adjacent to the starting PC, and in this case the adjacency happens to be a step
forward in the scale ordering. It would require 10×gd to reach a PC a step back in the
scale ordering (10 and 3 are complementary mod 13, so moving up by 10 is the same
as moving down by 3). e progression II-VI-X-I might carry approximately the same
Constructing Generalized Diatonic Scale Cycles For Prolongation 113
valence in this system as ii-V-I does in a major key.20 e cycle of orderings, with tonic
and dominant PCs in bold italics, is:
S : 0 1 2 3 4 5 6 7 8 9 10 11 12 03→ H : 2 5 8 11 1 4 7 10 0 3 6 9 12 23→ G: 8 4 0 9 5 1 10 6 2 11 7 3 12 83→ S : 0 1 2 3 4 5 6 7 8 9 10 11 12 0
Notice that the penultimate column contains 12 (the leading tone) in each ordering.
I will show why I have arranged the cycles this way when I show how chromatic em-
bedding works with scale orderings. e short explanation is that once the orderings
are embedded in an equal temperament, it is not a simple matter to determine which
PC should be tonic. It turns out that choosing the last PC in the generator chain as
the leading tone guarantees that the dominant harmony has the appropriate chromatic
intervals, and it is convenient to order the cycles to reflect this.
e bottom half of figure 2.4 is also a Q system (S→H = 3), but here H →G = 2
and gd = 3× 2= 6. Here it would take 11×gd to reach a PC one step clockwise from
the starting point (0 + 11 × 6 = 66 ≡ 1 mod 13). However, it only takes 2×gd to
move a step counterclockwise from the starting point (0+ 2× 6 ≡ 12 mod 13). Since
2< 11, this system contains the operator G→ S = 2. To see why this matters requires
some explanation. To begin, note that a multiplicative permutation of any generator
chain which does not saturate the chromatic always contains exactly two interval types
between adjacent members of the ordering. is is a consequence ofMyhill ’s Property, or
MP. If a scale has MP, then every scale interval comes in two different chromatic sizes.21
For instance, in a major scale, which has MP, there are two kinds of steps (chromatic
intervals 1 and 2), two kinds of third (3 and 4), two kinds of fourth (5 and 6) and so
forth. A generator chain G automatically has this property: consider a chain of five20I have used all capital roman numerals for the 13-tone progression because I do not know anything
about the intervallic disposition of the harmonies.21After the mathematician and logician John Myhill. See Clough and Myerson, “Musical Scales and
the Generalized Circle of Fifths,” 1986, page 698.
P E T 114
perfect fourths in 12, with wraparound: < 0,5,10,3,8,1− 0 >. Clearly there are
exactly two sizes of chromatic interval between adjacent PCs: 5, the generator, and 11,
the wraparound interval. ere are also two sizes of chromatic interval between pitches
2 spaces apart in the ordering: namely 10 (0 to 10, 5 to 3, etc.) and 4, (8 to 0, 1 to 5). is
pattern continues for all generic intervals with respect to the G ordering. erefore, a
multiplicative permutation of G, which rearranges the ordering by successive G intervals
of the same size according to the skip factor, will always have two sizes of chromatic
interval.
One of these intervals occurs more than the other; let that interval be called the
ordering’s primary interval and the scarcer one the ordering’s secondary interval.22 e
generator chain (imagined as a cycle) has only one instance of its secondary interval:
the one between the last member and the first member of the chain, as the cycle is
completed.23 For any other multiplicative permutation, the primary interval is always a
sum of a set of generators or the mod c complement where c is the chromatic cardinality,
and the secondary is always a sum of generators plus the secondary interval from the G
ordering or their complements; in both cases the number of addends is the same.24
e tonic harmony consists of the same diatonic PCs in both the top and bot-
tom halves of figure 2.4, namely [0,3,6,9]. e dominant harmonies are different,
[9,12,2,5] for the top and [6,9,12,2] for the bottom. We can reason from this that for
the bottom example, the diatonic interval of 6 must be the same chromatic interval –
the generator – between 0 and 6, and between 6 and 12: in order for the dominant to be
a transposition or pitch-class inversion of the tonic, 12 must be the dominant-of-the-
dominant PC. is is the crucial point: since the chromatic interval between 0 and 622For a major scale in scale order, the whole step is the primary interval and the half step the secondary.23For a major scale in a 12 chromatic, gc is 5 or 7, and is the primary interval of the G ordering;
the secondary interval is 6.24For a major scale, the primary stepwise interval is 2 ≡ 7+ 7 mod 12, and the secondary stepwise
interval is 1 ≡ 7+ 6 mod 12. In its H ordering, the primary interval is 3 ≡ 5+ 5+ 5 mod 12 and thesecondary is 4≡ 5+ 5+ 6 mod 12.
Constructing Generalized Diatonic Scale Cycles For Prolongation 115
and between 6 and 12 are the same, the step between 12 and 0 in the S ordering must be
the primary interval rather than the secondary, in contradistinction to a traditionally es-
sential property of leading tones that if they are not a secondary interval from the tonic,
they must be altered to make it so. is can not happen for the present scale system
because to do so would damage the intervallic integrity of the dominant harmony. We
saw this in the 9-PC system from part one, which had the following chromatic PCs
in order (mod 12): < 0,2,4,6,5,7,9,11,10− 0 >, with 10 as leading tone, a primary
interval below the tonic.
Nevertheless the leading tone is distinct from other PCs in a scale system in two
ways. First, because the leading tone is always at the rightmost end of the generator
chain, it always participates in one secondary interval in every multiplicative permuta-
tion. In ascensive systems (i.e. those with a G→ S operator), this secondary interval in
the S ordering will be between the leading tone and the PC to its right, the tonic. By
contrast, in descensive systems (i.e those with a G → S operator), there will be a sec-
ondary interval between the leading tone and the interval to its left. Second, although
all the constraints laid out so far entail that the leading tone must be a member of the
dominant harmony and that the dominant harmony must be a transposition or pitch-
class inversion of the tonic harmony, the chord(s) clockwise from the dominant on the
H cycle which have both the leading tone and the leftmost PC of the generator chain as
members will be of different set classes from all of the other harmonies of the system.25
e leading tone is unique for these reasons, and it is convenient to rotate all multiplica-
tive permutations in order to place the leading tone in the same order position in each.
is must all be kept in mind later when we attempt applied dominants and “minor
mode” analogues. As a side note, one may also notice that the tonic and dominant PCs25is specifies the exact number of primary and “odd” harmonies for a given system, and provides the
necessary refinement to the fourth sub-condition of Straus’s second condition, which requires the numberof primary harmonies in a system to be “nearly equal” to the cardinality of the scale. See page 39 in Part1 above.
P E T 116
occur in the left side of the G ordering in ascensive (see the list of orderings on page
113), while they occur in the right side of the generator chain in descensive ones:
S : 0 1 2 3 4 5 6 7 8 9 10 11 12 03→ H : 2 5 8 11 1 4 7 10 0 3 6 9 12 22→ G: 5 11 4 10 3 9 2 8 1 7 0 6 12 52→ S : 11 10 9 8 7 6 5 4 3 2 1 0 12 11
To summarize the constraints I have explored so far, I started by restricting the
S→H operator to either 2 or 3 to create T(tertian) or Q (quartal) systems, respectively.
I showed that cardinality of a referential harmony (labeled h) varies with the cardinality
of the diatonic scale (labeled d ) and inversely with the S→H operator. e exact rela-
tionship is h = [ d(S→H )]where the square brackets indicate truncation of fractional values
to the largest integer not greater than the fractional number. Because I will be using
the generator as the interval between tonic and dominant, and because in Schenkerian
systems the dominant attains its function from its participation in the arpeggiation of
the tonic harmony, the generator must appear as an interval between one or more pairs
of PCs in a referential harmony. is means that the value of h determines the range
of possible values for the H →G operator, such that 1< (H →G)< h (these are strict
inequalities). is latter restriction eliminates a number of hypothetical systems, such as
a Q system where d = 11, h = 3, and the ordered triple of S→ H →G→ S operators
is < 3,5,3>. In that system, since h = 3, the H →G operator (5) is too large.
Is there any reason to restrict the possibilities for a G→ (S⊻S) operator? To answer
this question requires a change of perspective. Heretofore I have used consecutive in-
tegers mod d to represent the S ordering. For the following, consecutive integers mod
d represent the G ordering, so intervals between adjacent members of the S and H or-
derings are expressed in terms of generator steps rather than diatonic scale steps. For
example, consider all of the multiplicative permutation classes for d = 11, keeping 10
(the leading tone) in the penultimate column:
Constructing Generalized Diatonic Scale Cycles For Prolongation 117
M1 0 1 2 3 4 5 6 7 8 9 10 0
M2 1 3 5 7 9 0 2 4 6 8 10 1
M3 2 5 8 0 3 6 9 1 4 7 10 2
M4 3 7 0 4 8 1 5 9 2 6 10 3
M5 4 9 3 8 2 7 1 6 0 5 10 4
11 is prime so it is coprime to every nonzero integer mod 11; this list only extends to
M5 because M6 is just the retrograde of M5. Without embedding this system in any
particular equal temperament, it is still possible to glean some information about the
chromatic intervals, since we know the relationship between the G ordering’s primary
and secondary intervals, and the primary and secondary intervals of the other multi-
plicative orderings. To instantiate these orderings in a chromatic, one would simply
multiply each integer by the desired chromatic generator interval.26
e interval succession of M1 is < 1,1,1,1,1,1,1,1,1,1,−10> (expressed in terms
of a gc ). M2’s interval succession is < 2,2,2,2,−9,2,2,2,2,2,−9 >. It is clear that
each time the ordering crosses 0 (e.g. from 9 to 0 and 10 to 1 in M2), the result is a
secondary interval. If I list the interval succession for each ordering using pairs of letters
for primary and secondary intervals (a,b for M1, c,d for M2, etc.), the following pattern
emerges:
M1 a a a a a a a a a a b
M2 c c c c d c c c c c d
M3 e e f e e e f e e e f
M4 g h g g h g g h g g h
M5 i j i j i j i j i i j
e skip factor for each row is equivalent to the number of secondary intervals in the
succession. e M1 row is the interval succession associated with the G ordering. After
that, as the skip factor increases, the interval successions move from what might be
called “scale-like” or “S-like” successions to “harmony-like” or “H -like” successions.26See From Diatonic to Chromatic on page 170, below.
P E T 118
In other words, the rule of thumb is that with respect to the G ordering, small skip
factors are most likely to generate S orderings, while larger skip factors are most likely
to generate H orderings. e skip factors here are in fact skip-factor classes, where
complementary integer skip factors belong to the same class, which is labeled with the
smaller of the two numbers.
It is preferable to require that the S skip-factor class (with respect to G) be smaller
than the H skip-factor class. e reason is that because the Straus conditions require
intervallic regularity among the harmonies used as Stufen, and because h < d2 in all
cases, intervals in the interval succession of H need to alternate quickly. For instance if
the M2 ordering above were used as H , even in a T system where h = 5, there may be
too many chord types for satisfactory intervallic consistency, and in addition there are
no unique harmonies for the leading tone to participate in. More specifically, the 5-PC
harmony types would be members of the following interval succession types: c-c-c-c,
c-c-c-d, c-c-d-c, c-d-c-c, and d-c-c-c. In the examples below, the H orderings produce
much more consistent harmonies.
I now have enough information to find all of the usable 11-PC diatonic systems.
First, I am going to rotate the cycle of multiplicative order operations to begin on G
rather than S : G → (S⊻S) → H → G. Next, I want to find a set of 3 integers the
product of which ≡ 1 or 10 mod 11, and which comply with the constraints outlined.
Starting with the T/Q constraint, we know that the ordered triple of operations must
contain at least one 2 or one 3. One such set is 2 × 2 × 3 = 12 ≡ 1 mod 11, and
another is 3× 3× 5= 45≡ 1 mod 11. e first of these triples can be ordered in three
ways: < 2,2,3 >, < 2,3,2 >, and < 3,2,2 >. Two of these are T systems, and one is
a Q system. Of the three orderings of the other triple – < 3,3,5 >, < 3,5,3 >, and
< 5,3,3 > – the one whose S → H operator is 5 (viz. < 3,5,3 >) can be eliminated
because this would be neither a T nor a Q system.
Constructing Generalized Diatonic Scale Cycles For Prolongation 119
In mod 11, h = 5 for T systems and h = 3 for Q systems. All three orderings of
the first triple comply, but since the second triple can only generate Q systems, the one
whose H →G operator is 5 (viz. < 3,3,5>) can be eliminated because this value must
be less than h, which is 3. Finally, we need the skip-factor class of S with respect to
G to be smaller than that of H . e following maps the operation triples to triples
of skip-factor classes on G, which must be strictly ascending to comply with the final
constraint: < 2,2,3 >→< 1,2,4 >, < 2,3,2 >→< 1,2,5 >, < 3,2,2 >→< 1,3,5 >,
and < 5,3,3>→< 1,5,3>. Only this last one is not strictly ascending, so it is out.27
us there are 3 possible systems for d = 11. Mapping out the orderings and the
interval succession patterns gives:
G→ S→H →G =< 2,3,2> (Q system)
G 0 1 2 3 4 5 6 7 8 9 10 0
S 1 3 5 7 9 0 2 4 6 8 10 1
H 5 0 6 1 7 2 8 3 9 4 10 5
G a a a a a a a a a a b
S c c c c d c c c c c d
H e f e f e f e f e f e
G→ S→H →G =< 3,2,2> (T system)
G 0 1 2 3 4 5 6 7 8 9 10 0
S 2 5 8 0 3 6 9 1 4 7 10 2
H 5 0 6 1 7 2 8 3 9 4 10 5
G a a a a a a a a a a b
S c c d c c c d c c c d
H e f e f e f e f e f e
27Refer to the interval successions on page 117 to see what<G, S, H > skip-factor classes of< 1,5,3>would look like.
P E T 120
G→ S→H →G =< 2,2,3> (T system)
G 0 1 2 3 4 5 6 7 8 9 10 0
S 1 3 5 7 9 0 2 4 6 8 10 1
H 3 7 0 4 8 1 5 9 2 6 10 3
G a a a a a a a a a a b
S c c c c d c c c c c d
H e f e e f e e f e e f
e interval successions for S and H are of primary interest. One of the scales,
like the major scale, has two spans of a repeated primary interval punctuated by the
secondary interval. e other scale, like the major scale, is made from primary-interval
spans of length 2 and 3, separated by secondary intervals. In an 11-PC T system. ere
are two different H interval successions, and one is used for both a T system and a Q
system. e Q system harmonies have three members, with two intervals separating
successive members. ere are therefore two main qualities of harmony in the system:
five harmonies with the interval succession e-f, and four with f-e. ese are analogous
to major and minor triads in traditional tonality. With wraparound, there is one odd
harmony with interval succession e-e, which is analogous to the leading-tone triad in a
major key.
In the T system with the same H interval succession (the second one listed), there
are again two main qualities of harmony; four with interval succession e-f-e-f and four
with f-e-f-e. As with the Q system and in traditional tonality, these qualities of har-
mony are pitch-class inversions of one another. With wraparound there are now three
odd harmonies: e-f-e-e, f-e-e-f, and e-e-f-e; the first and last are related by inversion,
and the middle is inversionally symmetrical. Something analogous to the V7 chord in
traditional tonality is possible with the PCs < 3,9,4,10,5,0 > from the H ordering,
with interval succession f-e-f-e-e; this has a primary harmony in the bottom, and an
Constructing Generalized Diatonic Scale Cycles For Prolongation 121
odd harmony on the top, which distinguishes it from all the other 6-PC chords from
the H ordering.
e final system is a T system with an interesting interval succession for H . ere
are now three primary chord qualities: 3 of e-f-e-e, 3 of f-e-e-f, and 3 of e-e-f-e – the
same interval configuration as the odd harmonies of the previous system. Similarly, the
odd harmonies here have the same interval configuration as the primary harmonies of
the previous system: e-f-e-f and f-e-f-e. A system with three primary chord qualities
which can be used as maximal consonances is quite different from the traditional tonal
system, and it provides the opportunity for more harmonic variety within a rigorously
prolongational system.
Similar diatonic prolongation system templates which comply with all of the con-
straints outlined in this section are provided in figure 2.5, from d = 7 to d = 23, 34
diatonic systems in total. Note that I have switched back to the S → H →G→ (S⊻S)orientation. In the next section (a second musical interlude) I show how four of these
systems might be used to make prolongational structures. After that I systematize the
chromatic embedding I perform intuitively in the interlude examples.
P E T 122
d T/Q h gd S → H → G → (S⊻S)
7 T 3 4 2 2 2 S
9 T 4 4 2 2 2 S
11 T 5 4 2 2 3 S
11 T 5 6 2 3 2 S
11 Q 3 6 3 2 2 S
13 T 6 4 2 2 3 S
13 T 6 6 2 3 2 S
13 Q 4 6 3 2 2 S
13 Q 4 9 3 3 3 S
14 Q 4 9 3 3 3 S
15 T 7 4 2 2 4 S
15 T 7 8 2 4 2 S
17 T 8 4 2 2 4 S
17 T 8 8 2 4 2 S
17 T 8 6 2 3 3 S
17 Q 5 6 3 2 3 S
17 Q 5 9 3 3 2 S
d T/Q h gd S → H → G → (S⊻S)
19 T 9 4 2 2 5 S
19 T 9 10 2 5 2 S
19 T 9 6 2 3 3 S
19 Q 6 6 3 2 3 S
19 Q 6 9 3 3 2 S
21 T 10 4 2 2 5 S
21 T 10 10 2 5 2 S
22 Q 7 9 3 3 5 S
22 Q 7 15 3 5 3 S
23 T 11 4 2 2 6 S
23 T 11 12 2 6 2 S
23 T 11 6 2 3 4 S
23 T 11 8 2 4 3 S
23 Q 7 6 3 2 4 S
23 Q 7 12 3 4 2 S
23 Q 7 9 3 3 5 S
23 Q 7 15 3 5 3 S
Figure 2.5: Diatonic prolongation templates from d = 7 to d = 23.
2.2 M I II: Règle De L’Octave
In classroom harmony (that is, harmonic theory more in the Rameauian and Riemann-
ian traditions and less in the Schenkerian), diatonic voice leading, spacing, and doubling
is far more systematically sophisticated than what is perceived as the more advanced
chromatic harmony. e primary reason for this is that diatonic harmony voice leading
is prototypical: chromaticism in the form of tonicization, mixture, and chromatic in-
flection (e.g. augmented-6th chords) is fashioned after diatonic models. It can therefore
be pedagogically profitable to investigate diatonic theory on its own without reference
to an underlying chromatic.28
28In a slightly different context, John Clough makes the following points about leaving out chromaticinformation in Clough, “Diatonic Interval Sets and Transformational Structures,” 1979, pages 466-467.
[…] we will be concerned with the effects of suppression, in diatonic contexts, of preciselythat information which is conveyed by clef and key signature. […] It is as though thepitch information […] were given in the form of a succession of scale degrees where the
Musical Interlude II: Règle De L’Octave 123
I propose the following exercise: given a diatonic scale system, one may create a
corresponding grand staff system on which the tonic pitch is declared by fiat, in order to
explore the voicing, doubling, and voice-leading properties of the system. By omitting
information about how the scale is embedded in an equal temperament, it is possible to
see what features of a 9- or 11-PC scale (e.g.) will hold among different temperaments.
In so doing, one is assuming that the embedding complies with Straus conditions 1,
2, and 4 (i.e. the consonance/dissonance, scale-degree, and voice-leading conditions).
e purpose of this exercise is to explore the characteristic voice-leading possibilities
of a system that may let it comply with the third Straus condition (the embellishment
condition). I show how this approach applies in four examples, each of which is based
on the diatonic theory established in the previous section, and each of which has a line
in the soprano, the bass, or both which traverses a modulus (an octave in three of the
four cases), providing a règle de l ’octave of sorts for each example.29
T Diatonic Staff
When you first learned diatonic harmony and voice leading, the preliminary examples
and exercises were most likely written in C major. is is such a prevalent practice that
many students habitually give explanations in C major or a combination of C major and
abstract terms. e following is typical: “you use IV before a V if you have a C in the
designation scale degree 1 is arbitrarily assigned to the first note without any connotationof tonality or scale type.[…] musical constructs may be usefully analyzed and compared on the basis of less thancomplete information. I argue further […] that certain pitch-structural features can bemore clearly observed when intervallic information is distilled in a particular way. I believeit is a useful form of inquiry to ask: Exactly what things are revealed through study of alimited set of pitch-interval characteristics?
29A règle de l ’octave, first devised by Campion, is a harmonized ascending and descending bass scalethrough an entire octave. e harmonies used for a given scale degree and direction of bass motion areconsidered normative for the purposes of learning thoroughbass. e examples in this interlude extendto both soprano and bass octave traversals (or more technically, modulus traversals), and I am concernedmore with possibility than with normativity.
