a theory of voice-leading sets for post-tonal music
TRANSCRIPT
A Theory of Voice-leading Sets for Post-tonal Music
by
Justin Lundberg
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor David Headlam
Department of Music Theory
Eastman School of Music
University of Rochester
Rochester, NY
2012
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Curriculum Vitae
Justin Lundberg was born in Eau Claire, Wisconsin on May 4th
, 1982. He received a
Bachelor of Music in Music Theory from the University of Wisconsin-Eau Claire in
2004, where he was a national merit scholar. In 2006 he received a Master of Arts in
Music Theory from the Pennsylvania State University, where he was a Graham
fellowship recipient. His Master’s thesis, on sonata form in three late works by
Joseph Haydn, was supported by a summer residency at the Institute for the Arts and
Humanities. He began pursuing the Doctor of Philosophy in Music Theory at the
Eastman School of Music in 2006, and received a Raymond N. Ball dissertation
fellowship in 2010-2011.
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Acknowledgements
I wish to thank my friends and family for their support during the preparation of this
dissertation, as well as throughout my graduate studies. In addition, I am grateful to
my professors and classmates at Eastman who offered their encouragement, criticism,
comments, and suggestions during various stages of my work. This dissertation was
also supported in part by a Raymond A. Ball fellowship. I wish to thank the scholars
to whom the ideas presented in this dissertation are most indebted: John Roeder,
Joseph Straus, and Dmitri Tymoczko. Finally, I would like to thank my advisor,
Dave Headlam. I first began to explore the subject of post-tonal voice leading in
earnest in an independent study with him in the summer of 2007, and am grateful for
the enthusiasm, support, and patience he continues to provide.
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Abstract
This dissertation develops tools for the analysis and representation of voice leading in
post-tonal music. A voice leading is defined as an ordered series of mappings of one
pitch-class set onto another. Because voice leadings are represented by ordered sets,
they are independent of the harmonic structure of the pitch-class sets they span.
Further, each mapping in a voice-leading set represents an individual pitch-class
voice. The dissertation presents a theory of voice leading that first asserts that voice-
leading sets may be organized into set-classes under transposition, inversion, and/or
rotation, then, following a discussion of the groups formed by voice-leading sets,
demonstrates the construction of voice-leading spaces based on individual voice-
leading sets, as well as reduced spaces which represent voice-leading classes. A
number of distinct spaces are explored, with an emphasis on the complete spaces, that
is, spaces which include all voice-leading sets or classes in a given cardinality.
Selected analyses demonstrate that the diverse pitch and intervallic material of post-
tonal pieces may be unified by the consistent use of a small number of voice-leading
classes. The voice-leading spaces formed by these classes may then be used to model
the voice-leading transformations of specific pieces. The dissertation concludes by
suggesting a few avenues of future research.
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Table of Contents
Chapter 1 Introduction 1
Introduction to Voice-leading Sets 3
Background 9
Chapter 2 Voice-leading Set Theory 19
Contrapuntal Motion in Pitch-class Space 22
Voice-leading Set-classes 26
Voice-leading Group Theory 33
Group Representations 39
Chapter 3 Voice-leading Spaces 44
Dyadic Spaces 47
Trichordal Spaces 65
Chapter 4 Analysis 74
Arnold Schoenberg’s Op. 11 n. 1 79
Anton Webern’s Op. 5 n. 2 82
Anton Webern’s Op. 5 n. 3 89
Alban Berg’s Op. 5 n. 1 96
Alban Berg’s Op. 5 n. 2 101
Chapter 5 Summary, Conclusion, and Future Work 106
Bibliography 221