P E T 124
·‚·‚··‚· º œ œ œ ª œ œ œ — º œ œ œ ª œ œ œ — º
º œ œ œ ª œ œ œ — º œ œ œ ª œ œ œ — º1
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9
1
1 2
2 3
3 4
4
5
5 6
6 7
7 8
8 9
9 1
1
·‚··‚· º œ œ œ ª œ — º œ œ œ ª œ — º
º œ œ œ ª œ — º œ œ œ ª œ — º
Figure 2.6: A 7-PC scale on diatonic staves.
soprano and you want it to go to B, but you can use a ii6 if you want a D in the soprano
instead. e rest of the voices do the same thing as they do in the IV-V progression;
you just make it a D instead of a C.” is kind of thinking can be systematized by
introducing a new kind of grand staff, in which the tonic and dominant pitches are used
as clefs, tonic indicated by open diamond noteheads and dominant by filled diamond
noteheads, as in example 2.6 (the leading tone gets a triangle notehead). I call these
diatonic staves: there is deliberately no way to notate chromatic information on them.
is notation does not represent any specific pitches, PCs, temperaments, or even
contours explicitly. It is a way of abstracting the relevant functional aspects of a har-
monic and contrapuntal system that might apply in a number of contexts. Movement
up and down the scale on the diatonic staff corresponds with movement clockwise and
counterclockwise on a scale cycle.30 Any system which complies with the 7-PC system
in figure 2.5 can be realized on the diatonic staff, as they operate at the same level of
abstraction prior to chromatic embedding. Figure 2.7 contains examples of chromati-
cally embedded diatonic systems that would apply. For all of them, d = 7, the diatonic
generator interval gd = 4 and S → H → G→ S =< 2,2,2 >, yielding an ascensive T
system.31
30e technical relationship is an epimorphism, in which two sets are related by a surjective (a many-to-one and onto) function. e relationship between vertical direction on a diatonic staff and clock-wise/counterclockwise direction around a scale cycle is the same kind as that between pitch and pitch-class, or between all integers and integers mod n.
31I remind the reader that ascensivemeans that there is a G→ S order operator, as opposed to descensive,which indicates there is a G→ S operator. e chromatic generator for a system is labeled gc .
Musical Interlude II: Règle De L’Octave 125
S 0 2 4 5 7 9 11 0c = 12, gc = 7 H 2 5 9 0 4 7 11 2
G 5 0 7 2 9 4 11 5
S 0 2 4 11 1 3 5 0c = 12, gc = 1 H 2 11 3 0 4 1 5 2
G 11 0 1 2 3 4 5 11
S 0 3 6 5 8 11 1 0c = 13, gc = 8 H 3 5 11 0 6 8 1 3
G 5 0 8 3 11 6 1 5
S 0 4 8 5 9 13 3 0c = 14, gc = 9 H 4 5 13 0 8 9 3 4
G 5 0 9 4 13 8 3 5
S 0 10 3 12 5 15 8 0c = 17, gc = 5 H 10 12 15 0 3 5 8 10
G 12 0 5 10 15 3 8 12
S 0 3 6 8 11 14 17 0c = 19, gc = 11 H 3 8 14 0 6 11 17 3
G 8 0 11 3 14 6 17 8
S 0 2 4 11 13 15 17 0c = 24, gc = 13 H 2 11 15 0 4 13 17 2
G 11 0 13 2 15 4 17 11
Figure 2.7: Various systems modeled by the 7-PC diatonic staff.
e first is, of course, the traditional diatonic. e second is its M7 dual, and the
c = 19 system is its 19 cognate. With respect to PC and PC-interval ordering, the
systems are all isomorphic with one another. Each follows the intervallic pattern:
S a a b a a a b
H c d c d c d c
G e e e e e e f
Because of the epimorphism between movement up/down on the diatonic staves with
movement clockwise/counterclockwise on a scale cycle, and because of the PC/PC-
interval ordering isomorphism among all compliant chromatic embeddings, I can con-
vert any event on the staves to an event in the PC-space corresponding to a given chro-
P E T 126
matic embedding. To the extent that I am able partition the notes on the diatonic staff
into voices, I can also attempt to preserve those voices in a pitch-space realization of
the mapping from diatonic staff to PC-space.32 A scale ordering whose PC ordering
is strictly ascending through one modulus and whose primary and secondary intervals
are small and similar in size (e.g. the first c = 12 or the c = 19 system in figure 2.7) is
contour preserving with respect to a set of events on the diatonic staves. Others, like the
M7 system I explored in section (e Goldberg Canon Under the M5/M7 Transform,
page 59), as well as the c = 13 and the c = 24 systems in figure 2.7, which have long
stretches of small strictly ascending primary intervals separated by negative secondary
intervals or secondary intervals that are much different in size from the primary, are
partially contour preserving. Finally, systems like the c = 17 system in figure 2.7, the
primary and secondary intervals of which are large, does not preserve contour at all.33
I shall save discussion of whether or not contour preservation is required in order
to create prolongational systems for later. My aim in this interlude is to illustrate two
ideas using only contour-preserving systems. e first is to show how the voice-leading
properties of a given diatonic system of size d can be clearly and definitively modeled
on diatonic staves, and the second is show how a prolongational structure on a diatonic
staff may be converted to an isomorphic structure in a given equal temperament. In the
process I also show how one might extend the PC-space of a chromatically embedded
diatonic system by means of mixture and tonicization.
How can I apply the diatonic-staff construct to sacles with more than 7 PCs? e
traditional 5-line staff is designed so that the top line and bottom space represent the
same PC letter-name. e seven letter names correspond with the seven PCs of a di-
atonic scale, so a staff for a 9-PC scale would need six lines in order to retain the top-32Recall that voice partitioning is the primary purpose of Straus’s fourth condition, and that for the
purposes of this exercise we are hypothesizing that a given system complies with this condition.33However, it is interesting that this system’s H ordering is in fact strictly ascending through one
modulus.
Musical Interlude II: Règle De L’Octave 127
·‚·‚··‚· º œ œ œ ª œ œ œ — º œ œ œ ª œ œ œ — º
º œ œ œ ª œ œ œ — º œ œ œ ª œ œ œ — º1
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9
1
1 2
2 3
3 4
4
5
5 6
6 7
7 8
8 9
9 1
1
·‚··‚· º œ œ œ ª œ — º œ œ œ ª œ — º
º œ œ œ ª œ — º œ œ œ ª œ — º
Figure 2.8: A 9-PC scale on 6-line diatonic staves.
line/bottom-space feature. An interesting feature of the grand staff is its mirror symme-
try with respect to the C-lines/spaces: flipping a grand staff around the horizontal axis
while retaining treble clef on top and bass clef on bottom maps all Cs to other Cs; mid-
dle C remains the same. I can use that to my advantage in making staves for other scale
cardinalities, by declaring the “middle-C” ledger line in both the top and bottom staff
to be tonic. Figure 2.8 illustrates such a pair of 6-line diatonic staves for a 9-PC scale.
Note that the tonic pitches (open diamond noteheads) on the two staves are where the
C-spaces would be with respect to the outermost lines of the grand staff. I have markedb5 as the dominant (filled diamond noteheads), following the template in figure 2.5 (page
122), which shows that gd= 4 in the 9-PC diatonic system.
With a scale in place, a set of voice-leading principles presents itself in figure 2.9.
Because the harmonic norm is tetrachordal, the corresponding contrapuntal norm re-
quires five voices, and a normative voice leading for root-position tetrachords can be
described in terms of how the upper voices are forced to move to support different root
motions in the bass. With tertian harmony, we expect root motion by 3rd to provide the
most parsimonious voice leading, with movement by step in one upper voice (illustrated
with open noteheads) and common tones in the other three.34 Expanding root motions
by successive thirds causes more voices to move: root motion by a 5th (i.e. two thirds)
or 6th requires stepwise movement in two voices, root motion by 7th or 4th moves three
voices, and motion by 2nd or 9th requires that all voices move.34Note that the complement of the 3rd in this system is the 8th rather than the 6th.
P E T 128
·‚·‚··‚·œœœœ œœœ œœœœ œœœ œœœœœ œœ œ œœ œ
3rds & 8ths
œœœœ œœ œœœœ œœ œœœœœ œœ œ œœ œ
5ths & 6ths
œœœœ œ œœœœ œ œœœœœ œœ œ œœ œ
4ths & 7thsORœœœœ ˙ œœœœ ˙ œœœœ
œ œœ œ œœ œœœœœ ˙ œœœœ ˙ œœœœœ œœ œ œœ œ
2nds & 9ths
·‚·‚··‚·ºœªœ —œªœ œœœœ —œœœ
ºœœœ œ—œª œºœª ºœœœ —œªœ œœœœ —œªœ ºœªœº ª œ œ œ ª º œ ª œ ª ºI V II vii* VI V I iv V II V I
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œœœœœœœœ
œœœœ œ œœœœ œœœ œœœ œœœ œœ œœœ œœœœœœ œœœœ
˙ ˙ œ œ œ œ œ œ œ œ œ ˙
º ª œ œ œ ª º œ ª œ ª º
10 9 8 7 6 5 4 3 2 1
15 6 7
8 9 16 7 8 9 1
·‚·‚··‚·œ œ œœœœ œœœ œœœ˙ ˙ ˙
œ œ œœœœ œœœ e œœœ˙ ˙ ˙
·‚·‚··‚·
œ œ œ jœ œ œ œ œ œœœœ œœœ œ œ œœœœœœ œœœ
œ ˙ œ œ Jœ œ œ œ œ ˙˙ œ8 8 8
œ œœ œjœœœœ œœœ œœœ œœœ œœœ˙ ˙ ˙œ œ
œ œ œœœœ œœœ œœœ˙ ˙ ˙
e œ œœœœ œœœ œœœ˙ ˙ ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Figure 2.9: Normative voice leading for the 9-PC diatonic.
ere is much to notice even in just this little voice-leading exploration, but I would
point to three things. First, root motion by 2nd requires all four voices to move, and
following traditional voice-leading practices I have made them move in contrary motion
to the bass. is requires that three voices move by step and one by third, and is the
most parsimonious voice-leading that is not simply all five voices moving in tandem.
Second, the most parsimonious voice leading for root motion by 4th or 7th is to retain
one common tone and have all other voices move by step; in some cases, however, it can
be useful to allow all of the upper voices move in the opposite direction from how they
would normally, which requires two voices to move by step and two to move by 3rd. is
principle applies to the more parsimonious root motions as well, but usually because
of some constraint on the soprano voice (of which there are many in prolongational
systems). ird, because the diatonic generator gd= 4 (i.e. the diatonic 5th), rootmotion
by 5th is the prototype for the primary bass arpeggiation from tonic to dominant and
back. e characteristic voice leading for this progression involves two lower neighbors,
with soprano moving from tonic to leading tone and back in a b1−b9−b1 line, and another
upper voice moving b3−b2−b3. But what happens if I want, say, a b3−b2−b1 line in the soprano?
In that case I can borrow the initial b3−b2 from the other voice and move them to the
soprano, allowing the soprano to make a full b3−b2−b1 descent. See figure 2.10.
Parallel octaves in such a system are conceptually obvious, but is there anything anal-
ogous to parallel fifths? e answer to this is complicated because there are few consis-
tent and agreed-upon explanations for why parallel fifths are prohibited in traditional
Musical Interlude II: Règle De L’Octave 129
·‚·‚··‚·œœœœ œœœ œœœœ œœœ œœœœœ œœ œ œœ œ
3rds & 8ths
œœœœ œœ œœœœ œœ œœœœœ œœ œ œœ œ
5ths & 6ths
œœœœ œ œœœœ œ œœœœœ œœ œ œœ œ
4ths & 7thsORœœœœ ˙ œœœœ ˙ œœœœ
œ œœ œ œœ œœœœœ ˙ œœœœ ˙ œœœœœ œœ œ œœ œ
2nds & 9ths
·‚·‚··‚·ºœªœ —œªœ œœœœ —œœœ
ºœœœ œ—œª œºœª ºœœœ —œªœ œœœœ —œªœ ºœªœº ª œ œ œ ª º œ ª œ ª ºI V II vii* VI V I iv V II V I
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œœœœœœœœ
œœœœ œ œœœœ œœœ œœœ œœœ œœ œœœ œœœœœœ œœœœ
˙ ˙ œ œ œ œ œ œ œ œ œ ˙
º ª œ œ œ ª º œ ª œ ª º
10 9 8 7 6 5 4 3 2 1
15 6 7
8 9 16 7 8 9 1
·‚·‚··‚·œ œ œœœœ œœœ œœœ˙ ˙ ˙
œ œ œœœœ œœœ e œœœ˙ ˙ ˙
·‚·‚··‚·
œ œ œ jœ œ œ œ œ œœœœ œœœ œ œ œœœœœœ œœœ
œ ˙ œ œ Jœ œ œ œ œ ˙˙ œ8 8 8
œ œœ œjœœœœ œœœ œœœ œœœ œœœ˙ ˙ ˙œ œ
œ œ œœœœ œœœ œœœ˙ ˙ ˙
e œ œœœœ œœœ œœœ˙ ˙ ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Figure 2.10: 9-PC diatonic tonic-dominant-tonic progressions.
strict counterpoint. One possible generalization from parallel fifths would be to parallel
generators. As plausible as this sounds in the abstract, it is far too restrictive. For in-
stance, one important feature of traditional voice leading is that with but few exceptions,
in a progression from one root-position triad to another, the upper voices are totally in-
vertible. In the 9-PC example, however, root motion by second (the last example in
figure 2.9) requires parallel generators (i.e. fifths) in some spacings, and prohibiting this
would greatly reduce the flexibility required to manage upper voices in prolongational
structures, which frequently require voice inversion and other kinds of respacing.
A less strict approach to traditional voice leading allows occasional parallel fifths
between upper voices in some circumstances, especially when their occurrence is not
between two essential harmonies. For instance, in a ii6−V8−76−54−3−I progression, a soprano
which moves b6−b5−b4−b3 will necessarily make a parallel fifth with an alto or tenor voice
moving b2−b1−b7−b1, between the ii6 and the cadential 64, which is only a passing harmony.
e offending parallel would be a parallel fourth if the two voices were inverted, which
is one reason why placing the root of a first-inversion triad in the soprano is a rule of
thumb in tonal pedagogy. It may be possible to prohibit parallel generators between
the bass and an upper voice in a 9-PC system, but in my view even this is a mistaken
generalization which will limit flexibility. e following argument shows why.
In traditional tonality, the maximal consonances (major and minor triads) each con-
tain one generator interval; if a generator standing alone (e.g. a “bare fifth” in Mus-
sorgsky) implies a consonance, that consonance can have only one possible root. We
P E T 130
might say that the generator is root definitive. is is not necessarily the case in any
system where the maximal consonance has more than three pitches, because the gen-
erator may occur more than once in each harmony. In a consonant tetrachord in the
9-PC system, for example, a generator alone is not sufficient to define a root because
it could be either the root and 5th or the 3rd and 7th of a harmony. I believe that the
prohibition of parallel fifths – especially stepwise fifths – in tonal music between bass
and upper voice and over two essential harmonies, stems from the fact that the fifths
are root definitive and thereby badly disrupt the collection of simultaneous constraints
on voice independence, parsimonious voice leading, proper spacing and doubling, and
syntactically well-formed harmonic progression.
ese constraints are interdependent, and a disruption in one can trigger a disrup-
tion in the others. For instance, given a stepwise parallel fifth, an attempt to retain
parsimonious voice leading creates a loss of voice independence if they all move in par-
allel with the bass, or forces an improper doubling (a missing third, say) if one or more
voices are led in contrary motion to the perfect fifth. If voice independence is to be re-
tained, then the voice leading requires unparsimonious skips. e reader may remember
difficult past attempts to salvage a harmony exercise marred by an inadvertent parallel;
the reason these errors are difficult to correct is that the confluence of constraints is bro-
ken at that point. If this argument is sound, then the generalized prohibition should
be of parallel generators between the bass and an upper voice over two consonances,
only when those harmonies are in root position (that is, only when the generators do
in fact define the bass as the root). If this seems an arcane rule, I would suggest this
is because parallel fifths and octaves in traditional counterpoint are best thought of as
symptoms rather than causes of a breakdown in the balance of higher-level contrapuntal
constraints. e proper concept here is degrees of freedom: adding voices by increasing
the cardinality of the harmonic norm increases in the dimensionality of the system with
Musical Interlude II: Règle De L’Octave 131
&?
œ
œ
8
I
?œ œbœn
œb œ
7
10
V
? œ œb
œb œ
6
10
VI
œb œbœn œ
œ œn
5 4ˆ ˆ
10
V 65
œn œnœ œn
3
10
I
œbœb6
IV 6
œn œbœ
2
6
V 6
œœœ
1
I
&?
œ
œ
8
I
œ œbœn
œb œ
7
10
V
œ œb
œb œ
6
10
VI
œb œbœn œn
œ œn
5 4ˆ ˆ
10
V 65
œn œn
œ œn
3
10
I
œbœb6
IV 6
œnœbœ
2
6
V 6
œn œœ
1
I
&?
.œ
œ œ œ œ œb œ
.œn œ œ œ œ# œ#
œ# œ# œ# œn œ œ#
œ œn œ œn œ#.œbœn œ œ œ œ œ
.œ# œ œn œn œ œnœn œ œ œ œ œ
œ œb œn œn œ.œ#œb œ œ œ œb œ
.œn œ œ# œb œn œnœ# œb œ œn œ œn
œ œ# œ# œb œn.œœ œ œ œ# œ œ
.œb œ œn œb œ œ œœ œ œ# œ# œb œn
.œ .œ
.œn
&?
œ œn œ# œ œ œn œ# œ œ# œn œ œ œœœœ œœœ œœœ# œœœ#n œœ œœœn œœœ œœœ# œœœn œœœ## œœœnn œœœœ
˙ ˙ œ œ œ# œn œ œ œ œ# œn ˙
10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I V II 86 vii* VI V 865 I iv 8 V 8 II 864 V 86 I
&?
˙
œ œ œ œ œ œ# œ œ
˙œ œ œ# œ œ œ œ#
œn œ# œn œ œ œ# œ œ
œ œ œn œ# œ œ œn˙
œ œ œ œ œ# œn œ# œ
˙# œ œ œ œn œ# œ œœn œ# œ œ# œn œ œ œ
œ œ œ œ œn œ# œ˙nœ œ œ# œn œ œ œ œ#
˙œ œ# œ œ œ œn œ#œ œ œn œ œ œ œ œn
œ œnœ œ œ œ
4
Figure 2.11: e middleground level from figure 1.36 (Goldberg analogue).
corresponding gains in the degrees of freedom for the voice leading of any given voice.
A case could be made that higher-cardinality systems are flexible enough that no par-
allel prohibition rule need be enforced, or that prohibiting types of parallel trichords is
a better analogue. I am currently agnostic about these matters, but nonetheless I try to
(tacitly) follow the rule I derived above in the examples below.
A Goldberg P R
Figure 2.11 brings back figure 1.36 from page 84. is example was meant to illustrate
a structure analogous in form to the middleground structure controlling the first phrase
of the canon at the 2nd from Bach’s Goldberg variations. Recall that the scale system
underlying this example is not strictly ascending: the stretches from b3−b4−b5 and b7−b8−b9ascend by whole tone and then descend by semitone.
Recomposing this passage on diatonic staves requires unwrinkling the scale system,
so to speak, to make it strictly ascending. I might first start by writing all of the chords
defined by the roman numerals in the example as root-position chords on the diatonic
staves, while also attempting to follow the normative voice leading paradigms from
figure 2.9. is will help me to spell the chords correctly, check progression syntax, and
to look for likely faults in the voice leading. e result is the top part of figure 2.12.
P E T 132
·‚·‚··‚·œœœœ œœœ œœœœ œœœ œœœœœ œœ œ œœ œ
3rds & 8ths
œœœœ œœ œœœœ œœ œœœœœ œœ œ œœ œ
5ths & 6ths
œœœœ œ œœœœ œ œœœœœ œœ œ œœ œ
4ths & 7thsORœœœœ ˙ œœœœ ˙ œœœœ
œ œœ œ œœ œœœœœ ˙ œœœœ ˙ œœœœœ œœ œ œœ œ
2nds & 9ths
·‚·‚··‚·ºœªœ —œªœ œœœœ —œœœ
ºœœœ œ—œª œºœª ºœœœ —œªœ œœœœ —œªœ ºœªœº ª œ œ œ ª º œ ª œ ª ºI V II vii* VI V I iv V II V I
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œœœœœœœœ
œœœœ œ œœœœ œœœ œœœ œœœ œœ œœœ œœœœœœ œœœœ
˙ ˙ œ œ œ œ œ œ œ œ œ ˙
º ª œ œ œ ª º œ ª œ ª º
10 9 8 7 6 5 4 3 2 1
15 6 7
8 9 16 7 8 9 1
·‚·‚··‚·œ œ œœœœ œœœ œœœ˙ ˙ ˙
œ œ œœœœ œœœ e œœœ˙ ˙ ˙
·‚·‚··‚·
œ œ œ jœ œ œ œ œ œœœœ œœœ œ œ œœœœœœ œœœ
œ ˙ œ œ Jœ œ œ œ œ ˙˙ œ8 8 8
œ œœ œjœœœœ œœœ œœœ œœœ œœœ˙ ˙ ˙œ œ
œ œ œœœœ œœœ œœœ˙ ˙ ˙
e œ œœœœ œœœ œœœ˙ ˙ ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Figure 2.12: A diatonic-staff transcription of the middleground level in figure 2.11.
Since there are two paradigms for motion by fourth, I must choose which to use in each
instance; here I have made those decisions so that the example begins and ends with the
same registration.
e transcription of the prolongational structure is in the bottom part of figure 2.12,
with scale-degrees in the bass and soprano labeled. I have also included a fundamental
bass that is a copy of the bass from the top of the figure. While this might be somewhat
suspect in Schenkerian circles, I do not think of it as a tool for explaining structure. Like
the normative-voice-leading part of the figure, the fundamental bass is instead a guide
for spelling and progression syntax that is especially useful in this novel tonal environ-
ment. A further reduction of this middleground is given in figure 2.13. It conceives of
the b3 in the soprano line over b1 in the bass as something analogous to a cadential 64 chord,
in that it is spelled the same as the tonic harmony but is an appoggiatura within a wider
dominant prolongation. e Ursatz level at the end of the sequence is almost identical
to the tonic-dominant-tonic progressions in figure 2.10. is is highly consonant with
Musical Interlude II: Règle De L’Octave 133
·‚·‚··‚·œœœœ œœœ œœœœ œœœ œœœœœ œœ œ œœ œ
3rds & 8ths
œœœœ œœ œœœœ œœ œœœœœ œœ œ œœ œ
5ths & 6ths
œœœœ œ œœœœ œ œœœœœ œœ œ œœ œ
4ths & 7thsORœœœœ ˙ œœœœ ˙ œœœœ
œ œœ œ œœ œœœœœ ˙ œœœœ ˙ œœœœœ œœ œ œœ œ
2nds & 9ths
·‚·‚··‚·ºœªœ —œªœ œœœœ —œœœ
ºœœœ œ—œª œºœª ºœœœ —œªœ œœœœ —œªœ ºœªœº ª œ œ œ ª º œ ª œ ª ºI V II vii* VI V I iv V II V I
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œœœœœœœœ
œœœœ œ œœœœ œœœ œœœ œœœ œœ œœœ œœœœœœ œœœœ
˙ ˙ œ œ œ œ œ œ œ œ œ ˙
º ª œ œ œ ª º œ ª œ ª º
10 9 8 7 6 5 4 3 2 1
15 6 7
8 9 16 7 8 9 1
·‚·‚··‚·œ œ œœœœ œœœ œœœ˙ ˙ ˙
œ œ œœœœ œœœ e œœœ˙ ˙ ˙
·‚·‚··‚·
œ œ œ jœ œ œ œ œ œœœœ œœœ œ œ œœœœœœ œœœ
œ ˙ œ œ Jœ œ œ œ œ ˙˙ œ8 8 8
œ œœ œjœœœœ œœœ œœœ œœœ œœœ˙ ˙ ˙œ œ
œ œ œœœœ œœœ œœœ˙ ˙ ˙
ORe œ œœœœ œœœ œœœ˙ ˙ ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Figure 2.13: Reduction sequence for the diatonic middleground in figure 2.12.
the Schenkerian idea that tonal backgrounds are examples of simple, normative voice
leading paradigms that prolong the tonic harmony.
Now that I have this abstract middleground and the corresponding sequence of
background levels, I can use it as the model for embedding the structure in an equal tem-
perament. I mentioned above that I shall only be concerned with contour-preserving
systems in this section, so I need to find a temperament that will work it in a contour-
preserving way. e easiest way is to look at the 9-PC diatonic system from the per-
spective of the G cycle, infer the intervallic structure of the scale, and from there choose
a temperament. Here is a complete listing, using integers that ensure that tonic is 0:
d = 9
S→H →G→ S = < 2,2,2> (descensive T system)
G 6 5 4 3 2 1 0 -1 -2 6
S 5 3 1 -1 6 4 2 0 -2 5
S 0 2 4 6 -1 1 3 5 -2 0
H 2 6 1 5 0 4 -1 3 -2 2
e goal is to find a temperament c and a multiplication operator on pitch-class M
mod c so that the S cycle strictly ascends through one modulus. A simple heuristic
makes this easier than it may sound: I can see that the interval succession for the S cy-
cle follows the pattern <a,a,a,b,a,a,a,b,a>. To make this strictly ascending, I can replace
the primary and secondary intervals with two different intervals, say 2 and 1 respec-
tively: < 2,2,2,1,2,2,2,1,2>, which is analogous to a pattern of whole steps and half
P E T 134
steps. I know that the sum of all the numbers in the interval succession must equal c
to complete a modulus. For this scale, then, 16 should be contour-preserving, with
S =< 0,2,4,6,7,9,11,13,14− 0>. I can infer that the M mod 16 operator must be 9,
because b5 is −1 in the listing and 7 in the mod 16 scale cycle, and −1× 9≡ 7 mod 16.
Or I could simply write out the H and G cycles by reordering the mod 16 scale to arrive
at the same result. Here is the transformed listing:
d = 9; c = 16; gc = 7
S→H →G→ S = < 2,2,2> (descensive T system)
G 6 13 4 11 2 9 0 7 14 6
S 13 11 9 7 6 4 2 0 14 13
S 0 2 4 6 7 9 11 13 14 0
H 2 6 9 13 0 4 7 11 14 2
I will discuss the principles involved with choosing scale intervals more systemati-
cally in the next section, but I should point out now that letting the primary and sec-
ondary intervals of an S cycle equal 2 and 1 always yields a contour-preserving scale
within the resulting c cardinality. Furthermore, the G → (S⊻S) operator always gives
the number of times a secondary interval occurs in the S cycle, so there is a simple
formula to derive c from d when the primary and secondary intervals are 2 and 1:
c = 2d − (G → (S⊻S)). is formula also shows that with primary intervals 2 and
1, d is always just over half of c , which has important implications for modulation to
other harmonic regions. Finally, when G → (S⊻S) = 2, then c is always a multiple of
four when d is odd. Since prolongation scales are possible for all odd numbers greater
than or equal to 7, every c that is a multiple of 4 greater than or equal to 12 can be used
for embedding contour-preserving odd-cardinality diatonic scales.
To continue with the present investigation, I need to make a notation for 16.
In the first musical interlude I introduced the notation created by Easley Blackwood,
which uses the accidentals s and f to raise and lower pitch, respectively. Figure 2.14 is
Musical Interlude II: Règle De L’Octave 135
& œn œs œn œs œf œn œn œs œf œn œs œf œn œn œs œf œn0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0
& œn œn œf œn œs œn œf œn œs œn0¢ 150¢ 300¢ 450¢ 525¢ 675¢ 825¢ 975¢ 1050¢ 1200¢
1 2 3 4 5 6 7 8 9 10
&?
ºœªœffs —œªœssf œœœœnn —œœœsf ºœœœn œ—œªssfœºœªsfnf ºœœœnn —œªœssf œœœœnn —œªœssf
ºœªœfnfs
ºn ªs œn œf œn ªs ºn œn ªs œn ªs ºnV II vii* VI V I iv V II V II
&??
œ œs œn œf œ œs œn œf œ œœœœffs œœœsf œœœn œœœs œœf œœœsf œœœsfœœœœn
œœœœssnsœœœœnn œœœssf œœœsffœ
˙ ˙s œ œf œ œs œ œ œf œn œs ˙
œn œs œn œf œn œs œn œn œs œn œs œn
&?
œ œs œs jœf œn œs œ œ œœœœffs œœœsf œn œ œœœsf œœœfss œœœsff˙ œs Jœ œ œf œn œs ˙˙ ˙s œ œn
8 8 8
œ œœs œjœf
˙ ˙s ˙œ œœœœffs œœœfs œœœfs œœœsfs œœœffs
&?
œ œs œœf œf œfœœsf œœs œœsf˙ ˙s ˙OR ef œ œœ œs œfœœfs œœfs œœsf˙ ˙s ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Chromatic
Diatonic
Figure 2.14: 16 chromatic and 9-PC diatonic scales..
one of many ways one might notate a 16 scale using Gn as tonic using Blackwood’s
accidentals on regular 5-line staves. e full chromatic is listed on the top with slurs in-
dicating the diatonic scale; the diatonic is on the bottom with slurs indicating the tonic
tetrachord. It is important to note that in most n- chromatic scales, the question of
how to spell each PC is handled in a rather ad hoc fashion, although I do try to make it
look as much as possible as it sounds. It takes some experience with this notation not to
see A s and B f as enharmonic. is particular notational strategy also implies an upper
limit for chromatic cardinality: if there are seven note names and three possible acci-
dentals, then 21 is the largest chromatic cardinality it can handle without using s s
and f f or some combination of traditional and novel accidentals; both are cumbersome.
e exact tuning in ¢ for the diatonic is provided at the bottom of figure 2.14 for
reference. Because 16 is divisible by 4, a 16 scale’s IC4 is identical with 12’s
IC3, and the set-class [0,4,8,12] sounds exactly the same as a fully diminished 7th chord
in 12. In a contour-preserving system, a diatonic scale is maximally even with re-
spect to the chromatic and a maximal consonance is maximally even with respect to
the diatonic; this makes maximal consonances second-order maximally even sets with
respect to the chromatic. e H cycle of the present diatonic is an RI chain that alter-
nates pitch-class intervals 3 and 4, making the two primary consonances in the system
the set-classes [0,4,7,11] and [0,3,7,10], which we might call “large” and “small” re-
spectively, due to the difference in intervallic span from the root to the highest chord
P E T 136
& œn œs œn œs œf œn œn œs œf œn œs œf œn œn œs œf œn0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0
& œn œn œf œn œs œn œf œn œs œn0¢ 150¢ 300¢ 450¢ 525¢ 675¢ 825¢ 975¢ 1050¢ 1200¢
1 2 3 4 5 6 7 8 9 10
&?
ºœªœffs —œªœssf œœœœnn —œœœsf ºœœœn œ—œªssfœºœªsfnf ºœœœnn —œªœssf œœœœnn —œªœssf
ºœªœfnfs
ºn ªs œn œf œn ªs ºn œn ªs œn ªs ºnV II vii* VI V I iv V II V II
&??
œ œs œn œf œ œs œn œf œ œœœœffs œœœsf œœœn œœœs œœf œœœsf œœœsfœœœœn
œœœœssnsœœœœnn œœœssf œœœsffœ
˙ ˙s œ œf œ œs œ œ œf œn œs ˙
ºn ªs œn œf œn ªs ºn œn ªs œn ªs ºn
&?
œ œs œs jœf œn œs œ œ œœœœffs œœœsf œn œ œœœsf œœœfss œœœsff˙ œs Jœ œ œf œn œs ˙˙ ˙s œ œn
8 8 8
œ œœs œjœf
˙ ˙s ˙œ œœœœffs œœœfs œœœfs œœœsfs œœœffs
&?
œ œs œœf œf œfœœsf œœs œœsf˙ ˙s ˙OR ef œ œœ œs œfœœfs œœfs œœsf˙ ˙s ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Chromatic
Diatonic
Figure 2.15: 16 transcription of the diatonic-staff middleground structure in figure2.12.
member, namely and 11 and 10. Since they both almost divide the octave evenly, they
sound quite similar to fully diminished 7th chords. e first and fourth Straus conditions
require a strict distinction separating consonant and dissonant intervals from harmonic
and stepwise intervals. Of 16’s 8 ICs, the ones which appear in large and small tetra-
chords are 3, 4, 5, 6, and 7. 1 and 2 are stepwise intervals, and 8 is a dissonance that
only appears in the odd harmonies containing both b9 (the leading tone) and b4. Figures
2.15 and 2.16 are 16 transcriptions of the corresponding diatonic-staff structures in
figures 2.12 and 2.13. e reader is invited to compare these structures both with their
diatonic-staff counterparts and with the original contour-nonpreserving middleground
in figure 2.11.
M
Matthew Brown has made a convincing case that Schenker conceived his tonal universe
in chromatic terms.35 Instead of presupposing chromaticism in tonal music to be em-35Brown, “e Diatonic and Chromatic in Schenker’s ‘eory of Harmonic Relations,”’ 1986
Musical Interlude II: Règle De L’Octave 137
& œn œs œn œs œf œn œn œs œf œn œs œf œn œn œs œf œn0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0
& œn œn œf œn œs œn œf œn œs œn0¢ 150¢ 300¢ 450¢ 525¢ 675¢ 825¢ 975¢ 1050¢ 1200¢
1 2 3 4 5 6 7 8 9 10
&?
ºœªœffs —œªœssf œœœœnn —œœœsf ºœœœn œ—œªssfœºœªsfnf ºœœœnn —œªœssf œœœœnn —œªœssf
ºœªœfnfs
ºn ªs œn œf œn ªs ºn œn ªs œn ªs ºnV II vii* VI V I iv V II V II
&??
œ œs œn œf œ œs œn œf œ œœœœffs œœœsf œœœn œœœs œœf œœœsf œœœsfœœœœn
œœœœssnsœœœœnn œœœssf œœœsffœ
˙ ˙s œ œf œ œs œ œ œf œn œs ˙
œn œs œn œf œn œs œn œn œs œn œs œn
&?
œ œs œs jœf œn œs œ œ œœœœffs œœœsf œn œ œœœsf œœœfss œœœsff˙ œs Jœ œ œf œn œs ˙˙ ˙s œ œn
8 8 8
œ œœs œjœf
˙ ˙s ˙œ œœœœffs œœœfs œœœfs œœœsfs œœœffs
&?
œ œs œœf œf œfœœsf œœs œœsf˙ ˙s ˙OR ef œ œœ œs œfœœfs œœfs œœsf˙ ˙s ˙
Normative Voice-Leading
Prolongation
Fundamental Bass
Chromatic
Diatonic
Figure 2.16: Reduction sequence for the 16 middleground in figure 2.15.
bellishment of an essentially diatonic background, it is possible instead to use pitches
and indeed entire harmonies from outside the home key even in the Ursatz, by employ-
ing mixture and tonicization in composing out triads. I think Brown overstates his case
when he writes more recently:
… Modes and scales have, of course, played a pivotal role in shaping ournotions of music theory. Over the centuries, music theorists have expendedconsiderable energy discussing a broad spectrum of modes and scales, rang-ing from the so-calledChurchmodes tomajor, minor, and chromatic scales,and even a plethora of exotic scales. ey have primarily been guided by thesimple belief that the behavior of a particular piece is determined by inter-vallic properties of some source scale. In some cases, they have assumedthat harmonic systems derive from scales. We will refer to this particularassumption as ‘e Myth of Scales.’36
Brown is right to point out that tonality in general, and Schenker’s conception of it
in particular, incorporates the PCs of every mode with the same final into one major-
minor system. However, I believe I have shown conclusively that harmonic systems do
in fact derive from scales (or more appropriately, scales and harmonic systems together36Brown, Explaining Tonality, 2005, “Schenker and ‘e Myth of Scales,”’ page 141
P E T 138
comprise a larger framework), and that the intervallic properties of source scales are
compositional determinants. In tonal music, it matters that there are seven distinct
scale degrees, as opposed to say, six or eight. It also matters that there is an intervallic
regularity in the traditional diatonic scale that allows a given scale degree to perform
its function in relation to the rest of the scale degrees. In short, Schenker’s harmonic
framework, in which Stufen are necessary features, relies implicitly on broader features
of the diatonic scale, if not on the exact modal pattern of tones and semitones. What
Brown gets right is that the Schenkerian voice-leading paradigms he appeals to, which
are largely written “in C major,” in fact serve to represent a vast array of chromatic
potentials by means of mixture and tonicization. He misses the logic of the diatonic
paradigms taken as diatonic (that is, ignoring their chromatic instantiation completely).
at is the purpose of my diatonic-staff notation; it is meant to show that there is
a harmonic, contrapuntal, and prolongational logic that is determined in part by the
properties of the diatonic cardinality d . e structures created on diatonic staves are
paradigmatic in the same sense that Schenker’s are, and await further specification in
whichever chromatic milieu they find themselves. e last example never deviates from
the “major” 9-PC collection embedded in 16. e next two examples show how
generalized mixture can work given a purely diatonic paradigm to start with. Each
example employs an 11-PC diatonic, which provides ample opportunities for mixture.
In the previous section I discussed two T systems for d = 11. Both are ascensive,
but for the first G → S = 2 and for the second G → S = 3. is means that the first
resulting scale will contain two instances of its secondary interval, and the second will
have three. Similarly, their H → G operators are 3 and 2, respectively, which, since
they are T systems, imply that gd= 6 for the first and gd= 3 for the second (that is, the
dominant is a diatonic 7th above the tonic in the first and a diatonic 5th above the tonic
Musical Interlude II: Règle De L’Octave 139
in the second). Here are the listings and associated interval-cycle forms; the listings are
given in terms of the generator, adjusted here so that 0 represents tonic:
d = 11; h = 5; gd= 6
S→H →G→ S =< 2,3,2> (ascensive T system)
G -1 0 1 2 3 4 5 6 7 8 9 -1
S 0 2 4 6 8 -1 1 3 5 7 9 0
H 2 6 -1 3 7 0 4 8 1 5 9 2
G a a a a a a a a a a b
S c c c c d c c c c c d
H e f e e f e e f e e f
d = 11; h = 5; gd= 4
S→H →G→ S =< 2,2,3> (ascensive T system)
G -2 -1 0 1 2 3 4 5 6 7 8 -2
S 0 3 6 -2 1 4 7 -1 2 5 8 0
H 3 -2 4 -1 5 0 6 1 7 2 8 3
G a a a a a a a a a a b
S c c d c c c d c c c d
H e f e f e f e f e f e
As before, in order to create contour-preserving scales for each, I can assign 2 and 1
to the primary and secondary intervals respectively, and use c = 2d − (G→ S) to come
up with the moduli for the two systems. For the first, c = 2 · 11− 2 = 20, and for the
second c = 2 ·11−3= 19. While the 20 scale I use has octave equivalence, in order
to experiment a bit with tuning possibilities for the 19 scale I will take advantage
of the fact that 12p
2 ≈ 19p
3. is allows me to use a just perfect 12th, which has a 3:1
frequency ratio, as the interval of equivalence, and to divide it into 19 equally tempered
semitones that are almost exactly the same size (ca. 101.1¢) as the traditional 12
P E T 140
semitones. I label non-octave temperaments with the frequency ratio of the interval of
equivalence: 19{3:1} divides the 3:1 “tritave” into 19 equal intervals, 8{ 1+p
52 :1}
divides the interval derived from the golden ratio (ca. 833.1¢) into 8 equal intervals (of
ca. 104.1¢), and so forth. Otherwise, {2:1} is implied if it is left out of the label.
Applying the proper multiplier to the listing for the 20 scale system (the first of
the two given above) yields:
d = 11; h = 5; c = 20; gd= 6; gc= 11
S→H →G→ S = < 2,3,2> (ascensive T system)
S 0 2 4 6 8 9 11 13 15 17 19 0
H 2 6 9 13 17 0 4 8 11 15 19 2
G 9 0 11 2 13 4 15 6 17 8 19 9
S 2 2 2 2 1 2 2 2 2 2 1
H 4 3 4 4 3 4 4 3 4 4 3
G 11 11 11 11 11 11 11 11 11 11 10
An interesting feature of this scale is the interval succession of its H cycle. An
11-PC T system has pentachordal harmonies, so by inspection this system has three
primary chord types rather than just the usual two. I give the following labels to each Tn
type: [0,4,8,11,15] is “major,” [0,4,7,11,15] is “minor,” and [0,3,7,11,14] is “small.” e
major and minor pentachords are pitch-class inversions of one another, and the small
pentachord is inversionally symmetrical (and fully contained in a smaller interval than
the other two, viz. 14 instead of 15). Because they nearly divide the octave evenly into
five, each of the pentachords has a vaguely traditional pentatonic sound. e odd chords
containing the leading tone and the leftmost PC on the generator chain (that is, 9), are
of Tn type [0,4,7,11,14] and [0,3,7,10,14]. Figure 2.17 gives a Blackwood notation for
20, an 11-PCmajor scale with pentachords, and scale tunings in ¢. Roman numerals
for major and minor pentachords are indicated using the usual majuscule and minuscule.
Musical Interlude II: Règle De L’Octave 141
&??
œn œs œf œ œs œf œn œs œf œs œf œf œnœf œf œn œs œf œs œn œs œf œf œn˙n œf œn œn œf œs ˙n œ œf œs œn œf œnœn ˙nº ª ª º
12 11 10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&??
œn œs œf œ œs œf œn œs œf œs œf œf œnœœnf œœfn œœnn œœs œœf œœs œœn œœsf œœfn œœss œœsn œœss œœfs
œœsf œœsf œœff œœsn œœsf œœfs œœsf œœsf œœns œœsf œœnf œn œœn œf œn˙n œf œn œn œf œs ˙n œ œf œœss œ œ ˙
ºn œf œf œn ºn œs ºn ªn œf œs œs ªn ºnm:i s:v s:x s:iv m:i m:iv* M:I M:VII m:viii M:IX* m:vi M:VII M:I
& œn œs œf œn œf œn œs œf œn œs œf œn œs œf œn œs œf œn œs œf œn0 1 2 3 4 5 6 7 8 9 10 11 13 13 14 15 16 17 18 19 0
&&?
œs œn œf œn œf œf œs œn œs œn œf œsœn œs œn œf œs œn œf œn œf œf œs œn
œn œf œs œn œf œn œf œf œs œn œs œnœf œs œn œs œn œf œs œn œf œn œf œf
œn œf œf œs œn œs œn œf œs œn œf œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
I ii iii iv v VI VII viii IX* x xi* I
minor small
0¢ 120¢ 240¢ 360¢ 480¢ 540¢ 660¢ 780¢ 1200¢1140¢1020¢900¢
Figure 2.17: 20 notation and the embedded 11-PC major scale.
e small pentachords are indicated with underlined minuscule, and the two odd are
given asterisks and upper or lower case depending on the ascending interval order.
One reason generated scales are so useful for tonal structures is their ample means
for position finding, which is gleaning the Stufe value (scale degree or chord built on it)
based on its intervallic context with respect to the rest of the scale.37 I discuss this in
detail below (Properties of Generated Scales and the Straus Conditions on page 170),
but here I need to point out that this position-finding context leads directly to the de-
termination of which consonances in the system may be used as tonics, and therefore
what resources are available for mixture.
Consider the traditional diatonic, the interval vector of which is < 254361>. One
way position finding works is in the rarest intervals. A given tritone, for instance, can
belong to only two diatonic collections and their two complements with respect to those
two diatonic collections are mutually exclusive. Any tritone plus a third PC, then, de-
fines a diatonic collection.38 e tritone in a given diatonic collection has important37Richmond Browne, “Tonal Implications of the Diatonic Set,” 1981 is a classic account of this phe-
nomenon for the traditional diatonic collection.38Browne (1981), page 7.