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List of Tables
Table 2-1 Subgroups of the group of dyadic vlsets 138
Table 2-2 Group table for the subgroup generated by <17> 139
Table 2-3 Subgroup <02> with inversion 141
Table 2-4 Examples of four-member subgroups of [C12 C12] C2 142
Table 2-5 Examples of two-member subgroups of [C12 C12] C2 143
Table 2-6 C12 C12 subgroups combined with inversional vlsets 143
Table 2-7 Group table for {<04>, <40>} + <61>i 146
Table 3-1 Dyadic vlset spaces 164
Table 3-2 Trichordal voice-leading spaces 177
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List of Figures
Figure 1-1 Mm. 11-13 of Arnold Schoenberg’s Op. 11 n. 2 119
Figure 1-2a Fuzzy transposition as in Straus 2003 119
Figure 1-2b Pitch voice leadings as in Roeder 1994 119
Figure 1-2c Vlsets from chord a to b and d to e 120
Figure 1-3 Values for the moving voice when retaining two common tones 121
Figure 1-4a The trichordal orbifold 122
Figure 1-4b One-quarter of the trichordal orbifold 123
Figure 2-1 Number of vlsets by cardinality 124
Figure 2-2 Vlsets spanning reordered pcsets 125
Figure 2-3 Parallel pitch-class voice leading 126
Figure 2-4 Oblique pitch-class voice leading 126
Figure 2-5 Contrary pitch-class voice leading 126
Figure 2-6 Similar pitch-class voice leading 127
Figure 2-7 Vlset transposition 127
Figure 2-8a Mm. 1-13 of Arnold Schoenberg’s Op. 11 n. 1 128
Figure 2-8b Vlclass [002] between ordered trichords 128
Figure 2-9 Ordered pcset tn-classes paired by ordered vlclasses 129
Figure 2-10 Ordered tn-class cycles 131
Figure 2-11 An ordered tn-class [04] tn-class cycle 132
Figure 2-12 Ordered pcset and tn-class cycles formed by vlset <14> 132
Figure 2-13 Transpositions of vlset <01>i and <0e>i 133
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Figure 2-14 Ordered tn-class paired by ordered vlclass [01]i and [10]i 134
Figure 2-15 Ordered pcset-classes paired by ordered vlclasses [04]i and [40i] 135
Figure 2-16 Tn-class pairs 136
Figure 2-17 Ordered tn-classes paired by ordered vlclass [025] 137
Figure 2-18 Subgroup <17> with three sets of pcsets as objects 140
Figure 2-19 Cyclic groups and rotational symmetry, C3: T0, T4, T8 148
Figure 2-20 Dihedral groups, D3: T0, T4, T8, In, In+4, In+8 148
Figure 2-21 T/I group representation 149
Figure 2-22 The dodecagonal torus 150
Figure 2-23 Interlocking dodecagons 151
Figure 2-24 Vlset <t27> on interlocking dodecagons 152
Figure 3-1 Straus’ Trichordal set-class space 153
Figure 3-2a The dyadic orbifold 154
Figure 3-2b Dyadic permutational equivalence 154
Figure 3-2c Dyadic Moebius strip 155
Figure 3-3 Torus knot of singleton dyads 156
Figure 3-4a Moebius strip dyadic orbifold 157
Figure 3-4b Dyadic set-class line 158
Figure 3-4c Dyadic tn-class circle 158
Figure 3-5a <05> space 158
Figure 3-5b <05> Moebius strip 159
Figure 3-5c Adjacent voice leadings along the <05> Moebius strip 160
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Figure 3-5d Adjacent and compound moves along the <05> Moebius strip 160
Figure 3-6a <05> tn-class circle 161
Figure 3-6b <05> set-class line 161
Figure 3-7a <2e>, {00} toroidal voice-leading space 162
Figure 3-7b <2e>, {00} toroidal voice-leading space 163
Figure 3-8 <02>, {00} toroidal voice-leading space 165
Figure 3-9 <02>, {11} toroidal voice-leading space 165
Figure 3-10 <02>, {01} toroidal voice-leading space 166
Figure 3-11 <02>, {10} toroidal voice-leading space 166
Figure 3-12 <02>, {11} Moebius strip 167
Figure 3-13 Vlset <02> spaces 168
Figure 3-14 <68>, {00} toroidal voice-leading space 168
Figure 3-15 <67>, {00} toroidal voice-leading space 169
Figure 3-16 <67> Moebius strip 170
Figure 3-17 <12>, {47} toroidal voice-leading space 171
Figure 3-18a <42>, {11} toroidal voice-leading space 171
Figure 3-18b <42>, {11} toroidal voice-leading space 172
Figure 3-19 <12>, {47} Moebius strip 173
Figure 3-20 <42>, {11} permutation space 174
Figure 3-21 <06>, {11} permutation space 174
Figure 3-22 <06>, {01} voice-leading space 174
Figure 3-23 <57>, {02} permutation space 174