P E T 142
harmonic functions which are essentially position-finding: a dominant seventh chord
implies exactly one key (hence its ubiquitous use in toniczation), and we like to say that
the PCs of a tritone have “tendency” (the leading tone “wants” to resolve to b1). ose
two PCs occur at the far left and right ends of a diatonic collection’s generator chain,
which means their intervallic contexts within the collection are inversionally related.
e tonic triad of either a major or minor key must be a locus of stability, and it is
suitable that they each exclude PCs in the collection that participate in a tritone (in the
C-major collection, only the C-E-G and A-C-E triads exclude F and B).39 Another
way of saying this is that none of the PCs in a tonic triad should have tendency.40
We can extend the same logic to the 11-PC 20 diatonic collection to find the
pentachords aside from the tonic major that can be used as tonics. e PCs at the
far ends of the generator chain are 9 (b6) and 19 (Ó11, the leading tone), so they form
boundaries within the H cycle: 2 6 9 [13 17 0 4 8 11 15] 19 2. e 7 bracketed PCs
comprise three overlapping pentachords of three qualities: major [0,4,8,11,15], minor
[13,17,0,4,8], and small [17,0,4,8,11], so each could be used as a tonic for another scale.
In figure 2.17 I drew rectangles around these minor and small pentachords as they occur
in the major scale. It is worth noting that for all three of these pentachords, the chord
whose root is a diatonic 7th (the dominant interval) above the pentachord’s root has the
same quality as that pentachord. In keeping with features with the traditional minor
system, I alter the two non-major dominant chords so they are major and contain the
appropriate leading tone. Figure 2.18 shows the resulting “natural minor” and “natural
small” scales. For my experiments with mixture I will potentially be able to use any of39is is implied in Browne (1981), andmade explicit in Carey andClampitt, “Aspects ofWell-Formed
Scales,” 1989, pages 204-205.40is is not an obligatory requirement, but it may explain why the Church modes were eventually
filtered from practice as functional tonality came into its own. Only the Phrygian mode retains an identityseparate from the major-minor system, and this is because the bb2 does not arise as the result of simplemixture or tonicization of a nearby region. Its use in the bII6 chord has the trappings of mixture, though,and Schenker seemed to think of it this way. See Brown, “e Diatonic and Chromatic in Schenker’s‘eory of Harmonic Relations,”’ 1986.
Musical Interlude II: Règle De L’Octave 143
&&?
œs œf œs œn œf œf œs œf œs œn œf œsœf œs œn œf œs œf œs œn œf œf œs œf
œn œf œs œf œs œn œf œf œs œf œs œnœf œs œf œs œn œf œs œf œs œn œf œf
œn œf œf œs œf œs œn œf œs œf œs œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
i II* iii iv* V vi vii viii ix X XI I0¢ 120¢ 240¢ 360¢ 420¢ 1200¢1080¢960¢900¢540¢ 660¢ 780¢
&&?
œn œf œs œn œf œn œn œf œs œn œs œnœf œs œn œs œn œf œs œn œf œn œn œf
œn œs œn œf œs œn œf œn œn œf œs œnœn œn œf œs œn œs œn œf œs œn œf œn
œn œf œn œn œf œs œn œs œn œf œs œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
i ii* III iv v vi vii VIII IX x XI* i0¢ 120¢ 180¢ 300¢ 420¢ 540¢ 660¢ 720¢ 840¢ 960¢ 1080¢ 1200¢
&?
œœœœ## œœœœ### œœœœ## œœœœ œœœœœ# œœœœœ# œœœœœœ# œœœœœœ
œœœœœœ# œœœœœœœ
# œœœœœœœ# œœœœœœœœ
#
œœœœœœœ###œœœœœœœ### œœœœœœœ# œœœœœœœ
# œœœœœœœœœœœœ œœœœœ œœœœœ œœœœœ œœœœ œœœœ œœœ
&?
��
&?
��
Minor
Small
Figure 2.18: 11-PC minor and small scales in 20.
the pentachords arising in the major, minor, and small scales. is does not exhaust
every possibility for mixture, but it provides a good foundation.
e same procedure works for the 19{3:1} system. is is the transformed list-
ing:
d = 9; h = 5; c = 19; gd= 4; gc= 7
S→H →G→ S = < 2,2,3> (ascensive T system)
S 0 2 4 5 7 9 11 12 14 16 18 0
H 2 5 9 12 16 0 4 7 11 14 18 2
G 5 12 0 7 14 2 9 16 4 11 18 5
S 2 2 1 2 2 2 1 2 2 2 1
H 3 4 3 4 3 4 3 4 3 4 3
G 7 7 7 7 7 7 7 7 7 7 6
Now I only have two primary pentachordal Tn types: major [0,4,7,11,14] and minor
[0,3,7,10,14]. e PCs on the extreme ends of the G chain are 5 (b4) and 18 (the
P E T 144
leading tone Ó11). Pentachords containing both of those are of Tn types [0,4,8,11,15],
[0,4,7,11,15], and [0,3,7,11,14]. I can use the two extreme G-chain PCs as boundaries
within the H cycle to find possible tonic pentachords, as I did above for the 20
scale: 2 5 [9 12 16 0 4 7 11 14] 18 2. is time there are eight PCs comprising four
overlapping potentially tonic pentachords: [9,12,16,0,4] (minor), [12,16,0,4,7] (major),
[16,0,4,7,11] (minor), and the major-scale tonic [0,4,7,11,16] (major). Figure 2.19 pro-
vides the major scale and the two possible minor scales based on the two minor penta-
chords. I have called them “high minor” and “low minor” for no other reason than that
the bracketed part of each scale happens to form a traditional natural minor scale in this
temperament, and it occurs in the high end of the high minor and the low end of the
low minor. ese scales are identical except for b6, which also changes the quality of a
few chords.
By analogy, the major scale in figure 2.19 could be called a “low major” due to the
first 8 scale degrees forming a traditional major scale. But what of the mode starting
on this low major’s b8, which we might call “high major”? I have not included it among
the scales for mixture because, like the two minor scales, they would only differ by one
scale degree, b4, and the # b4 of the high major scale is in fact the PC that would be used to
tonicize the dominant, which brings it out of the realm of mixture properly conceived.
I am taking it for granted that the 3:1 interval of equivalence is cognitively sound –
obviously a problematic assumption – though I thinkwith training it is. For the purposes
of this investigation I am only concerned about whether the structural logic works out.
e reader is invited to play through a few of the chords over the entire three-tritave
range given in the scales to hear how registral substitutions change the sonic quality
of a given harmony. ere is a large body of literature that discusses how tuning and
timbre interact; the cognitive status of this system may be different with some timbres
than with others. e details are beyond the scope of this essay, but William Sethares’s
Musical Interlude II: Règle De L’Octave 145
&&?
œ# œb œ# œn œb œb œ# œb œ# œn œb œ#œb œ# œn œb œ# œb œ# œn œb œb œ# œbœn œb œ# œb œ# œn œb œb œ# œb œ# œnœb œ# œb œ# œn œb œ# œb œ# œn œb œb
œn œb œb œ# œb œ# œn œb œ# œb œ# œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
i II* iii iv* V vi vii viii ix X XI I0¢ 120¢ 240¢ 360¢ 420¢ 1200¢1080¢960¢900¢540¢ 660¢ 780¢
&&?
œn œb œ# œn œb œn œn œb œ# œn œ# œnœb œ# œn œ# œn œb œ# œn œb œn œn œbœn œ# œn œb œ# œn œb œn œn œb œ# œnœn œn œb œ# œn œ# œn œb œ# œn œb œn
œn œb œn œn œb œ# œn œ# œn œb œ# œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
i ii* III iv v vi vii VIII IX x XI* i0¢ 120¢ 180¢ 300¢ 420¢ 540¢ 660¢ 720¢ 840¢ 960¢ 1080¢ 1200¢
&?œ œ# œ# œ œ
œ# œ
œ## œ
œ## œ
œœœ#
# œœ#
# œœ#
# œœ
#œœ
œœ
# œœ#
# œœ œ œ# œ# œ œ œ# œ# œ
1 2 3 4 5 6 7 8 9 10 11 12ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
I ii iii IV V vi VII* VIII IX* x xi* I
high minor low minor
&?œ œ# œ œ œ
œ# œ
œ## œ
œœœ
œœ#
# œœ
œœ
œœ
#œœ
œœ
# œœ
œœ œ œ# œ œ œ œ œ œ
1 2 3 4 5 6 7 8 9 10 11 12ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
i II* III IV* v vi* VII viii ix X XI i
&?œ œ# œ œ œ
œ# œ
œœœ
œœ
œœ#
# œœ
œœ
œœ
#œœ
œœ
# œœ
œœ œ œ œ œ œ œ œ œ
1 2 3 4 5 6 7 8 9 10 11 12ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
i ii* III iv v VI VII viii IX* X XI* i
Minor
Small
High Minor
Low Minor
Major
Figure 2.19: e major and two minor 11-PC scales in19{3:1}.
P E T 146
·····
œœœœœ œœœœ œœœœœ œœœœ œœœœœœ œ
œ œ œœ œ
3rds & 10ths œœœœœ œœœ œœœœœ œœœ œœœœœœ œ
œ œ œœ œ
5ths & 8ths œœœœœ œœ œœœœœ œœ œœœœœœ œ
œ œ œœ œ
6ths & 7ths œœœœœ œ œœœœœ œ œœœœœœ œ
œ œ œœ œ
OR4ths & 9ths
œœœœœ˙ œœœœœ ˙ œœœœœ
œ œœ œ œ
œ œœœœœœ
˙ œœœœœ ˙ œœœœœœ
œœ œ œ
œœ
2nds & 11ths
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ jœ œ œ œ œ jœ œ œœ œ œ œ œœœœœœœœ œœœœ
œ œ Jœ œ œ œ œ Jœ œ œ˙ ˙ œ œ ˙º ª º ª œ ª œ ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ
œœœœ œ …œœœ œ œ…œœ œ œœœœ
œœœœ œœœœœ…œœ œ œœœœœ …œœœ œ œœœœœ œ œ œ œ œ œ œ œœ œ jœ œ œ
˙ ˙ ˙œ œ Jœ œ œ Jœ œ œ œº ª º œ e ª œ ª œ º œ ª º
Figure 2.20: Normative voice leading for d = 11.
Tuning, Timbre, Spectrum, Scale41 is a good and fairly comprehensive introduction to the
problems and various solutions.
e two examples I’m going to compose will feature an octave/tritave descent in the
soprano along with an octave/tritave ascent in the bass. Before I embark I must look at
the normative voice leading for 11-PCT systems, which is the same for both the 20
and the 19{3:1} scales (see figure 2.20). is proceeds exactly as it did for the 9-PC
voice leading above, but now the staves have seven lines to accommodate the extra PCs,
and because the harmony is pentachordal, a six-voice texture is the norm. e clef at the
beginning of the line only shows tonic; this is because the two systems under inspection
have different generator intervals. e tonic-dominant interval for the 20 scale is
the diatonic 7th and it is clear from the voice-leading paradigm that root motion by 7th
requires movement of three upper voices. For the 19{3:1} scale the tonic-dominant
interval is the diatonic 5th, the root motion of which requires only two upper voices to
move.
Figure 2.21 incorporates this information into two Ursatz paradigms for each scale,
onewith a b1−Ó11−b1 lower-neighbor figure, and the other with a b3−b2−b1Urlinie. e 20
Ursätze are on the left of the figure, and the 19{3:1} ones are on the right. One
interesting feature of the b3−b2−b1 paradigms is that each can accommodate a passing tone
from the dominant PC in the soprano (which doubles the bass) down to the member
of the tonic triad a third below the dominant – the 5th in the 20 example on the41Sethares (2005).
Musical Interlude II: Règle De L’Octave 147
·····
œœœœœ œœœœ œœœœœ œœœœ œœœœœœ œ
œ œ œœ œ
3rds & 10ths œœœœœ œœœ œœœœœ œœœ œœœœœœ œ
œ œ œœ œ
5ths & 8ths œœœœœ œœ œœœœœ œœ œœœœœœ œ
œ œ œœ œ
6ths & 7ths œœœœœ œ œœœœœ œ œœœœœœ œ
œ œ œœ œ
OR4ths & 9ths
œœœœœ˙ œœœœœ ˙ œœœœœ
œ œœ œ œ
œ œœœœœœ
˙ œœœœœ ˙ œœœœœœ
œœ œ œ
œœ
2nds & 11ths
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ jœ œ œ œ œ jœ œ œœ œ œ œ œœœœœœœœ œœœœ
œ œ Jœ œ œ œ œ Jœ œ œ˙ ˙ œ œ ˙º ª º ª œ ª œ ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ
œœœœ œ …œœœ œ œ…œœ œ œœœœ
œœœœ œœœœœ…œœ œ œœœœœ …œœœ œ œœœœœ œ œ œ œ œ œ œ œœ œ jœ œ œ
˙ ˙ ˙œ œ Jœ œ œ Jœ œ œ œº ª º œ e ª œ ª œ º œ ª º
Figure 2.21: Ursatz paradigms for d = 11. Left: gd = 6. Right: gd = 4.
left and the 3rd in the 19{3:1} on the right. In each case the resulting dominant
hexachord is analogous in function to a V7 chord in traditional tonality.
Using the b3−b2−b1 Ursätze as scaffolding for the full-modulus ascents and descents
will prove useful, since the bass can ascend in parts through its primary arpeggiation
and the soprano can descend over and converge with an alto line containing b3−b2−b1. I
will start with a hypotheses for the 20 version on the diatonic staff, in figure 2.22,
top. I know already that the bass arpeggiation from b1 to b7 cuts the excerpt about in
half, and so I include the intervening pitches of the bass ascent before and after the
primary b7, noting that the passage will end with a dominant-tonic cadence. For this
cadence I allow the bass to ascend into the tenor voice, while prolonging the structural
dominant pitch to the penultimate chord. Now I have to decide which pitch of the
soprano descent to support with the structural dominant, and b7 is the obvious choice
because it is also in about the middle of the run and doubling the bass will give me a lot
of flexibility with handling the inner voices. b2 of the soprano descent is supported by
the leading tone in the bass ascent and the prolonged structural dominant. e b3−b2−b1alto line provides some extra structural support.
Next, I need to figure out how the five pitches between b1 and b7 in the bass ascent
line up with the corresponding four pitches in the soprano descent, and likewise for
the bass and soprano pitches after the structural dominant. e bottom of figure 2.22
is one possible solution. I notice that a chord built on Ó10 also contains b1 and b3, so I
can support the Ó12−Ó11−Ó10 part of the soprano with a voice exchange between the b1−b2−b3
P E T 148
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ jœ œ œ œ œ œœ œ œ œ œ œ œ œ œ œœ œ œ œ œ œjœœ œœœ œ œ œ˙ ˙ œ ˙
º œ œ œ º œ º ª ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ œœœœ œœœœ
œœœœœœœ œœ œœœœ
œœœœœœœœ
œœœœœœœœœ œœœœœ
jœœ
œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙œ Jœ
º œ œ œ º œ º ª œ œ œ ª º
Figure 2.22: Background hypothesis and development in the d = 11, gd = 6 system, tobe embedded in 20.
Musical Interlude II: Règle De L’Octave 149
of the bass and an alto line. I notice also that a chord built on b4 can support b4, b6, b8,and Ó10, which fits nicely with the first and last pitches of the bass b4−b5−b6, and with a
tie from the previous voice exchange, the Ó10−b9−b8 of the soprano. I add b6−b5−b4 over the
bass in an inner voice to support the whole thing with another voice exchange. e first
voice exchange prolongs tonic and leads to pre-dominant function, the second prolongs
a pre-dominant area, which in turn prepares the structural dominant.
To figure out the second half of the excerpt, I know first that I am going to be
prolonging dominant from the structural dominant to the penultimate chord, which
means the b8−b9−Ó10 stretch of the bass ascent either fills in an arpeggiation of the domi-
nant from the root to the leading tone, or it prolongs a pre-dominant stretch within the
dominant. Keeping that in mind, I remember from theUrsatz paradigm that b7−b6−b5 can
occur over dominant-tonic in the bass, using b6 as a passing motion making a dominant
hexachord (analogous in function to V7 in tonal music). I do not have a tonic to land on
immediately in the bass after the structural dominant, but the dominant’s upper neigh-
bor b8 could work as a resolution of the dominant in something resembling a deceptive
cadence.
I also notice that I can continue the middle-voice line from the second voice ex-
change in a stepwise descent giving me a sonority spelled as a tonic pentachord over the
structural dominant, which will function very similarly to a cadential 64 in tonal music,
which then resolves to the dominant pentachord and its deceptive cadence. If I look
to continue that inner voice down until it converges with the leading tone ascending
bass, that might give me some direction for how to use the remaining soprano and bass
pitches. e naïve note-to-note solution works well enough; after the deceptive cadence
I have, from bass to soprano, the following two simultaneities: b9,b2,b4 and Ó10,b1,b3, both
of which work as incomplete pentachords I can fill in at the next level. In fact both of
them are compatible with b6, which as the lower neighbor to b7 would serve as an excel-
P E T 150
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ jœ œ œ œ œ œœ œ œ œ œ œ œ œ œ œœ œ œ œ œ œjœœ œœœ œ œ œ˙ ˙ œ ˙
º œ œ œ º œ º ª ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ œœœœ œœœœ
œœœœœœœ œœ œœœœ
œœœœœœœœ
œœœœœœœœœ œœœœœ
jœœ
œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙œ Jœ
º œ œ œ º œ º ª œ œ œ ª º
Figure 2.23: Foreground in the d = 11, gd = 6 system, to be embedded in 20.
lent pre-dominant preparation for the final dominant. I incorporate this into the final
harmonized example in figure 2.23.
&??
œn œs œf œ œs œf œn œs œf œs œf œf œnœf œf œn œs œf œs œn œs œf œf œn˙n œf œn œn œf œs ˙n œ œf œs œn œf œnœn ˙nº ª ª º
12 11 10 9 8 7 6 5 4 3 2 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
&&??
œn œs œf œ œs œf œn œs œf œs œf œf œnœœnf œœfn œœnn œœs œœf œœs œœn œœsf œœfn œœss œœsn œœss œœfs
œœsf œœsf œœff œœsn œœsf œœfs œœsf œœsf œœns œœsf œœnf œn œœn œf œn˙n œf œn œn œf œs ˙n œ œf œœss œ œ ˙
ºn œf œf œn ºn œs ºn ªn œf œs œs ªn ºnm:i s:v s:x s:iv m:i m:iv* M:I M:VII m:viii M:IX* m:vi M:VII M:I
& œn œs œf œn œf œn œs œf œn œs œf œn œs œf œn œs œf œn œs œf œn0 1 2 3 4 5 6 7 8 9 10 11 13 13 14 15 16 17 18 19 0
&&?
œs œn œf œn œf œf œs œn œs œn œf œsœn œs œn œf œs œn œf œn œf œf œs œn
œn œf œs œn œf œn œf œf œs œn œs œnœf œs œn œs œn œf œs œn œf œn œf œf
œn œf œf œs œn œs œn œf œs œn œf œn1 2 3 4 5 6 7 8 9 10 11 12ˆˆˆˆˆˆˆˆˆˆˆ
I ii iii iv v VI VII viii IX* x xi* I
minor small
0¢ 120¢ 240¢ 360¢ 480¢ 540¢ 660¢ 780¢ 1200¢1140¢1020¢900¢
Figure 2.24: 20 realization of the foreground from figure 2.23.
Now it is time to realize this in 20 with mixture. For the sake of simplicity I
engage a traditional strategy by beginning in minor and ending in major. For each har-
mony I am going to choose a counterpart from one of the three modes (major, minor,
and small), and try to also produce smooth voice leading while avoiding the pitfalls of
mixture, such as prominent augmented seconds. I have to be careful with the begin-
ning of the soprano descent and the end of the bass ascent, by employing something
analogous to descending and ascending melodic minor fragments; the ascending frag-
ment borrows from major, so it leads naturally into a major ending. Voice exchanges
are especially exploitable by exchanging different versions of scale degrees. Figure 2.24
Musical Interlude II: Règle De L’Octave 151
is the finished realization. I have labeled every chord with a roman numeral that gives
its quality, as well as the scale it came from: M for major, m for minor, and s for small.
It is written on three staves for ease of reading.
I now present the 19{3:1} prolongation with less detailed explanation (see figure
2.25). b5 is dominant in this scale, but I still place b7 in the soprano for the structural
dominant as it keeps the number of pitches per run about the same in the bass and
soprano, and provides the opportunity for a b7−b6−b5 over b5−b6−b7 voice exchange with the
bass. I open with the same voice exchange as in the previous example, but now lead the
soprano through a 4th over it (rather than a 3rd as before) to prolong the tonic pentachord.b8 in the soprano over b4 in the bass is an ideal pre-dominant for preparing the structural
dominant. e end proceeds similarly to the previous example. One matter of interest is
that Ó10 in the bass ascent is supported by a chord that behaves very much like a cadential64, but does not consist of the tonic pentachord PCs. e degrees of freedom associated
with a pentachordal norm and a large diatonic cardinality allow for many subtle shades
of function that are not available in traditional tonal music.
e completed diatonic-staff foreground appears in figure 2.26. is example is a
bit less complex harmonically than its 20 counterpart. Some of that may have to do
with the fact that fewer voices move in the dominant-tonic progression. To embellish
the texture I have added suspensions (indicated with slashed noteheads) and a variety
of extra passing tones. e dominant pedal at the end is a nice touch, but it may be too
much dominant considering the dominant voice exchange that just ended. is is one
reason mixture is useful: it can add variety to passages that are otherwise very similar in
function.
I am ready to realize this in 19{3:1} with generous mixture. e primary chords
sound as 9th chords do in 12, so the surface has something of a Stravinskian fla-
vor. Also Stravinskian are the pseudo-cross-relations like the Cn in the bass against the
P E T 152
·····
œœœœœ œœœœ œœœœœ œœœœ œœœœœœ œ
œ œ œœ œ
3rds & 10ths œœœœœ œœœ œœœœœ œœœ œœœœœœ œ
œ œ œœ œ
5ths & 8ths œœœœœ œœ œœœœœ œœ œœœœœœ œ
œ œ œœ œ
6ths & 7ths œœœœœ œ œœœœœ œ œœœœœœ œ
œ œ œœ œ
OR4ths & 9ths
œœœœœ˙ œœœœœ ˙ œœœœœ
œ œœ œ œ
œ œœœœœœ
˙ œœœœœ ˙ œœœœœœ
œœ œ œ
œœ
2nds & 11ths
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ jœ œ œ œ œ jœ œ œœ œ œ œ œœœœœœœœ œœœœ
œ œ Jœ œ œ œ œ Jœ œ œ˙ ˙ œ œ ˙º ª º ª œ ª œ ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ
œœœœ œ …œœœ œ œ…œœ œ œœœœ
œœœœ œœœœœ…œœ œ œœœœœ …œœœ œ œœœœœ œ œ œ œ œ œ œ œœ œ jœ œ œ
˙ ˙ ˙œ œ Jœ œ œ Jœ œ œ œº ª º œ e ª œ ª œ º œ ª º
Figure 2.25: Background hypothesis and development in the d = 11, gd = 4 system, tobe embedded in 19{3:1}.
·····
œœœœœ œœœœ œœœœœ œœœœ œœœœœœ œ
œ œ œœ œ
3rds & 10ths œœœœœ œœœ œœœœœ œœœ œœœœœœ œ
œ œ œœ œ
5ths & 8ths œœœœœ œœ œœœœœ œœ œœœœœœ œ
œ œ œœ œ
6ths & 7ths œœœœœ œ œœœœœ œ œœœœœœ œ
œ œ œœ œ
OR4ths & 9ths
œœœœœ˙ œœœœœ ˙ œœœœœ
œ œœ œ œ
œ œœœœœœ
˙ œœœœœ ˙ œœœœœœ
œœ œ œ
œœ
2nds & 11ths
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚·œ œ œœœœœ œœœœ œœœœ˙ ˙ ˙
œ œ œœœœœ œœœœ e œœœœ˙ ˙ ˙
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œœ œ œœ œœ œœœœ œœœœœ œ œ œ œ œ œ œ œ œ œ œ˙ ˙ œ ˙º ª ª º
·‚·‚··‚··‚·
œ œ œ œ jœ œ œ œ œ jœ œ œœ œ œ œ œœœœœœœœ œœœœ
œ œ Jœ œ œ œ œ Jœ œ œ˙ ˙ œ œ ˙º ª º ª œ ª œ ª º
·‚·‚··‚··‚·
œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ
œœœœ œ …œœœ œ œ…œœ œ œœœœ
œœœœ œœœœœ…œœ œ œœœœœ …œœœ œ œœœœœ œ œ œ œ œ œ œ œœ œ jœ œ œ
˙ ˙ ˙œ œ Jœ œ œ Jœ œ œ œº ª º œ e ª œ ª œ º œ ª º
Figure 2.26: Foreground in the d = 11, gd = 4 system, to be embedded in 19{3:1}.
Musical Interlude II: Règle De L’Octave 153
&??
œ# œ œ œ# œ œ œ# œ# œ œ# œ# œœ œ# œœ# œœ# œœœœ## œœœœ###œ œ œn œ œ œ# œ# œ œ œ# œ# œ˙ ˙ œ ˙º ª ª º
&??
œ# œ œ œ# jœ œ œ# œ# œ jœ# œ# œœ œ# œ œ# œœœ# œœœœ## œœœœ###œ œn Jœn œ# œ# œ œ Jœ# œ# œ
˙ ˙ œœ œ ˙º ª º ª œ ª œ ª º
&??
œ# œ œ œ# œ œ œ# œ# œ œ# œ œ# œœœœœ## œœœœ### œœœœ## œ …œœn#œ œ œ…œ# œ# œœœ## œœœ### œœn œ… œ# œœœ# …œ# œ# œœ##
œ œ œ œ# œ œœ œœa# œœ# œœn œœ# œœ# œœ œœ#˙ ˙ œ œ œ#
˙œ œn Jœn œ# œ# Jœn œ œ œ œjœ#
º ª º œ e ª œ ª œ º œ ª ºm:i m:v m:i hm:II* (hm:IV*) m:v M:IX* M:V lm:iv M:I M:VIII M:V M:I
&?
��
��
��
��
��
��
��
��
Figure 2.27: 19{3:1} realization of the foreground from figure 2.26.
C# in the soprano in the third chord, which arise from the non-octave modulus. e
dominant voice exchange begins minor and ends major, but a deliberate Fn in the bass,
an augmented 2nd lower than the preceding G#, is an assertive minor pre-dominant in
preparation for the major dominant pedal and final cadence.
is example shows how rather traditional-sounding harmony and voice leading
could work in a tritave universe. In my opinion it is more successful than the 10-PC
Bohlen-Pierce scale, for the simple reason that the 10-PC diatonic cardinality has too
few coprimes to create G, H , and S orderings, and therefore its voice leading possibilities
are already stunted. e 19{3:1} system has an added advantage, which is that it is
playable on standard 12 instruments. One final observation about this example
serves as a segue to the next set: in the structural-dominant voice exchange, the passing
chord written with an An at the top of the bass staff is a IX* chord from minor, with an
extra third added above the 9th. With an A#, it is an applied dominant of V.
T
Alongside mixture, the other primary way to expand the PC world of a tonal system
from diatonic to chromatic is by tonicizing Stufen other than the tonic. To illustrate how
this might work, I construct our first Q system, using d = 13 and the order permutation
P E T 154
triple < 3,2,2 >. Since G→ S = 2, the c = 2d − 2 formula for the simplest contour-
preserving embedding applies, yielding c = 24, and a scale with two runs of the primary
scale interval separated by two occurrences of the secondary scale interval. As with the
d = 9; c = 16 system above, this is a descensive system, and so we expect the secondary
scale interval to occur below the leading tone in the scale ordering. Putting this all
together yields the following orderings:
d = 13; h = 4; c = 24; gd= 6; gc= 11
S→H →G→ S = < 3,2,2> (descensive Q system)
S 0 2 4 6 8 10 11 13 15 17 19 21 22 0
H 4 10 15 21 2 8 13 19 0 6 11 17 22 4
G 10 21 8 19 6 17 4 15 2 13 0 11 22 10
Figure 2.28 displays the 13 Stufen for this system. Since this is a system of ordi-
nary quarter tones, I drop the Blackwood notation in favor of traditional quarter-tone
notation. A Q presents a notational difficulty for diatonic staff notation. For T sys-
tems, “stacked third” notation corresponds intuitively with the notational concerns of
the common-practice staff, as it places all of the notes belonging to a tightly-spaced
root-position chord on lines or on spaces. But this is not so for Q systems, since stacked
fourths alternate lines and spaces and are a little harder to see by eye immediately. My
solution for this is to widen the spaces so that they accommodate two diatonic pitches,
one directly above a line and one directly below. is way, notes belonging to tightly-
spaced primary chords all sit on a line or in one of the space positions.42 is nota-
tion also reflects some voice-leading concerns as I show below. Figure 2.28 provides
all 13 Stufen with a Gn tonic. ere are two qualities of maximal consonance: major
[0,6,11,17], and minor [0,5,11,16]. IC 11 is the generator. As with the h = 4, d = 9,42I have also made ledger lines a bit wider to make them easier to place visually.
Musical Interlude II: Règle De L’Octave 155
·‚··‚·
œœœœ œœœœœœœ
œœœœœœœ
œ œ œ œ œ
4ths & 11ths
œœœœ œœ œœœœœœ
œœœœœ œ
œ œ œœ œ
7ths & 8ths
œœœœ œ œœœœœ˙
œœœœœ œ œ œ œ
5ths & 10ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œ œ œ œ
2nds & 14ths
œœœœ ˙˙
œœœœ˙˙
œœœœœ œ œ œ œ
3rds & 12ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œœ œ œ
œ œ
6ths & 9thsORœœœœ ˙
˙œœœœ
˙˙
œœœœœ œ
œ œ œœ œ
&?
·‚··‚·
·œªœœœœœ
œœœœœ—œª
œ·œœœœœœ
ªœ—œœœ·œ
œœœœœœœ—
œªœ·œœœœ
—œœœ·œªœ
· œ œ œ œ œ ª œ œ œ œ œ — ·
·œªœnbìñ œœœœnbñì œœœœnnññ œ—œªñb#ì œ·œœnìnñ œœœœñnbñ ªœ—œñìn# œœ·œñìbn œœœœññnb œœœ—#ìnn œªœ·ñnñb œœœœñbìn —œœœ#nñn ·œªœnbìñºn œb œn œb œn œn ªñ œì œñ œì œñ œñ —# ºnI II III iv v vi VII VIII IX X* xi xii xiii* I
I II III iv v vi VII VIII IX X* xi xii xiii* I
&?
·‚··‚·
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙n ˙# ˙nœœœbìñ œœœnìñ œœœbìñ˙n ˙ñ ˙n
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙b ˙n ˙nœœœñnì œœœñ#ì œœœbìñ˙n ˙ñ ˙n
Figure 2.28: Stufen for a descensive Q system with h = 4, d = 13, c = 24.
c = 16 system above, since the maximal consonances are tetrachords, they almost divide
the octave into traditional diminished 7th chords.43
Normative voice leading for this system is given in figure 2.29. is is a Q system
with tetrachordal maximal consonances, so normative counterpoint contains five voices
and intervals increase by fourths from left to right in the figure instead of by thirds
as in T systems. As expected, root motion by fourth is the most parsimonious, and
root motions by all diatonic intervals that are not members of a maximal consonance
require motion in all voices. e diatonic generator is the 7th (diatonic IC 6), so root
motion by 7th and 8th constitutes the tonic-dominant relationship and the primary bass
arpeggiation prototype and holds two common tones through the progression.
is leads to an interesting feature of any Q system: supposing we want to use the
tonic-dominant-tonic paradigm as the basis for Ursätze, a descending line from b4 to b1must use b3 as a passing tone, but not b2, because b3 occurs in the dominant tetrachord andb2 is not consonant with either the dominant PC (Cñ in this case) or the leading tone.
43is system is in fact quite similar to one developed by Ivan Wyschnegradsky. See Skinner, “Toward aQuarter-Tone Syntax,” 2007, pages 144-232 for a thorough investigation of Wyschnegradsky’s treatment.My approach differs from Wyschnegradsky’s in a number of key details which result from my broadermethodology.
P E T 156
·‚··‚·
œœœœ œœœœœœœ
œœœœœœœ
œ œ œ œ œ
4ths & 11ths
œœœœ œœ œœœœœœ
œœœœœ œ
œ œ œœ œ
7ths & 8ths
œœœœ œ œœœœœ˙
œœœœœ œ œ œ œ
5ths & 10ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œ œ œ œ
2nds & 14ths
œœœœ ˙˙
œœœœ˙˙
œœœœœ œ œ œ œ
3rds & 12ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œœ œ œ
œ œ
6ths & 9thsORœœœœ ˙
˙œœœœ
˙˙
œœœœœ œ
œ œ œœ œ
&?
·‚··‚·
·œªœœœœœ
œœœœœ—œª
œ·œœœœœœ
ªœ—œœœ·œ
œœœœœœœ—
œªœ·œœœœ
—œœœ·œªœ
· œ œ œ œ œ ª œ œ œ œ œ — ·
·œªœnbìñ œœœœnbñì œœœœnnññ œ—œªñb#ì œ·œœnìnñ œœœœñnbñ ªœ—œñìn# œœ·œñìbn œœœœññnb œœœ—#ìnn œªœ·ñnñb œœœœñbìn —œœœ#nñn ·œªœnbìñºn œb œn œb œn œn ªñ œì œñ œì œñ œñ —# ºnI II III iv v vi VII VIII IX X* xi xii xiii* I
I II III iv v vi VII VIII IX X* xi xii xiii* I
&?
·‚··‚·
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙n ˙# ˙nœœœbìñ œœœnìñ œœœbìñ˙n ˙ñ ˙n
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙b ˙n ˙nœœœñnì œœœñ#ì œœœbìñ˙n ˙ñ ˙n
Figure 2.29: Normative voice leading in the 13-PC Q system.
·‚··‚·
œœœœ œœœœœœœ
œœœœœœœ
œ œ œ œ œ
4ths & 11ths
œœœœ œœ œœœœœœ
œœœœœ œ
œ œ œœ œ
7ths & 8ths
œœœœ œ œœœœœ˙
œœœœœ œ œ œ œ
5ths & 10ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œ œ œ œ
2nds & 14ths
œœœœ ˙˙
œœœœ˙˙
œœœœœ œ œ œ œ
3rds & 12ths
œœœœ˙˙ œœœœ ˙
˙œœœœ
œ œœ œ œ
œ œ
6ths & 9thsORœœœœ ˙
˙œœœœ
˙˙
œœœœœ œ
œ œ œœ œ
&?
·‚··‚·
·œªœœœœœ
œœœœœ—œª
œ·œœœœœœ
ªœ—œœœ·œ
œœœœœœœ—
œªœ·œœœœ
—œœœ·œªœ
· œ œ œ œ œ ª œ œ œ œ œ — ·
·œªœnbìñ œœœœnbñì œœœœnnññ œ—œªñb#ì œ·œœnìnñ œœœœñnbñ ªœ—œñìn# œœ·œñìbn œœœœññnb œœœ—#ìnn œªœ·ñnñb œœœœñbìn —œœœ#nñn ·œªœnbìñºn œb œn œb œn œn ªñ œì œñ œì œñ œñ —# ºnI II III iv v vi VII VIII IX X* xi xii xiii* I
I II III iv v vi VII VIII IX X* xi xii xiii* I
&?
·‚··‚·
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙n ˙# ˙nœœœbìñ œœœnìñ œœœbìñ˙n ˙ñ ˙n
˙ ˙ ˙œœœ œœœ œœœ˙ ˙ ˙
˙b ˙n ˙nœœœñnì œœœñ#ì œœœbìñ˙n ˙ñ ˙n
Figure 2.30: Possible Ursätze for the quarter-tone Q system.
erefore we must be prepared to take diatonic seconds and thirds as stepwise intervals
in Q systems, which in the present quarter-tone system can be chromatic ICs 1, 2, 3, or
4 (sounding a quarter-tone, semitone, three-quarters tone, and whole tone). As shown
in figure 2.30, the descending line b4−b3−b1 in this system (analogous to major) sounds
identical to a b3−b2−b1 line in a traditional minor scale.
My development strategy for this example is to retain the b4−b3−b1 line in the soprano,
and attempt an octave ascent in the bass as a prolongation of the first chord in the Ur-
satz. As we have seen, since both diatonic seconds and thirds count as steps, the octave
ascent need not instantiate all 14 scale degrees; which ones are used is a question of har-
Musical Interlude II: Règle De L’Octave 157
&??
·‚··‚··‚·
˙ ˙ ˙œœœ œ œœ œ œ œœœœ œ
œ œ œ œ˙ ˙ ˙
· ‚ ·
˙b ˙n ˙nœœñì œn œn œœñì œn œ# œœœbìñœn œb œ# œn œn œn˙n ˙ñ ˙n
·n ‚ñ ·n
˙ œ œ ˙ ˙œœœœ œ œœœœ
œ œ œœœ œœœœ œ œœœ
˙ œ œ œ œ œ œ œ œ˙ ˙ ˙
· ‚ · ‚ ·
˙b œñ œñ ˙n ˙nœœœnñì œn œn œœœœñn#ì œñ œñ œœœñìb œœœnìñ œn œ# œœœììñ
˙ œn œb œñ œn œ# œn œn œn˙n ˙ñ ˙n·n ‚ñ ·n ‚ñ ·n
( )
( )
&??
·‚··‚··‚·
œ œ œ œœ œ œ œ œ œ œœœœ
œœœœœœ
œœœœœœ
œ œ œœœ œœœ œœœ œœœ œœœ œœœ˙ œ œ œ œ œ œ œ œ˙ ˙ ˙Jœ Jœ
· ‚ · œ ‚ ‚ · ‚ · ‚ ·
œb œn œn œ œñ œ œ œn œn œ# œnœœñì œœñì œœñì œœ#ñ œœì œñ œñ œœìñ œœ œœñì œœñì œœñì œœñì˙ œn œb œñ œn œ# œn œn œn˙n ˙ñ ˙nJœñ Jœ#œn œ# œn œn œ œn œb œn œb œn œb
·n ‚ñ ·n œ# ‚ñ ‚ñ ·n ‚ñ ·n ‚ñ ·n
*
*( )
( )
Figure 2.31: Background structures derived from the second Ursatz.
monic structure and compatibility with upper voices. An interesting result is that the
easiest harmonic paths favor some scale degrees over others, creating a kind of secondary
diatonic over above the primary 13-PC diatonic. I begin with a motivic repetition of
the b4−b3−b1 line in the soprano to prolong the first chord, and then b3−b1−Ó13 to prolong the
second (see the left half of figure 2.31). Already, then, a feature of the piece might be
the motivic interplay of these three-note motives which combine both diatonic thirds
and seconds. To support these motives, simple voice exchanges with the bass and tenor
work nicely.
e bass component of the first voice exchange also serves as a good beginning
for the planned octave ascent. To sketch out this ascent, I make a secondary tonic-
dominant-tonic arpeggiation in the bass and continue the soprano descent into an inner
voice from b4 (Bb) to Ó10 (Eì), which is a member of both the tonic and dominant tetra-
chords and serves as a good connection from the end of the initial tonic prolongation
P E T 158
to the structural dominant (see the right half of figure 2.31). I do not yet know whether
the stretch from Ó13 to Ó10 passes through Ó11 or Ó12 or both, so I leave both options open in
parentheses. While the soprano makes its descent to an inner voice, I also want to start
thinking about the phrase structure, and specifically whether to aim for a high climax
in the soprano before a final descent. Assuming I do, the easiest way to do this is to
arpeggiate in the soprano from b4 (the initial Bb) up to b7 (Cñ), which will act as a cover
tone during the descent to an inner voice. is cover tone may be embellished as desired
or left as a structurally unimportant part of the linear counterpoint.
e next steps are to harmonize the voice exchanges and to figure out what pitch to
use in the bass to pass from b4 (Bb) to b7 (Cñ). I want to keep the harmonization of the
voice exchanges simple and well within the primary Gn scale (see figure 2.32). To that
end, using a passing chord spelled as a dominant tetrachord in the tonic voice exchange
and a passing chord spelled as a tonic tetrachord in the dominant voice exchange makes
the most sense; this tonically spelled passing tetrachord is analogous in function to a
cadential 64 sonority. e penultimate sonority is a pentachord analogous in function
to a dominant 7th chord in traditional tonality. It is simply the dominant tetrachord
extended by another fourth above b3 (An) to b6 (Cn).e bass passing tone will have pre-dominant function, falling as it does between
the tonic and dominant portions of the bass’s octave ascent. b5 or b6 are the possible can-
didates from the diatonic scale. b5 appears in Stufen II, v, Ix, and xii, and b6 appears in III,
vi, X*, and xiii*. X* and xiii* are dominant in function, so using either of them before the
dominant tetrachord would be more like a dominant arpeggiation than a pre-dominant
to dominant progression. My solution is to use none of these, and instead tonicize the
dominant tetrachord using Bñ as a leading tone (see figure 2.32). My reason for this is
to parallel what will likely have to be at the end of the octave ascent, namely a toniciza-
tion of the tonic with F# in the bass (also provided in figure 2.32). is provides some
Musical Interlude II: Règle De L’Octave 159
&??
·‚··‚··‚·
˙ ˙ ˙œœœ œ œœ œ œ œœœœ œ
œ œ œ œ˙ ˙ ˙
· ‚ ·
˙b ˙n ˙nœœñì œn œn œœñì œn œ# œœœbìñœn œb œ# œn œn œn˙n ˙ñ ˙n
·n ‚ñ ·n
˙ œ œ ˙ ˙œœœœ œ œœœœ
œ œ œœœ œœœœ œ œœœ
˙ œ œ œ œ œ œ œ œ˙ ˙ ˙
· ‚ · ‚ ·
˙b œñ œñ ˙n ˙nœœœññnœn œn œœœœñn#ì œñ œñ œœœñìb œœœnìñ œn œ# œœœììñ
˙ œn œb œñ œn œ# œn œn œn˙n ˙ñ ˙n·n ‚ñ ·n ‚ñ ·n
( )
( )
&??
·‚··‚··‚·
œ œ œ œœ œ œ œ œ œ œœœœ
œœœœœœ
œœœœœœ
œ œ œœœ œœœ œœœ œœœ œœœ œœœ˙ œ œ œ œ œ œ œ œ˙ ˙ ˙Jœ Jœ
· ‚ · œ ‚ ‚ · ‚ · ‚ ·
œb œn œn œ œñ œ œ œn œn œ# œnœœñì œœñì œœñì œœ#ñ œœì œñ œñ œœìñ œœ œœñì œœñì œœñì œœñì˙ œn œb œñ œn œ# œn œn œn˙n ˙ñ ˙nJœñ Jœ#œn œ# œn œn œ œn œb œn œb œn œb
·n ‚ñ ·n œ# ‚ñ ‚ñ ·n ‚ñ ·n ‚ñ ·n
*
*( )
( )
Figure 2.32: Deep middleground with applied dominant.
contextual support for a hypothetical analyst to arrive at the same structure from the
foreground. It does mean that the bass ascent employs pitches from outside the main
diatonic, but in Schenkerian terms this is not a problem: an applied dominant still has
pre-dominant function and is part of the chromatic repertoire available at any point
in the prolongation. Indeed, it is clear in the diatonic staff notation that the applied
dominant has the same diatonic function as the unaltered leading-tone tetrachord, em-
ploying as it does Ó13−b3−b6−b9. e chromatic alteration of b6 from Cn to Bñ is what gives
it its pre-dominant function. Notice also that b5 cannot exist in a diatonically spelled
tetrachord with Ó13, b3, and b9, so the Bñ is unequivocally an altered b6. As a final bit of
bookkeeping, the cover-tone from the previous level is revealed to be a reaching over
from an alto voice.