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Figure 3-24 <26>, {11} toroidal voice-leading space 174
Figure 3-25 <001>, {000} hypercube 175
Figure 3-26 The trichordal orbifold 176
Figure 3-27a Vlclass [047] ordered tn-class space 179
Figure 3-27b Motion among adjacent ordered tn-classes in [047] ordered
tn-class space 179
Figure 3-28 Vlclass [047] ordered set-class space 180
Figure 3-29 Examples of complete ordered set-class spaces 181
Figure 3-30 Vlclass [002] ordered set-class spaces 184
Figure 4-1a <442>, {000} voice-leading space 185
Figure 4-1b <442>, {001} voice-leading space 186
Figure 4-1c Vlclass [002] whole-tone ordered set-class space 187
Figure 4-1d Vlclass [002] mixed whole-tone ordered set-class space 188
Figure 4-1e Ordered set-class interpretations from Figure 2-8 189
Figure 4-2a [026] in Webern’s op. 5 n. 2 190
Figure 4-2b Network of [026]s in Webern’s op. 5 n. 2 190
Figure 4-3 Melodic statements of vlclass [043] in Webern’s op. 5 n. 2 191
Figure 4-4 Network of melodic vlsets in Webern’s op. 5 n. 2 192
Figure 4-5 Vlclass [043] in the accompaniment in Webern’s op. 5 n. 2 193
Figure 4-6 Vlclass [043] transformation graph 195
Figure 4-7a Webern’s op. 5 n. 3, mm. 1-4 196
Figure 4-7b Vlset <t16> mappings 196
Figure 4-8 Variations of A 197
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Figure 4-9 B, mm. 9-10 198
Figure 4-10a [038] voice-leading space 199
Figure 4-10b Melody B’s complete voice-leading {2t1} – {0et} – {t59} 200
Figure 4-11a Reduction of opening [034]s in Violins and Viola, mm. 1-3 201
Figure 4-11b Reduction of Violin II, Viola, and Cello, mm. 15-17 201
Figure 4-12 Berg’s op. 5 n. 1 opening clarinet gesture 202
Figure 4-13a Berg’s op. 5 n. 1 clarinet melody 203
Figure 4-13b Vlclass [002] ordered set-class space 204
Figure 4-14 Vlclass [022] in m. 2 LH 205
Figure 4-15 Vlclass [022] in m. 2 RH 206
Figure 4-16 Opening clarinet gesture in the accompaniment, m. 9 206
Figure 4-17 Accompaniment, mm. 3-5 207
Figure 4-18a Vlclass [002] in mm. 5-6 208
Figure 4-18b Vlclass [002] in m. 7 208
Figure 4-19 Berg’s op. 5 n. 2 209
Figure 4-20 Berg’s op. 5 n. 2 melody 210
Figure 5-1 Pentachordal vlsets in Berg’s op. 5 n. 2 211
Figure 5-2 Normal order interpretation of Berg’s op. 5 n. 2 melody 213
Figure 5-3a Vlsets on the trichordal orbifold 214
Figure 5-3b Vlsets on the trichordal orbifold 215
Figure 5-3c Vlsets on the [013] ordered set-class space 216
Figure 5-4 Hyper-vlset graph 217
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Figure 5-5 Alban Berg’s Lyric Suite, VI, mm. 1-6 218
Figure 5-6 Lyric Suite, VI, mm. 39-46 219
Figure 5-7 4-3 suspension as diatonic vlset 220
Figure 5-8 Transposition of 4-3 diatonic vlset 220
1
Chapter 1: Introduction
The study of voice leading has long been a prominent concern of music
theory. Although there are many differences between theories of voice leading in
tonal and post-tonal music, almost all emphasize either or both the harmonic
constraints and the minimal melodic distances available. In Schenkerian theory,
analytically-defined voices demonstrate the prolongation of tonic harmony at multiple
levels of structure. Neo-Riemannian theory focuses on harmonic successions which
have been described by Richard Cohn (and others) as parsimonious, or smooth voice
leadings; this voice-leading smoothness is inextricably linked with the harmonic
structure of tonal materials—triads and seventh chords.1
In post-tonal theory, studies of voice-leading typically focus on harmonic
similarity, as in the transformational voices formed by transposition and inversion, or
on total interval-class displacement.2 The more recent geometrical approach to voice
leading, as shown in the work of Clifton Callender, Ian Quinn, and Dmitri Tymoczko,
creates “voice-leading spaces,” by mapping pitch or pitch-class sets onto dimensional
coordinates with the semitone as a metric.3 This mapping produces minimal voice
1 “Maximally Smooth Cycles, Hexatonic Systems and the Analysis of Late-Romantic