P E T 160
I notice now that every sonority in my middleground except the new applied domi-
nant is spelled either as a tonic or dominant; the Dñ and Bñ from the applied dominant
are the only two PCs not found in the other chords. It is time to expand the PC content
and use more of the scale at hand. I have enough in place that I can make a decision
about the parenthesized pitches – should I use Ó12 (Fñ) or Ó11 (Eñ) between Ó13 and Ó10 in
the soprano’s descent to an inner voice? Whichever one is used will also be used in the
bass as part of the bass ascent. I have opted for Ó11 (Eñ) for two reasons (see figure 2.33).
First, the xi tetrachord contains Cñ in common with the two dominant-function chords,
so that the cover-tone can stay in place. If I were to use Fñ, the cover tone would have
to change (Cñ is not consonant with Fñ), and I would have to take that into account in
any subsequent embellishment; letting it stay gives me more flexibility. Second, and
more subtly, the pitch chain Bb-Bñ-Cñ has the chromatic interval succession +3, +2, as
does the pitch chain Eñ-F#-Gn. is further strengthens the motivic parallel between
the dominant tonicization and the tonic one that follows.
Figure 2.33 contains two other important embellishments. e first is a pre-dominant
chord before the structural dominant, built on b3 (An). Its inclusion here serves as the be-
ginning stage of a pre-dominant prolongation that will be used to prepare the structural
dominant. It is common in a traditional major key to tonicize the ii chord in preparation
for a dominant with b2 in the soprano, both to contextually highlight b2 as a structural
entity and to allow the composer to displace b2 from the actual dominant in favor of the
leading tone or a cadential 64 figure. Given that b3 in this system is similar in function
to b2 in traditional tonality, tonicizing it will provide similar contextual support for the
ensuing structural dominant. e second embellishment is a prolongation of the final
tonic tetrachord by means of an upper-neighbor figure in inner voices which yield an
VIII chord. In traditional tonality this is analogous to a neighboring 64 chord over a tonic
Musical Interlude II: Règle De L’Octave 161
&??
·‚··‚··‚·
˙ ˙ ˙ œ œ œ œ œ œœœœœœœ
œœœœœœ
œœœ
œœœ
œœœœ
œœ
œœœœ œœœœœœœ
œœœœ œœœœœœœ
œœœœœ œ œ Jœ œ œ œ œ œ œ œ œ
˙ ˙œ ˙ œ œ˙ œ œ
Jœ
· ‚ · œ ‚ œ ‚ · œ ‚ · ‚ · œ ·T D/D D P D T P D T (P) T
˙b ˙ ˙n œn œn œ œ œñ œœœñì œœñì œœñì œœ#ñ œœìñ œ œœì œ œœœnnñ œœìñ œœœnñì œœœ#ñì œœñì œœœñì œœœìñ
œn œb œñ Jœñ œ# œñ œ# œn œn œn œn œn˙n ˙ñ œñ ˙n œ œ n œñ œnJœnœn œ# œn œn œn œœbn œœnn œb œn œn œb œn œb œb œb
·n ‚ñ ·n œ# ‚ñ œñ ‚ñ ·n œn ‚ñ ·n ‚ñ ·n œì ·nT D/D D P D T P D T (P) T
*
*
Figure 2.33: Deep middleground with function analysis.
pedal, yielding a IV chord. e broad function of each chord is given underneath the
fundamental bass: T for tonic, D for dominant, and P for pre-dominant.
In order to finish the bass’s octave ascent, I need to develop the passage marked with
a bracket in figure 2.33. is development will also include composing out the b7 (Cñ)cover-tone, if possible. One way to check for options is to look for linear possibilities
that are already buried in the counterpoint and promote them to a higher voice. For
instance, in the bracketed passage, in the last three chords I see a descending tenor
line b6-b4-b3 (Cn-Bb-An), which would serve nicely as a passing motion from the b7 (Cñ)cover-tone in the soprano to the b3 (An) in the pre-dominant chord at the end of the
bracketed passage. Or I could notice that the tenor line in the last three chords of the
tonic prolongation (i.e. beginning with the second chord of the bracketed passage) a
descending b4-b3-b1 (Bb-An-Gn), that with one intervening chord I could connect to the
cover-tone to make a kind of “5-line,” b7-b6−b4-b3-b1 (Cñ-Cn-Bb-An-Gn).
P E T 162
&??
·‚··‚··‚·
œ œ œœœœ
œœœ
œœœœ
œœ
œœœœJœ œ œœ œ
Jœ‚ œ ‚ · œ
œ# œñ œœœìñ œ œœì œ œœœnnñJœñ œ# œñœñ œn
Jœnœn œœbn œœnn œb œn
‚ñ œñ ‚ñ ·n œn
œ œ œ œ œ œœœœœ œ œ œ œœœ œœ œœ œœ œœ
œ Jœ œ œœ œJœ
‚ — œ ‚ · œ
œñ œn œb œn œn œnœ# œ œn œì œœì œñ œñ œñ œñ œœnñœñ Jœñ œ# œñœñ œn
Jœnœn œn œb œn œb œn
‚ñ —# œñ ‚ñ ·n œn
œ œ œ œ œ œ œœœœœ œ œ œ œ œœœ œœ œœ œœ œœ œœ
œ œ Jœ œ œœ œJœ
‚ —œ œ ‚ · œ
œñ œn œb œn œn œn œnœ# œ œì œñ œì œœì œñ œb œñ œñ œñ œœnñœñ œì Jœñ œ# œñœñ œn
Jœnœn œn œb œb œn œb œn
‚ñ —# œb œñ ‚ñ ·n œn
*
*
Figure 2.34: Development of bracketed passage in figure 2.33 to complete octave ascentin bass.
Since I need an intervening chord for the bass ascent in any case, I attempt the
“5-line” development of the cover tone in figure 2.34, using a leading-tone chord as
the intervening chord containing b6 (Cn). I might change my mind, but I want a chord
with dominant function because the succession b6-b4 (Cn-Bb) is among other things a
resolution of a tendency tone (as we have seen in the dominant pentachords elsewhere
in the example), and the leading-tone chord is the only chord with dominant function
that also contains a b9 (Dñ) I can place between b7 (Cñ) and Ó11 (Eñ) in the bass ascent. is
makes the progression from dominant to pre-dominant function a kind of deceptive
cadence; in fact, the pre-dominant xi chord shares three PCs in common with the tonic
tetrachord, so in many ways it is like a vi chord in traditional harmony. I have one
problem, though, which is that because I probably want the leading tone to resolve up
to b1 in this deceptive cadence, and because the leading tone in the chord is part of the
earlier soprano descent to an inner voice, resolving the leading tone means disrupting
Musical Interlude II: Règle De L’Octave 163
that line (see the middle part of figure 2.34). Looking back, I may even have overlooked
this issue in themotion from the tonicized VII chord to the xi chord (the first two chords
of the bracketed passage), because this is also a dominant to pre-dominant deceptive
cadence and I probably should have resolved the leading tone back then. is would
have nullified the whole raison d’être for the xi chord in the first place, which was to
serve both in the ascent of the bass and descent of the soprano.
Is the soprano descent into an inner voice unsalvageable, and has a large part of my
structure up to now therefore been built on a mistake? e answer to these questions
is no, and the reason is due to the flexibility inherent in any sophisticated tonal system.
In traditional tonal music, when one needs the leading tone in a V-vi progression to
resolve down instead of up, the vi can just be tonicized so that the global leading tone
is recontextualized as a local b2. I can do something similar in the present example by
tonicizing the xi (Eñ) chord with a VII/xi, which is a major tetrachord spelled Bb-Db-Eì-Gì (see the right part of figure 2.34).44 I could have foreseen the leading-tone
problem and inserted the xi chord with its tonicization in an earlier level. So now my
leading-tone F# takes a small excursion to Gì on its way to Eñ, which in contrapuntal
terms is a kind of échappée figure. Meanwhile, b4 (Bb) in the “5-line” embellishment of
the cover-tone is displaced from the xi (Eñ) chord to its applied dominant. Placing the
development from figure 2.34, which completes the bass’s octave ascent, into the main
sketch yields figure 2.35.
e bracketed passage in this figure is the next to be developed. I want to fix one
thing immediately: the PCs in the bass and soprano of the first two chords form parallel
octaves (Gn-An). ere is an easy remedy which involves placing an intervening chord44Incidentally, one might notice that because this is a descensive system, the chromatic alteration nec-
essary to change the minor global iv chord, spelled Bb-Cñ-Eì-F#, to a major chord so that it can act asVII/xi, does not alter a pitch so that it may act as a tonicizing leading tone as in traditional tonality (an as-censive system). Instead, it alters a pitch so that it can function properly as the dominant upper-neighbor(Gì instead of F# in the present example, sounding a whole tone above the Eñ being tonicized).
P E T 164
&??
·‚··‚··‚·
˙ œ ˙ œ œ ˙ œ œ œ œ œœ œ œ œ œœ œ œ œ œ œœœœ
œœœœœœ
œœ œœœœœ
œœœ œœ œœ œœ œœœ œœœ œœœ œœœ œœœ œœœ œœœœ œ œ œ œ Jœ œ œ œ œ œ œ œ œ
˙ ˙œ ˙ œ œ œ œ
Jœ
· ‚ · œ ‚ — œ œ ‚ · œ ‚ · ‚ · œ ·
˙b œn ˙ œn œ# ˙n œ œ œn œn œœ# œ œ œì œñ œì œœñ œn œb œn œn œnœœñì œœñì œœñì œñ œì œñ œb œñ œñ œñ œœnñ œœìñ œœñì œœñì œœñì œœñì œœìñ
œn œb œñ œñ œì Jœñ œ# œñ œ# œn œn œn œn œn˙n ˙ñ œñ ˙n œ œ œñ œnJœnœn œ# œn œn œn œn œb œb œn œb œn œn œb œn œb œb œb
·n ‚ñ ·n œ# ‚ñ —# œb œñ ‚ñ ·n œn ‚ñ ·n ‚ñ ·n œì ·n
*
*
*
*
Figure 2.35: Middleground after development in figure 2.34, octave ascent complete.
Musical Interlude II: Règle De L’Octave 165
containing b2 (G#) to be used as a passing tone in the soprano (see the first system of
figure 2.36). Next, I notice that in the III chord (on An), the fourth above the root
(Cn) is in the tenor, which I can exploit as a voice exchange with the bass to tonicize b3as I had planned, in order to prepare the structural dominant (in the second system of
figure 2.36). I am dissatisfied with this for two reasons: first, the Gn-G#-An-G#-An-Gn-F# line in in the soprano is repetitive and reads much more like an inner voice; this can
be improved by moving the voice exchange from the tenor to the soprano. Second, the
chord intended to tonicize III, built on b9 (Dñ), is already a Stufe of the primary diatonic
scale, so its power to tonicize is diminished. is is analogous to the fact in traditional
tonality that it is difficult to tonicize the VI chord in minor with a III triad, since the
latter is already part of the system unaltered. Tonicization of VI in traditional minor
works much better if one employs a dominant 7th chord on III instead. So here, if I
want to tonicize the An tetrachord, I ought to use a dominant pentachord from the An-major system, which includes a Dn. is all comes together in the last stage of figure
2.36, with the advantage that now the soprano completes another 5-line. Now that III
is tonicized, I have also performed an intensive embellishment (i.e. one that deletes an
event from a higher level) by removing the first simultaneity of the structural dominant.
e last three chords are roughly analogous to a ii6−V6−54−3 progression in traditional
tonality.
One final point in figure 2.36 is worth mentioning: it is possible that by finishing
the octave bass ascent with a soprano descent from b7 to b1, I will have made this cadence
too final for the middle of the piece. I can remedy this by putting b7 (the cover-tone Cñfrom previous examples) back in the soprano at the resolution, to make it less final and
to move smoothly from the end of that prolongation to b6 (Cn) at the beginning of the
next one. is also highlights the sequential nature of the two 5-lines, as seen in figure
2.37. is figure highlights the linear counterpoint attained thus far by omitting most
P E T 166
&??
·‚··‚··‚·
œ ˙ œœœœœœœœœœœ œœœ œœœ
œ œ œ jœ œ œ ˙
œJœ
· œ ‚
œn ˙ œn œ#œœñì œœnñ œœìñ œœñì œœñìœn œn œñ œ# œœbn œœnn ˙ñ œñJœnœb œn œn
·n œn ‚ñ
œ ˙ œ œœœœœœœ œœœ œœœ œœœ œœœ
œ œœ œ œ œ œ ˙
œJœ
· œ œ ‚ · ‚
œn ˙ œn œ#œœñì œœññ œœññ œœ#ì œœñì œœñìœn œ#œn œñ œ# œn œn ˙ñ œñJœnœb œn œn œñ œb œn
·n œn œn ‚ñ ·n ‚ñ
&??
·‚··‚··‚·
œ œ œ ˙ œ œœœœœœœ œœœ œœœ œœœ
œœœ œœœ œœœœ œœ œ œ ˙
œJœ œ
· œ œ œ œ ‚ · ‚
œn œ# œn ˙ œn œ#œœñì œœññ œœññ œœññ œœññ œœ#ì œœñì œœñìœn œ#œn œñ œn ˙ñ œñJœn œnœb œn œn œn œn œn œb œn
·n œn œn œñ œn ‚ñ ·n ‚ñ
*
*
˙œœœœœœœœ
œœœœœœœœ œœœ œœœ œœœ
œ œ œ œ œ œœ œ œ ˙
œJœ œ
· œ œ œ œ · ‚
˙nœœìñ œœœñ#ñ œœœnññ œœœñ#ñ œœññ œœñì œœñìœn œn œ œn œn œ#œn œñ œn ˙ñ œñJœn œnœb œn œn œn œn œb œn
·n œn œn œñ œn ·n ‚ñ
*
*
ª ª
Figure 2.36: Development of bracketed passage in figure 2.35.
Musical Interlude II: Règle De L’Octave 167
&??
·‚··‚··‚·
œ œ ˙œ œ œœ œ œ œ œ œ œœœœœœœ œœ œœ
œœœœœœœœ
œœœœ œœœ œœœ œœœœ Jœ œ œ œJœ œ œ ˙
œ
œñ œn ˙nœñ œì œœb œn œn œn œn œn œ#œœñ# œœnñ œñ œñ œœœñ#ñ œœœñnñ œœœñ#ñ œœññ œœñì œœñìœì Jœñ œ# œn œì Jœn œn œn ˙ñ œñœb œb œn œb œn œn œn œn œb œn
œ œ œ ˙œ œœ œ œ œ œ œ œœœœœœœ œœ œ
œœœœœœœ
œœœœ œœœ œœœ œœœœ Jœ œ œ œJœ œ œ ˙
œ
œì œñ œñ n ˙nœñ œìœb œn œn œ œn œn œ#œœñ# œœnñ œñ œœñ# œœœñn ñ œœœñ#ñ œœññ œì œìœì Jœñ œ# œn œì Jœn œn œn ˙ñ œñœb œb œn œb œn œn œn œn œb œn
&?
·‚··‚·
œ œ
œ œ˙
˙b œn œnœn œb˙n
5 bars
œ œœ
œ œ
œ# œœñœñ œñ
*
*
œ œœ œ
œn œbœñ œì3 bars
*
*
œœ œJœ œ
œñ œn œnJœñ œ#
œ œ ˙œœœ œ
œ œJœ œ œ
œñ œn ˙œì œn œ œn œn œñ Jœn œn œn
4 bars
*
*œ œ˙
œ
œn œ#˙ñ œñ2 bars
˙ œ
˙ œ
˙n œ˙n œ
2 bars
œ
œ
œœ
Figure 2.37: Middleground after development in figure 2.36, linear counterpoint only.
of the harmony. e bass’s octave ascent, the soprano’s descent to an inner voice, and
the two sequential Züge are very clear in this view.
To conclude the example and this musical interlude, it will be useful to think of
phrase structure. At the bottom of figure 2.37 I have included suggestions for how to
make this example into a 16-bar phrase, perhaps in the manner of a Baroque prelude,
with 8 bars to finish the bass ascent tonic prolongation and 8 to prolong pre-dominant,
dominant, and final tonic. Composing out the first voice exchange over 5 bars, perhaps
with some sequential Fortspinnung, is a good way to establish the key. Increasing the
harmonic rhythm for the final three bars of the first half adds forward motion that will
be continued in the second half, and this also elides the final cadence of the first half.
e excerpt ends with two bars of dominant pedal followed by two bars of tonic pedal.
I have composed a foreground fulfilling these constraints in figure 2.38. I do not
wish to explain every detail, but there are a few things worth pointing out. First, I have
P E T 168
located the highest note in the piece at the beginning of the second half on the chord that
is at the furthest structural remove on the prolongation tree. Placing a climax at such
an unstable point makes it possible to discharge the structural tension through a long
descent over the second half. Second, the unfolding in the soprano from the end of the
first half into the beginning of the second further masks the harmonic phrase boundary
by means of motivic association, keeping the harmonic motion moving forward. Finally,
the strong applied-dominant counterpoint in the dominant and tonic pedals delay any
resolution to the very final bar. Of special interest is using a dominant pentachord on b1to tonicize b8 (Dì) over the tonic pedal; this is analogous to using a dominant 7th chord
on b1 to tonicize IV at the end of a prelude or fugue in Baroque music. is foreground
can now be embellished further with figuration, arpeggiation, extra passing figures, and
so forth, to make a final surface; the reader may wish to attempt this as an exercise.
is ends the musical interlude. e examples presented here show unequivocally
how prolongation is possible when the Straus conditions are met. Having established
the basis for Straus’s first, second, and fourth conditions elsewhere in the paper, my aim
here was to showcase problems that chiefly concern the third condition (i.e. the embel-
lishment condition), and to show how the primary methods of reaching outside the PC
space of the diatonic – mixture and tonicization – can work when the diatonic system
itself is robust enough to foster prolongational embellishment. In the next section I pull
together all of the strands to show how the various diatonic templates may be embedded
in various equal temperaments.
Musical Interlude II: Règle De L’Octave 169
& ?· ‚ ·· ‚ ·
œ œ œœ˙ ˙b œn
œn œnœb
˙n 5 bars
œœœ œœ œ#œœñ œñœñ* *
œ œ œœ œnœb œñœì
3 bars**
œœœ Jœœ œñœnœn Jœñœ#
œ œ˙
œœœ œ
œ œ Jœœœ
œñœn
˙œìœn
œ œn œn
œñJœn
œnœn4 b
ars
**œ œ ˙ œ œn œ# ˙ñ œñ2 b
ars
˙œ ˙œ ˙nœ ˙nœ 2 bars
œ œ œ œ
& ? ?· ‚ · · ‚ · · ‚ ·
˙ œ œ œ ˙ · ˙b œ œñì œ ·n
œ œ œ œ œ œ œ œ œì ì œœ œì
j œ œ œ œ œ œ j œn œ œìì œœ œn 5 bars
œ œ œ œ œ ‚ œ œ œì ñ œœ# ‚ñ
œ œ œ œ œ · œ œ œì ñ œbœ ·n
œ œœ
œœ
œœ œ
œœ
œ‚
œ#œ œñ
œñœ
œì œñœñœœ
œ#‚ñ
* *
œœ
œ œ œœ œ œ
œœ
—œ
œnœb œ œñ#œ œbì œñœìœœb
—#œb* *
3 bars
œœœ
œœ œ
œ œJœ
œ
œ‚ œì œñœnœ
ϖϖ
Jœñœ#œbœn
œñ·ñ
œœ œj œ œ œœ œ œ œ Jœ ·œ
ϖϖ ϓ
j œ œœ œñ œ Jœìœbœn ·nœn
œ œ œ œ œœ œ œ œœ œœ œñœ œ œñ#
œ œ œñœñœœn œnœñ** 4 b
arsœœ œ œ œ œœ œ œ Jœœ œœ œœ œ œ œñ ñœ œ Jœœœœ œnœn
œ˙œ œ œ œœ œ œœ œ œœœ œ
œœœ œ#˙
œ œ œñ ñœ œñ
œ œ
œœn œœn
œœ
œñœñ œn
* *
œœ
œ œ œœ œ œ
œ˙
·‚
œnœ#
œ œñìœ œ
œñ˙
œbœ
·n‚ñ 2 b
ars
œ œ œ œœ œ œ œ œœ œœ œ‚
œ œ œñ ññœ œì n œñœñœ œ œ# ñœ œ œ
œ#‚ñ
**
œ œœ œ ˙œ œ œœ œ œ ·œ œ œñ bœ œ œ œ œì ñ
œ œ œì ·nœn 2 b
ars
œ œ œœœ œ œœ œ œ œ — œ œbœ œ# œœœ œñ ì
œ œñ n œì—#
˙e œœ œ œ · ˙e œœ œ œì bñ ·n
Figure 2.38: Completed quarter-tone Q foreground.
P E T 170
2.3 F D C
Placing a diatonic system into an equal temperament may require some trial and error.
One might expect to be able to predict the outcome of matching a given diatonic system
to a given temperament, but modular arithmetic is ever subtle. Even something as
seemingly simple as finding the multiplicative inverse of a given member of a mod n
system’s multiplicative group requires an algorithm; there are no easy formulas. is
situation is often beneficent, in fact, because brute enumeration of possible solutions
and other trial-and-error methods gives one a more complete picture of the terrain than
a list of formulas ever would.45 Nevertheless, some guidance is useful. In this section
I provide some of the best strategies I have found to find desirable matches between
systems. Before doing so, I show why the generalized diatonic systems above – all of
which are generated scales – fulfill the Straus Conditions.
P G S S C
Our Gradus ad Parnassum has thus far not been a straight climb but rather a slow as-
cent along winding passes. Some review will be helpful to consolidate a map of the
terrain already covered. I began the essay discussing various approaches to generalized
prolongation, and settled on Joseph Straus’s four conditions as the most fruitful basis
for creating generalized prolongation systems. On page 35 I quoted Joseph Straus’s
description of his conditions:45In his foreword to his translation of Martin Buber’s I andou, Walter Kaufmann says the following
about Buber’s writing style: “Hemakes very difficult reading. He evidently did not wish to be read quickly,once only, for information. He tried to slow the reader down, to force him to read many sentences andparagraphs again, even to read the whole book more than once.” (Kaufmann (1970), page 43.) ere aremany good reasons for the reader to approach any work with the similar careful navigation they wouldapply to a difficult text. I think the same intuition applies to artists first learning new materials: thestructure of relations among different parts can sometimes only be internalized when one is forced topore over all of the parts and the whole a number of times over.
From Diatonic to Chromatic 171
First, there is the consonance-dissonance condition; we need a way based onpitch of distinguishing between structural and nonstructural tones. Second,there is the scale-degree condition; we need some kind of hierarchy amongthe consonant harmonies. ird, the embellishment condition; we need aconsistent model of voice leading that will enable us, for example, to tellan arpeggiation from a passing note. Fourth, there is the harmony/voiceleading condition; we need to be able to distinguish motions within a voicefrom motions between voices.46
In the context of the first Straus condition I introduced the notion of a maximal
consonance, which I took to mean a set-class containing the maximum number of mu-
tually consonant PCs in an equally tempered system. is definition has an intervallic
interpretation and a set-class interpretation. e former begins with a collection of con-
sonant ICs and then classifies maximal consonances as the set-classes with the largest
cardinality that contain only consonant ICs in their interval vector. e latter begins
with a set of SCs defined as consonant, and on that basis declares all of their constituent
ICs consonant. I believe that the former approach underdetermines consonant SCs
when only equal temperament is taken into account. For instance, if IC 4 is a conso-
nant interval in 12 then SC [048] should be a consonance. It is only when it is
compared to an underlying diatonic that it is revealed to be dissonant due to a necessary
chromatic alteration of the diatonic norm. erefore I have implicitly used the latter
approach rather consistently throughout the rest of the essay. It is the combination of a
consonant set-class and a regulating diatonic that now seems to constitute the maximal
consonance concept.
I modified the second Straus condition in such a way that it might be called the
“position-finding” condition instead. We do not need a hierarchy of scale degrees, but
merely a way to tell them one from another; similarly we do not need a strict hierarchy
of consonant harmonies built on the scale degrees, but rather a way to distinguish each46Straus (1987), page 7.
P E T 172
harmony on the basis of its intervallic relations to the rest of the scale, and to impose
a harmonic syntax on them. us far I have imported the notions of tonic, dominant,
and pre-dominant from tonal theory into my generalized prolongation scheme as part
of the harmonic syntax that regulates structural arpeggiations, filling-in, and other em-
bellishments.
e third of Straus’s conditions has acted like an end goal of fulfilling the other
three. Traditional Schenkerian theory already assumes the contents of Straus’s 1st, 2nd,
and 4th conditions by importing the major-minor system with seven Stufen. e novelty
of Schenkerian theory has always been in showing how a limited set of embellishment
types can connect surface to background through a series of generative and/or analytic
levels. In the two musical interludes (beginning pages 47 and 122, respectively), I have
shown how a set of embellishments nearly equivalent to those in Schenkerian theory can
work in other systems once the other three Straus conditions are met. Other methods
such as mixture and tonicization are also available.
e fourth of Straus’s conditions, in combination with the first two, has two primary
functions in generalization. First, it regulates the formation of generalized diatonic
systems so that they embed maximal consonances in maximally even ways. Second,
it requires that the diatonic systems meet certain other conditions (e.g. that they be
generated, and be of an appropriate cardinality) so that they may be ordered as Stufen.
My critique of Lewin’s generalized triad spaces (beginning page 66), and the first section
of Part 2 (Constructing Generalized Diatonic Scale Cycles For Prolongation, beginning
page 100 above) showed the fourth Straus condition in action. I argued that Lewin’s
approach could be improved with what I termed “wolf closure” (see figure 1.33 on page
78). Doing so guaranteed that the resulting collection was generated by an interval,
which I have called gc (the chromatic generator).
From Diatonic to Chromatic 173
ere is a strong connection between the properties of generated scales and the
Straus conditions. In most of the scale-theory literature, “generated scale” and “gen-
erated PC-set” have not been used interchangeably, the reason being an intuition that
scales have special properties not shared by all PC-sets. Eight of these features are
helpfully collected and discussed in Clough et al., “Scales, Sets, and Interval Cycles: A
Taxonomy,” 1999. Only five of them apply throughout my work, and I have construed
four of them in terms more general than Clough et al. I list them here with explanatory
description and indicate how they are generalized, if at all. e five properties in their
generalized forms together with the diatonic templates listed in figure 2.5 (page 122)
are sufficient to meet the Straus conditions, and all but one is either necessary for or a
consequence of the construction of S → H → G → (S⊻S) operations of the diatonic
templates.
1. GN (generated). is applies to a collection generated by a single interval. In
some contexts this is a chromatic interval, which I have labeled gc , and in others
this is a diatonic interval (an example of what Clough et al. refer to more generally
as generic intervals), which I have labeled gd . It is already sufficiently general
for my purposes. For a given chromatic or diatonic space with cardinality n, all
elements of the mod n multiplication group (and only those elements) may be
used as gc or gd .
2. WF (well-formed). A well-formed PC-set is generated by a gc (gd is not nec-
essary for this feature), and each instance of gc in the PC-set spans a constant
number of PCs in PC-space.47 I have generalized this concept so that my scale
systems are well-formed with respect to a cyclic ordering S . Let us call this gener-
alized well-formedness or WFγ . In the terms I have used throughout part 2 of this47When this is extended so that all ICs in the PC-set’s interval vector (not just gc ) spans a constant
number of PCs, the PC-set is said to be unambiguous. e traditional diatonic is not unambiguous becausethe tritone can span a generic 4th or generic 5th.
P E T 174
essay, for a given S , every gc span is a gd span over order positions in S . WF is
a special case of WFγ that holds if and only if S is contour-preserving (i.e. when
the S ordering is also S ’s PC-space ordering).
3. MP (Myhill’s property). A PC-set in which every generic interval has exactly two
corresponding chromatic sizes has Myhill’s property. Like WFγ , a generalized
Myhill ’s property MPγ extends the MP concept to orderings: a cyclic ordering
has MPγ when order-position spans of a given size yield one of exactly two PC
intervals. All orderings with MPγ also have WFγ and are generated sets. MPγ
automatically falls out of the multiplicative permutation operations S → H →G→ (S⊻S). e multiplicative permutation operators also guarantee that the PC
intervals of each of the three cycles are unique, which supports the harmony/voice
leading criterion (the fourth Straus condition).
4. ME (maximal evenness). A scale is maximally even if every generic interval comes
in either one chromatic size or two consecutive integer chromatic sizes. For ex-
ample, in the traditional 7-PC diatonic scale in 12, every diatonic 2nd is either
1 or 2 chromatic steps, every diatonic 3rd is either 3 or 4 semitones, and so forth.
e label “maximal evenness” was applied earlier48 to mean “a set whose elements
are distributed as evenly as possible around” the cycle of a given superset.49 In
fact the two definitions entail one another.
ere are two generalized forms of maximal evenness. e first, which I label
MEγ1, generalizes the second definition: a set with MEγ1 is a set whose elements
are distributed as evenly as possible around the cycle of an IC coprime to c . In
12, the 7-PC chromatic cluster has MEγ1 with respect to the circle of fifths.
Generally, if any set of size d in a chromatic of size c has ME, an Mn operator48Page 109.49Clough and Douthett, “Maximally Even Sets,” 1991, page 96.
From Diatonic to Chromatic 175
on PCs where n is coprime to c will yield a set that has MEγ1 with respect to
the n cycle of the chromatic.50 ME and MEγ1 are properties of sets. e second
generalization, MEγ2, extends the maximal evenness notion to orderings. An or-
dering has MEγ2 if every generic interval comes in either one chromatic size or
two chromatic sizes that differ by the same number. e 9-PC scale ordering I
used to create a Goldberg analogue in 12 is< 0,2,4,6,5,7,9,11,1−0>. Each
diatonic 2nd is either PC interval 2 or 11, each 3rd is 4 or 1, each 4th is 6 or 3, each
5th is 5 or 8, and so forth; each chromatic pair associated with a given diatonic
interval differs by 3. Every S , H , and G cycle in my theory, regardless of con-
tour preservation, has MEγ2. MEγ2, like WFγ and MPγ , results automatically
from the multiplicative permutation operations S → H → G → (S⊻S). Because
MEγ2 distributes the secondary intervals of S maximally evenly (but not perfectly
evenly) over the S ordering, this feature makes it possible to tell scale degrees from
one another and thereby fulfills the scale-structure requirements of the modified
second Straus condition.
5. DP (deep). A deep scale is one for which every IC of the chromatic system is
represented with a unique multiplicity in the scale’s interval vector. With the
exception of the tetrachord [0,1,2,4] in 6, every deep scale is also a generated
collection.51 A generated scale S is deep if and only if d = [ c2] or d = ([ c
2] + 1),
where the square brackets indicate truncation.52
An example will help demonstrate why this is the case. First, note that every
chromatic cluster in any equal temperament is generated by IC 1. e number of50e19 cognate scale of the traditionalmajor scale is< 0,3,6,8,11,14,17−0>; it hasMEbecause
its primary and secondary intervals differ by one. e 31 cognate < 0,5,10,13,18,23,28− 0 > hasprimary and secondary intervals which differ by 2, so it does not have ME. But it does have MEγ1: it ismaximally even with respect to the cycle of IC 15.
51Clough et al., “Scales, Sets, and Interval Cycles: A Taxonomy,” 1999, page 78. e 6 intervalvector for [0,1,2,4] is <2,3,1>.
52Clough et al., “Scales, Sets, and Interval Cycles: A Taxonomy,” 1999, page 79.
P E T 176
nonzero ICs in a given equal temperament is [ c2] (for instance, there are 6 nonzero
ICs in 12 and 13). In a chromatic cluster of size d , the largest PC-interval
is d − 1, between the lowest and highest member of the cluster. Suppose d =
([ c2]+ 1); then d − 1 is also the largest IC, and is represented once in the cluster.
e next-largest IC is represented twice in the cluster, and generally the number of
times an IC occurs in the cluster’s interval vector is d−IC. erefore for a cluster
of size d = ([ c2] + 1), each IC is represented at least once, each with a unique
multiplicity, and so the cluster is a generated set with DP. If d is one PC smaller
(d = [ c2]), then the largest IC is not represented in the interval vector, but all ICs
are represented with unique multiplicity so the cluster is also a generated set with
DP. Every cluster where d < [ c2] has more than one IC with zero multiplicity,
and for every cluster where d > ([ c2] + 1) at least one pair of ICs in its interval
vector has equal multiplicity. Every deep set in a given equal temperament is a
member of a class of deep sets all of which are related by PC multiplication using
operators from the multiplication group of c .
Deepness is not necessary for my prolongation schemes, but it is a virtue. First,
Richmond Browne points out that the unique multiplicity of ICs in deep scales
also helps distinguish scale degrees from one another in position finding.53 is
works because a corollary of deepness is that each PC of a deep set has a unique
PC-intervallic relationship with the other members; for instance, only the PCs
at the ends of the generator chain of a generated deep set participate in the rarest
intervals. is fulfills the position-finding constraints of the second Straus con-
dition. Second, deepness in a scale can ensure that the primary and secondary
intervals of S , H , and G are unique ICs (they are always unique PC intervals),
which fulfills the harmony/voice leading condition.54 ird, because deepness53Browne, “Tonal Implications of the Diatonic Set,” 1981, pages 6-7.54However, this is more about how much of the chromatic the diatonic saturates.
From Diatonic to Chromatic 177
regulates the value of d with relation to c , it also ensures that “keys” or “regions”
are related both by position on the full cycle of gc and PCs in common (measures
which are inversely correlated). is last virtue is closely related to the first, be-
cause one function of position finding is the accurate identification of region by
means of rare intervals.
Generalizing DP can be accomplished by admitting degrees of DP rather than
treating it as a binary property, by defining a generated deep set as one for which
all of its nonzero interval-vector entries are unique. en, supposing a scale S
meets this criterion, the degree of deepness is the number of nonzero interval-
vector entries, which is d − 1. is more general DP (let it be labeled DPγ) also
provides the first and second virtues of normal DP. e third virtue is provided
by DPγ when the degree of distance from one region to another on the gc cycle is
less than or equal to the degree of deepness. For instance, the ordinary pentatonic
scale in 12 is generated by IC 5. Its interval vector is <0,3,2,1,4,0>, so it has
DPγ with degree four. Transposing a pentatonic scale P by 1×gc yields a scale
with four PCs in common with P . Transposing it by 4×gc yields one with one
PC in common with P . Pentatonics a distance of five or six gcs away from P (in
either direction) have zero PCs in common with P ; these three regions do not
relate to P both by gc-cycle distance and PCs in common. e reader can work
out an analogous measurement of degree of DPγ for scales of d > ([ c2] + 1) by
finding the minimum number of shared PCs between regions of size d ; then for
a given S , gc-cycle distance negatively correlates with PC-intersection up to the
smallest distance with minimum overlap.
To summarize, GN is a necessary property for the system, and is in fact entailed by
WFγ , MPγ , MEγ2, and DPγ . MPγ also entails WFγ .55 e multiplicative permutation55e hierarchy of implication is discussed in detail in Clough et al., “Scales, Sets, and Interval Cycles:
A Taxonomy,” 1999, pages 77 ff.
P E T 178
operations S → H → G → (S⊻S) automatically generate orderings with WFγ , MPγ ,
and MEγ2. Straus condition 2 is fulfilled by the scale-structure features of MEγ2 and
the position-finding features DPγ when DPγ applies. e fourth Straus condition is
supported by MPγ , which along with the S→H →G→ (S⊻S) operations ensures that
the PC intervals between adjacent pitches of the S ordering – which are voice-leading
intervals – are different from the PC intervals of adjacencies in the H and G orderings
– which are harmonic intervals. e first Straus condition is fulfilled by this intervallic
partitioning and the constraints on constructing maximal harmonies of size h from the
H ordering. Finally, fulfilling Straus conditions 1, 2, and 4 provides the PC-space reg-
imentation necessary for applying consistent prolongational embellishments analogous
to those found in Schenkerian theory, fulfilling the third Straus condition. Adding con-
tour preservation as a further constraint moves the PC-space framework to pitch space
and allows unambiguous voice partitioning on the basis of pitch and time alone. In con-
tour nonpreserving systems, voices may need to be articulated by other means, such as
assigning different voices strictly to different octaves or to different timbres. Although
this maymake some embellishments that traverse register (e.g. motion to an inner voice)
unavailable, a restricted set of permissible embellishments nonetheless fulfills the third
Straus condition so long as the embellishments are unambiguously applicable.
In the next few sections I lay out some heuristics for matching a diatonic system
with an equal temperament.
R, I C, R B c d
Harmonic regions have figured very little in my theory thus far, but only because they are
a secondary feature of prolongational systems. Tonicization and modulation can be very
helpful in emphasizing points of harmonic stability, but they can be used just as easily to
From Diatonic to Chromatic 179
cloud the voice-leading picture.56 However, relations among harmonic regions clarify
how the structure of a diatonic scale system relates to the chromatic it is embedded
in. e notion of distance between regions being measured in multiples of a chromatic
generator gc is also built into the structure of the scale as a set generated by gc .
roughout this essay I have assumed that any gc must necessarily be coprime to
the chromatic cardinality c , so that a cycle of gc can generate the entire chromatic. If it
is, then a region is simply a d-sized segment of the full gc cycle, and there are therefore
c regions in such a system with measurable gc distance from one to any other. But
imagine if gc were not coprime to c : it would require two or more transpositions of agc cycle to yield the full chromatic, and distances between regions on different discrete
cycle transpositions would no longer be measurable in multiples of gc . Such a system is
not inconceivable; indeed, a number of composers have treated 24 as two separate
12 systems. But therein lies the conceptual problem: in a given equal temperament,
any cycle of an interval not coprime to c yields an independent equal temperament, and
so the behavior of scales and harmonies in this kind of system is identical with that
behavior in the daughter temperament. is is primarily why I have restricted gc to
coprimes of c .
An alternative measurement of distance between regions has traditionally been the
number of PCs in common between them, which is dependent on the set’s interval vec-
tor. is distance scale is applicable to Schoenberg’s inversionally combinatorial hexa-
chords, for example, and Schoenberg often works the ordering of his hexachords to re-
flect somemotivic intersection between closely related hexachords in order to strengthen
that relation. It is also applicable to traditional keys. Distance between keys on the cy-
cle of fifths correlates with distance measured in PCs in common due to the regularity
of the cycle of fifths and the fact that the traditional diatonic is a deep scale. ere is56See “Analysis By Key: Another Look at Modulation” in Schachter, Unfoldings, 1999 (pages 134-
160).
P E T 180
one ambiguity resulting from the transpositional symmetry of the tritone, which is that
keys both five and six fifths from a given key both share two PCs with that key. is
ambiguity is a part of any system where d = ([ c2] + 1) and c is even: all members of
the multiplication group of an even number are odd, so all values for gc in an even-
numbered temperament must be odd; also n · c2 ≡ 0 mod c when n is even, but ≡ c
2
when n is odd. us when c is even and d = ([ c2] + 1), all G chains begin and end on
PCs a half-modulus apart and will therefore have the half-modulus ambiguity.
As I discussed above, the DP property supports correlation of these two distance
measures to within the half-modulus ambiguity of even-numbered temperaments. In
systems where d < c2 there are multiple regions with zero PCs in common with a given
region, and in systems where d > ( c2+1), there are multiple regions with someminimum
number of PCs in common with a given region. e latter situation requires some extra
attention in establishing a region unambiguously. An extreme example is a 7-PC scale
embedded in 8, with scale order < 0,2,4,7,1,3,5− 0 > and gc= 1. is scale is
strictly ascending over two moduli, and has the properties G, WFγ , MPγ , and MEγ2.
Each of the eight regions has six PCs in common with every other region, so gc distance
is the only measure of distance between regions. At the other extreme, a region of a 7-
PC scale in 31 shares zero PCs in common with 18 regions.
Much more serious is that the 7-in-8 scale system fails to fulfill the Straus con-
ditions without extra care in disambiguation. Here is the entire scale system:
S 0 2 4 7 1 3 5 0
H 2 7 3 0 4 1 5 2
G 7 0 1 2 3 4 5 7
Its voice-leading ICs are 2 and 3, and its harmonic ICs are 1, 3, and 4. Given that 8
has only 4 ICs, the situation could be worse. It might be possible to distinguish the
voice-leading 3s from the harmonic 3s contextually, but extreme care would be needed
From Diatonic to Chromatic 181
G -1 0 1 2 3 4 5 -1S 0 2 4 -1 1 3 5 0
H 2 -1 3 0 4 1 5 2
G 6 5 4 3 2 1 0 -1 -2 6S 5 3 1 -1 6 4 2 0 -2 5S 0 2 4 6 -1 1 3 5 -2 0
H 2 6 1 5 0 4 -1 3 -2 2
Figure 2.39: Canonical templates for d = 7 and d = 9.
not to accidentally use [347], [470], [450], and [501] as consonant [014] harmonies.
Finally, the IC between the leftmost and rightmost members of the G chain is 2, which
is a voice-leading interval. None of this is a problem with a 7-in-31 scale, as it
would not come close to saturating the IC space.
is excursion brings up a subtlety in matching d with c . Figure 2.39 shows what
I will now call canonical templates for a given d and G → (S⊻S)→ H → G triple (one
template each for d = 7 and d = 9 in the figure; the triple is < 2,2,2 > in each case).
A canonical template begins with a G chain with gc= 1 in ascensive systems or gc=−1
in descensive systems. e integers are assigned and ordered so that tonic is 0. e
canonical template for a given diatonic system is universal, in that it can be embedded in
any chromatic with c > d through PCmultiplication by any integer in themultiplication
group of c ; the PCmultiplier becomes gc in ascensive systems, and in descensive systems
the PC multiplier’s mod c complement becomes gc .
One can learn a great deal about the interaction between d and c by retaining 1 as gc
in ascensive systems and−1 mod c as gc in descensive systems. Suppose that for d = 7,
I set c = 11; then the d = 7 template works out as follows:
P E T 182
G 10 0 1 2 3 4 5 10
S 0 2 4 10 1 3 5 0
H 2 10 3 0 4 1 5 2
PC intervals between adjacencies are 1 and 5 in G, 2 and 6 in S , and 8 and 4 in H .
ere is an ambiguity present here: the 5 from G and 6 from S are the same IC in
11. To see how to avoid this kind of ambiguity, recall that the presence of two PC
intervals in each cyclic ordering is a manifestation of Myhill’s property; one interval is
always results from a skip along the G chain that traverses only the gc interval, and the
traverses both gc and G’s secondary interval. In ascensive systems with gc= 1, the two
PC intervals for each ordering work out to be:
gc-only gc + secondaryG 1 c − d + 1S (G→ S) (G→ S)+ c − d
H (G→ S) · (S→H ) (G→ S) · (S→H )+ c − d
For descensive systems all of these intervals are the corresponding mod c comple-
ments. To look for an IC ambiguity between the secondary intervals of G and S , I can
set one equal to the complement of the other and then simplify:
c − d + 1 = c − ((G→ S)+ c − d )c = 2d − (G→ S)− 1
Any d and c pair with this relationship has G and S with the same IC as the secondary
interval. I can apply this formula to descensive systems as well because all of the intervals
are the mod c complements and the algebra works out the same. If I set d = 9, the
formula yields c = 15, and I can verify the ambiguity by inspection:
G 6 5 4 3 2 1 0 14 13 6
S 5 3 1 14 6 4 2 0 13 5
S 0 2 4 6 14 1 3 5 13 0
H 2 6 1 5 0 4 14 3 13 2
From Diatonic to Chromatic 183
PC intervals between adjacencies are 14 and 8 in G, 13 and 7 in S , 2 and 8 in S , and 4
and 10 in H .