Triadic Progressions” in Music Analysis, 15, n. 1 (1996): 9-40 and “Neo-Riemannian
Operations, Parsimonious Trichords and Their Tonnetz Representations” in Journal of
Music Theory 41, n. 1 (1997): 1-66. 2 See, for example, John Roeder’s “A Theory of Voice Leading for Atonal Music,”
(Ph.D. Diss., Yale University, 1984) and Joseph Straus’ “Uniformity, Balance, and
Smoothness in Atonal Voice Leading” in Music Theory Spectrum, 25, n. 2 (2003):
305-352. 3 Clifton Callender, Ian Quinn, and Dmitri Tymoczko “Generalized Voice-leading
Spaces” in Science, 320, (2008): 346-348, and Dmitri Tymoczko “The Geometry of
Musical Chords” in Science, 313 (2006): 72-74. Robert Morris “Voice-leading
2
leading between adjacent chords. In this dissertation, however, instead of basing the
metric for voice leading on a fixed unit, voice-leading distances will be redefined as
contextual distances and relative measures. This transformational approach to voice
leading follows David Lewin’s more generalized notions of “interval” as some
distance between two points in a musical space.4
The approach to post-tonal music in this dissertation defines voice leadings as
ordered sets of individual pitch-class mappings from one pitch-class set to another.
Although the possible voice-leading interpretations between two chords are
constrained by their pitch-class content, no particular interpretation is privileged a
priori. Voice leadings are thus extracted from their harmonic contexts and examined
on their own terms. Individual voice-leading sets are used to generate alternative
voice-leading spaces, in which the unit distance or metric is the generating set rather
than semitonal offset. The analytical goal of this theory is to define pitch-class voices
in post-tonal pieces. In the analyses shown here, voice-leading sets are used to unify
the pitch and intervallic diversity that characterizes the surface of many post-tonal
pieces. These motivic voice-leadings are then represented by motion within some
contextually-generated voice-leading space. After a brief introduction to voice-
leading sets, this chapter continues with a survey of secondary literature. Chapter 2 is
Spaces” in Music Theory Spectrum, 20, no. 2 (1998): 175-208. The authors’ build on
earlier work by Robert Morris and John Roeder, see Morris’ “Voice-leading Spaces”
in Music Theory Spectrum, 20, no. 2 (1998): 175-208 and Roeder’s “A Geometric
Representation of Pitch-class Series” in Perspectives of New Music, 25, no. 1/2
(1987): 362-409. 4 See, for example, Generalized Musical Intervals and Transformations (New Haven:
Yale University Press, 1987, reprint New York: Oxford University Press, 2007): 16.
3
devoted to the properties and group structure of voice-leading sets, and chapter 3
explores some of the voice-leading spaces these sets create. Chapter 4 demonstrates
the analytical applications of voice-leading sets and spaces, and chapter 5 suggests
some avenues for future research.
Introduction to Voice-leading Sets
In order to discuss voice leading, we must first describe the concept of a voice.
Traditionally, a voice is an actual voice-part, either a soprano, alto, tenor, or bass,
with a defined register and range. This concept is extended to instrumental music; a
voice is defined by the individual instrument or section. Just as in vocal music,
voices are distinguished by register and timbre.
There is a long history of analytical interpretations which differ from the
traditional definition of voices. Some of these analysts approach voices from a
harmonic perspective, demonstrating the parsimonious motions formed by harmonic
reductions. Others demonstrate linear connections formed over longer time-spans. In
Ernst Kurth’s discussion of polyphonic melody, he distinguishes between
Realstimmen, or actual voices, and Scheinstimmen, or apparent voices. Realstimmen
are formed by the actual sequence of notes on the score, and Scheinstimmen are
voices formed by registral connections. Kurth demonstrates, for instance, that in mm.