Other formulas like this can be devised for finding IC ambiguities between voice-
leading intervals and harmonic intervals, noting that each harmonic interval in a system
can be found as a skip on the G cycle. e easiest course of action to find ambiguities
among the more complex interactions is likely just simple inspection, but in any case it
is useful to know that this kind of ambiguity can arise for some combinations of c and
d .
F T S
Suppose I want to write music in a given equal temperament; I can pair a chromatic
with a canonical template for a given scale system, do the requisite PC multiplication
mod c , and evaluate the results. is is time consuming, so I may wish to know how
to determine whether a scale cycle that has some set of desired properties exists in that
temperament. To illustrate, I explore two desirable properties here: the first applies to
scales where primary and secondary intervals of a cycle are of similar size; the second is
contour preservation, which I take up in the next section.
In my discussion of maximal evenness above (page 174), I noted that in a given
system, the two PC intervals between adjacencies in the G, S , and H cycles differ by
the same number in each. Let this number, formally defined as gc minus G’s secondary
interval, be called the system’s ∆ value. In order for adjacency intervals to be of similar
size, ∆ must be small.
In a G cycle, there are d − 1 instances of gc and one instance of (gc−∆), which is
the secondary interval of G. Since the sum of all the intervals of a cycle must≡ 0 mod c ,
we can write the formula (d−1) · gc+(gc−∆)≡ 0 mod c . Simplifying and moving the
∆ term to the other side of the equivalence, the result is∆≡ d · gc mod c . Furthermore,
P E T 184
the possible∆ values for a system are the multiples of the greatest common divisor of c
and d ; therefore the smallest possible ∆-class for a system is gcd(c , d ). is is the case
because an additive series of an integer k mod n tours the residues of the multiples of
gcd(k , n) mod n. Only when c and d are coprime can ∆ ≡ 1 or c − 1. If d |c , then
∆≡ d .57
A specific desired ∆ value for a given combination of c and d can be found by
enumerating the ∆ values yielded by each potential gc (viz. the integers of the mod
c system which are coprime to c), by computing a mod c multiplication table, or by
employing the extended Euclidean algorithm to find the multiplicative inverse of d .58
Usually c is small enough that direct enumeration is easiest.
F S T
Finding a contour-preserving scale for a given temperament is less easy, and one may
not exist. Even if the temperament is known, it is easiest to work from scale structure to
a temperament. Recall that what I have called a contour-preserving scale is one which
is strictly ascending through one modulus. is occurs when the integer sum of all of
the PC intervals between adjacencies of the S cycle equals c .59
In order to help guide the search, I will list a few more definitions and identities.
Let the primary scale interval be a and the secondary interval be b , and the number of
each in a scale be α and β, respectively. Because the secondary interval results from a
skip on the G cycle that crosses G’s secondary interval, and G→ (S⊻S) determines the
number of times the S ordering crosses the secondary interval of G, it follows that:
β= (G→ (S⊻S)) and d = α+β
To find a contour-preserving scale, we solve the equation:57d |c means “d divides c evenly.”58A demonstration is beyond the scope of this essay, but may be found in any text on number theory,
and in many online resources.59e PC intervals are taken mod c but the sum is not.
From Diatonic to Chromatic 185
c = a ·α+ b ·β, a ≡ b mod c
An equivalent equation is:
c = a · d −β · (a− b ), a ≡ b mod c
Since (a− b )≡∆ mod c , the equation can also be expressed:
c = a · d −β ·∆, ∆ ≡ 0 mod c .
Without considering other properties like deepness, the values for a and b yielding
the smallest chromatic cardinality are respectively 1 and 2 with ∆ ≡ −1 mod c . us
the smallest possible c for a given diatonic template is c = d +β = d + (G→ (S⊻S)).Because d is odd in most systems and the smallest value of d for any template is 7, if I
keep a = 1 and let b vary as b ≥ 2, it becomes clear that there is at least one contour-
preserving scale for every odd c ≥ 9.
If I have already picked a diatonic scale system, I know the values of d , α, and β,
and can let a and b vary to see what values of c it is compatible with. If I pick the 11-PC
Q system, then (G→ S) = 2, α= 9 and β= 2. I can then create a table like the one in
figure 2.40 to find the possible values of c for which the scale will be contour-preserving.
In the figure, there are contour-preserving systems for all c ≥ 13, except multiples of d
(22, 33, and 44), and even numbers between 14 and 18, inclusive. Rows count up by 2
(the β value) and columns count up by 9 (the α value), and the first row begins on 11
(the d value), which is ×ed out. If desired, I can also restrict myself to small absolute
∆ values, which yield c values to the left and right of the × diagonal; in this case they
are 13, 20, 24, 31, 35, 42, and 46.
All of this suggests a strategy for finding a contour-preserving scale for a known
value of c . First, I decide whether deepness is a priority. If so, I begin the search with
values of d ≈ c2 . If small values of ∆ are also desirable, it is quick to check whether
the desired c is a small multiple of α or of β larger or smaller than a multiple of a d of
appropriate size. If neither deepness nor small ∆ is necessary, the easiest way to work
P E T 186
d = 11; S→H → G→ S = < 3,2,2>; ascensive Q system
a↓ b→ 1 2 3 4 5 6 7 8 9 10
1 × 13 15 17 19 21 23 25 27 29
2 20 × 24 26 28 30 32 34 36 38
3 29 31 × 35 37 39 41 43 45 47
4 38 40 42 × 46 48 50 52 54 56
Figure 2.40: Computation of c values for contour-preserving 11-PC Q systems.
is fixing β (probably to 2 or 3, since those are the most prevalent G → (S⊻S) values).en if I count down from c by β, I can inspect the resulting numbers for multiples
of known values of α that correspond with my β, recalling that the smallest possible
α is 5 when d = 7. For example, if I set c = 21 and β = 2, I can search the vector
< 19,17,15,13,11,9,7,5 > for multiples of possible α values, such as 5, 7, 9, 11, etc.
I need to watch out for values that set a = b , though. Suppose I inspect the value
15 = 5× 3 (this sets a = 3), the difference between 21 (the c value) and 15 is 6, and
since β= 2, b = 3 as well.
One final point is worth mentioning in connection with finding c from a scale struc-
ture. When I choose a diatonic system and values for a and b , and calculate the sum
of all the intervals, I will have found a c that is contour preserving. In addition, that
sum also ≡ 0 mod every divisor of c . For example, let d = 13, α = 10, β = 3, a = 5,
and b = 4. e c for which this is contour-preserving is 10× 5+ 3× 4 = 62, but this
combination of values also works out for c = 31, just not in a contour-preserving way.
e 31 template is:
S 0 5 10 15 19 24 29 3 7 12 17 22 27 0
H 10 24 7 22 5 19 3 17 0 15 29 12 27 10
G 7 19 0 12 24 5 17 29 10 22 3 15 27 7
From Diatonic to Chromatic 187
F H T S
If I want to find a temperament and scale system that accommodate a given harmony as
a maximal consonance, this is a far more difficult problem. An example will best serve to
show why. Suppose I want a scale system where a maximal consonance when built in its
most closed form sounds as close to a combination of the 4th, 5th, 6th, 7th, and 9th partials
of a given pitch as possible. Many of the usual methods of finding equal temperaments
that approximate just intonation of this kind will not necessarily be sufficient for my
purposes because it is not guaranteed that a generated scale which also embeds this
harmony with equal diatonic intervals can be formed in a given temperament.
I proceed as follows. First, to get a sense of what will count as a match, I should
calculate the intervals between adjacent pitches in cents. e formula to convert a fre-
quency ratio to cents is ¢ = 1200 · log(ρ)log(2) , where ρ is the frequency ratio and log(x) is the
base 10 or the base e logarithm.60 ese values (rounded to the nearest cent) are:
ρ ¢54 38665 31676 26797 435
is variation in interval size must be balanced against the fact that PC-intervals be-
tween adjacencies in H come in only two sizes, so that the equal-tempered interval
representing 54 will need to be the same as that representing 9
7 ; the ratios 65 and 7
6 must
also be tempered to the same interval.
So far I have used “primary interval” to indicate the PC-interval between adjacencies
that appears most often in an S , H , or G cycle, and “secondary interval” for the other
interval. is relationship can be misleading in H cycles. Above, I defined ∆ as gc
60e ratio of log() values calculates log2(ρ). In general, logn(x) =log(x)log(n) .
P E T 188
minus the secondary interval of G mod c , and that each cycle of a system has the same
∆. For a major scale in 12, gc is 7 and G’s secondary interval is 6, so ∆ ≡ 1.
is relationship holds for the S ordering: the primary interval 2 is 1 larger than the
secondary interval 1. In the H cycle, however, the interval 3 appears four times while
the interval 4 only appears three times. If 3 is the primary interval, then ∆ ≡ 11 mod
12. To solve this discrepancy, henceforth “primary interval” and “secondary interval”
for H cycles indicate the intervals that preserve the∆ value calculated from the G cycle
rather than the most and least prevalent intervals, when there is a discrepancy. is
would make 4 the primary interval and 3 the secondary interval in the major scale’s H
cycle.
In the scale systems under consideration, the number of maximally consonant Tn-
types is equivalent to H → G. H → G “unfolds” the H cycle into the G cycle in the
following way: every skip along the H cycle that yields a gc interval contains exactly
one of H ’s secondary interval, and the one skip containing two of H ’s secondary inter-
val yields G’s secondary interval.61 is means that the pattern made by primary and
secondary intervals in H along most of its length must occur with a period p equivalent
to H → G. A consequence of this is that every p-sized segment of H ’s interval cycle,
the chromatic span of which ≡ gc , can have the secondary interval in only one of the p
places. For instance, if p = 3 and the primary and secondary intervals of H are labeled e
and f , respectively, then there are three different possible p-sized segments e-e-f, e-f-e,
and f-e-e, placing the secondary interval f in each of the three places. ese p patterns
form the first p intervals of every primary harmony, and even if a harmony extends for
more intervals than p, the periodicity of H ensures that there are still only p Tn-types
that may be used for maximal consonances.61∆G ≡∆H , so including one less of H ’s primary interval and one more of its secondary interval will
necessarily yield G’s secondary interval.
From Diatonic to Chromatic 189
Using the tempered interval equivalences for my example harmony, I can see they
are in the pattern f-e-e-f, which means that for it to be used as a maximal consonance,
(H →G) = 3. e scale system I use not only requires (H →G) = 3, but also must set
h = 5. One 11-PC scale matches; it is the one I used for the 20 interlude above. It
is an ascensive T system, with canonical template:
G -1 0 1 2 3 4 5 6 7 8 9 -1
S 0 2 4 6 8 -1 1 3 5 7 9 0
H 2 6 -1 3 7 0 4 8 1 5 9 2
e form of H ’s interval chain is e-f-e-e-f-e-e-f-e-e-f, and since my target harmony
is the form f-e-e-f with smaller intervals in the middle, I must set e to the smaller of
the two intervals and f to the larger. e target harmony is of a different SC from the
other two primary harmonies (with intervals e-e-f-e and e-f-e-e). I might call e-e-f-e
“major,” e-f-e-e “minor,” and f-e-e-f “large,” since its containing interval is a ∆ larger
than the others. As such, the 0 value in the templates, which belongs to the major
harmony, is not actually the root of our target; a different mode is required as in the
mixture examples in the interlude above. Everything is now in place to start the search
for c .
e first thing to do is to imagine the entire H cycle written out in ascending pitch
in order to get a baseline measurement of the amount of tempering needed: assuming
octave equivalence, if the interval spanning this ascending H differs from an integer
number of octaves, the difference is the interval I will need to divide and distribute
among the primary and secondary intervals of H to reach an exact octave multiple. In
the current H I have 7 of the smaller intervals and 4 of the larger, and I can get low and
high limits by using the smallest and largest of the four ¢ values for the calculation. e
low limit is 7×267+4×386= 3413¢, and the high limit is 7×316+4×435= 3952¢.
ere is an exact multiple-octave interval between these two values, which is 3600¢, so
P E T 190
the eventual sizes of small and large intervals will need to produce an H cycle with a
three-octave range.
e next step is to note that the ratio between interval sizes in cents (not in fre-
quency) can be used to control the ratio of smaller equal tempered intervals in order to
test for a temperament that will match the desired tuning. A pair of intervals of the
appropriate ratio can be multiplied and summed according to number of occurrences of
the intervals in H ; and knowing that the final sum traverses three octaves, if the sum is
a multiple of three, then the sum divided by three yields c . e formula is 7e+4 f = 3c
where e and f are the primary and secondary intervals. e target ratio between inter-
vals must fall somewhere between 267435 ≈ 0.614 and 316
386 ≈ 0.819. Figure 2.41 provides
the relevant calculations and results if they exist.
e solution set for c is 16, 21, 31, 36, 47 for c < 50. Each chromatic cardinality has
a slightly different tuning for e and f , and a judgment of which to use could be made by
ear. ere are other practical considerations to make involving deepness and cognitive
constraints on pitch discrimination; contour-preservation is not an option because with
intervals of such similar sizes in H , the scale systems are cognates of one another. e
31 example is intriguing because 31 is often associated with just intonation;
its intervals are closer in size to the 9:7 major third and the 7:6 minor third than the
5:4 and 6:5 frequency ratios it is usually utilized for. 36’s interval sizes are even
closer. 16’s are the closest to 5-limit just intonation. A generated 11-PC set is deep
in 21. Completed templates for each system appear in figure 2.42. I have chosen
orderings for the S cycles that produce an overall ascent, even if the primary interval is
large with respect to c .
Another option comes to mind: recall that the total ascent of H must sweep a range
between 3413¢ and 3952¢. e tritave, with frequency ratio 3:1, spans about 1902¢;
two tritaves at 3804¢ also falls in the target range. I can amend my formula for finding e
From Diatonic to Chromatic 191
e fef 7e+ 4f c= 7e+4f
3 e ¢ f ¢
3 4 0.750 37 × × ×4 5 0.800 48 16 300 375
5 7 0.714 63 21 285 400
5 8 0.625 67 × × ×7 9 0.778 85 × × ×7 10 0.700 89 × × ×7 11 0.636 93 31 271 426
8 11 0.727 100 × × ×8 13 0.615 108 36 267 433
9 11 0.818 107 × × ×9 13 0.692 115 × × ×9 14 0.643 119 × × ×10 13 0.768 122 × × ×11 14 0.786 133 × × ×11 15 0.733 137 × × ×11 16 0.688 141 47 281 409
Figure 2.41: Table of calculations to find c for the target harmony.
P E T 192
c = 16G 3 0 13 10 7 4 1 14 11 8 5 3S 0 10 4 14 8 3 13 7 1 11 5 0
H 10 14 3 7 11 0 4 8 13 1 5 10
c = 21G 4 0 17 13 9 5 1 18 14 10 6 4S 0 13 5 18 10 4 17 9 1 14 6 0
H 13 18 4 9 14 0 5 10 17 1 6 13
c = 31G 6 0 25 19 13 7 1 26 20 14 8 6S 0 19 7 26 14 6 25 13 1 20 8 0
H 19 26 6 13 20 0 7 14 25 1 8 19
c = 36G 7 0 29 22 15 8 1 30 23 16 9 7S 0 22 8 30 16 7 29 15 1 23 9 0
H 22 30 7 15 23 0 8 16 29 1 9 22
c = 47G 9 0 38 29 20 11 2 40 31 22 13 9S 0 29 11 40 22 9 38 20 2 31 13 0
H 29 40 9 20 31 0 11 22 38 2 13 29
Figure 2.42: Diatonic templates derived from the calculations in figure 2.41.
From Diatonic to Chromatic 193
e fef 7e+ 4f c= 7e+4f
2 e ¢ f ¢
4 5 0.800 48 24 317 396
8 11 0.727 100 50 304 418
8 13 0.615 108 54 281 458
10 13 0.768 122 61 312 405
Figure 2.43: Table of calculations to find c for the target harmony in tritave equivalence.
and f to 7e+4 f = 2c , and look for multiples of 2 in the total-sum-of-intervals column
in figure 2.41. ose values are 48, 100, 108, and 122, with c values 24, 50, 54, and
61. Figure 2.43 has the details. e 24{3:1} system is the best from an intonation
standpoint, and also fits the 11-PC scale the best. Its scale template is:
c = 24{3 : 1}G 11 0 13 2 15 4 17 6 19 8 21 11
S 0 2 4 6 8 11 13 15 17 19 21 0
H 2 6 11 15 19 0 4 8 13 17 21 2
e scale is also contour-preserving, and would make another excellent acoustically mo-
tivated tritave alternative to the Bohlen-Pierce scale.
e theory-building part of this essay is now complete, but the theory itself extends
beyondmy present scope. In the final section I propose future research, with connections
to other areas of interest within compositional theory.
194
C I
M of prolongation in equal temperament explored in this essay brings to-
gether ideas from traditional and reconstructed theories of prolongation, neo-Riemann-
ian and tonal function theory, and scale theory. I can not possibly have addressed every
connection of my theory to those, and I have deliberately avoided most questions of
acoustics, aesthetics, and cognition. Some concluding remarks give me the opportunity
to suggest topics of future research in microtonal prolongation.
First, the generalized scale properties I introduced in the previous section (MEγ2,
MPγ , etc.) deserve a more formal treatment, and there are a number of interesting
features of the structure of my diatonic templates that will profit from a more detailed
account. For instance, I have implied that the dominant harmony is always the same
Tn type as the tonic in these systems, but have not proved this to be the case.62 I have
also set my templates up so that the rightmost member of the G chain is the leading
tone (lower or counterclockwise neighbor to the tonic PC), and have always preserved
its place at the end of the cycle in both S and H . ere are theoretical reasons for
these practices that extend beyond the heuristic; I have not yet pointed out that the
lower/counterclockwise neighbor of the dominant PC is always the leftmost member
of a template’s G chain, for instance. A full account of these properties will require a
substantial paper, and promises further integration of scale theory ideas into the formal
bases of generalized prolongation.62An intuitive approach would show that the roots of harmonies of the same quality form contiguous
segments on the G chain, and by definition the dominant and tonic roots are always separated by gc .
195
Second, there are thorny methodological and theoretical issues surrounding contour
non-preserving systems. Prolongation is in principle a system of transformations which
operate in pitch and time alone. e four voices of traditional voice leading are often
contextually defined in prolongational theory: they enter and exit as needed, and pitches
can belong to different voices in different levels even when they pass without change
from one level to another. is ability to distinguish voices contextually is one feature of
Schenkerian theory that has made it so productive and flexible as an analytical theory. A
Schenkerian approach to a Bach chorale, for instance, would show that the four surface
voices are themselves compound lines with many structural voices interacting at once,
and lines from a single structural voice can traverse many surface voices. is is even
more obviously the case in orchestrated music. And yet, the theory loses some of its
nuance when it does not take orchestration into account. For instance, orchestration
can help disambiguate which line is the structural soprano and which are middle voices
projected into an upper register, and basing an analysis on an overly literal keyboard
transcription is a problematic approach.
Future research on contour non-preserving systems will best approach the topic by
showing some examples of structural voices articulated by features of music other than
pitch and time in tonal music. A number of contour non-preserving systems can then
be investigated by showing points of voice ambiguity through an analysis of a precom-
posed surface generated prolongationally using such a system. en, once the prob-
lems are known, various systematic approaches to disambiguation and articulation of
structural voices can be investigated to solve these problems. A full account of the per-
missible transformations could then be devised for different kinds of related systems.
ere is also opportunity for connecting contour non-preserving systems to vakra ra-
gas in Carnatic music, which are ragas whose ascending and/or descending scale orders
take zig-zag patterns.63 Chitravina N. Ravikiran’s melharmony concept, which extends63Bhagyalekshmy, Ragas in Carnatic Music, 1990, pages 51-53
C I 196
the traditionally monophonic Carnatic tradition by developing harmonic theories for
Carnatic ragas, could also provide many points of connection.64
ird, my framework expands the space of usable scales and harmonies, so a natural
compositional question is whether more than one structurally different scale can be used
coherently in the same piece. For instance, could there be a background created using
one M-form of a scale, with local simultaneities composed out using a related M-form?
If so, this could be an approach to coherent prolongation of dissonance, and would
also provide a compositional approach to further hierarchization of tonal systems, and
possibilities for embedding one in another. Other questions might address changing
the n- of a piece – or even the modulus – over time or across levels.
Fourth, and finally, I anticipate that readers who are least convinced by the results
of this essay will base their criticism on cognitive grounds: to what extent is it possible
to hear the harmonic simultaneities that result from multiplicative permutation of a G
chain as consonances? Are there cognitive limits on the size of d or c that inhibit the
ability to perceive syntactic operations on them? I plan to address these and related
questions in a later paper. William Sethares and others have shown that the perception
of xenharmonic tuning has as much to do with the timbres used to realize a tuning
as the tuning itself.65 Because microtonal music is so often synthesized electronically,
and because synthesis also offers enormous timbral flexibility, this research will focus
on electroacoustic approaches to realizing the phonology of these systems so that they
make the syntactic structures as clear as possible from a cognitive standpoint. As with
the present essay, it is my hope that this integration of different areas of research will
give composers and scholars new ways to compose and to think about music.
64Morris and Ravikiran, “Ravikiran’s Concept of Melharmony,” 2006.65Sethares, Tuning, Timbre, Spectrum, Scale, 2005.
197
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