21-30 of the Allegro of Bach’s Violin Sonata n. 3, the solo violin portrays a G pedal
point as well as a lyrical melody in the upper register.5
5 Ernst Kurth: Selected Writings, ed. and translated by Lee Rothfarb, (New York:
Cambridge University Press, 2008), 81.
4
The concept of polyphonic melody is central to Schenkerian analysis. The
voices defined by analysis demonstrate the overarching prolongation of the tonic
triad. Voices are defined by step-wise motion, and a number of techniques and
concepts, such as reaching-over, register transfer, coupling, and cover-tone, allow for
octave equivalence. If the tonic triad at the highest level is viewed as having three
voices—root, third, and fifth—all foreground and middleground material expresses
these voices through the transformations shown in the analysis—arpeggiation,
neighbor motion, linear progressions, etc. The end result of analysis is to demonstrate
the unique way in which the tightly bound voice leading of the Ursatz is unraveled to
form the musical surface. 6
Joseph Straus incorporates the concept of polyphonic melody into his
discussion of voice leading in post-tonal music, distinguishing between a voice and a
line. For Straus, a line is determined by register, while a voice is formed by mapping
one pitch-class onto another through transposition or inversion. Straus describes this
mapping as transformational voice leading, which projects an underlying “pitch-class
counterpoint.” 7
William Benjamin discusses this concept in the context of tonal
music, stating that a harmonic progression is made up of pc lines, while traditional
6 An important distinction between Schenkerian and post-tonal voice leading is the
accompanying harmonic prolongation through the levels of a Schenkerian analysis.
Straus has argued that post-tonal voice leading is associational, rather than
prolongational, by which he means that we make associations between non-adjacent
notes, rather than connect them by prolongation, in “The Problem of Prolongation in
Post-tonal Music” Journal of Music Theory, 31, n. 1 (1987): 13-15. 7 As Straus 2003 points out, the concept of “pitch-class counterpoint” is from William
Benjamin’s “Pitch-class Counterpoint in Tonal Music” in Topics in Music Theory,
Richmond Browne, editor (New York: Academic Press, 1981): 4.
5
counterpoint is made up of pitch lines. A harmonic reduction, therefore,
demonstrates the pc counterpoint formed by harmonic motion underneath the musical
surface.8
In Straus’ distinction between voice and line, voices are analytical choices.
Because voices are formed by mappings, voice-leading analysis defines voices. The
idea of transformation, or a mathematical mapping, as voice leading originates in the
work of David Lewin. For example, in a discussion of the opening two chords of
Schoenberg’s Op. 19, n. 6, Lewin describes all of the transposition and inversion
operations that map at least two notes of the first chord onto two notes of the second
chord. The two chords are not members of the same set-class, so there is no
transposition or inversion that maps all three members onto three members of the
following chord. Lewin then shows how the potential of some of these near
transformations is realized later in the piece.9
In the analysis of post-tonal music, Straus follows up on Lewin’s earlier work
by proposing that voice leadings be described according to how transposition- or
inversion-like they are. Three criteria, uniformity, balance, and smoothness, evaluate
voice leadings. Transposition results in voice leading that is uniform, as all voices
move by the same interval, while inversion produces balanced voice leading, in that
all voices flip around the same abstract axis of symmetry. Uniformity and balance
8 Benjamin 1981: 1-32. For an extended analytical example, see Benjamin’s
“Debussy’s ‘pour les Sixtes’: an Analysis,” in Journal of Music Theory, 22, n. 2
(1978): 253-290. 9 “Transformational Techniques in Atonal and Other Music Theories” in Perspectives
of New Music, 21, n. 1/2 (1982-1983): 336-342; see especially examples 19 and 20 on
pp 339 and 340, respectively.
6
both involve the concept of fuzzy transformations, that is, transformations which are
some measurable distance from the crisp transformations they resemble.10
Smoothness is simply semitonal distance—the sum of the interval-classes traversed
by the voices. Pitch-class mappings that are near-transpositions, such as pcsets {014}
and {015}, have a high degree of uniformity; in this case, {014} maps onto {015} by
T0 with one semitone offset. Balance is similar: pcset {016} is nearly inverted by It
to pcset {48t}. Straus uses an asterisk to show that the transformation is not “pure,”
and lists the number of offsets in parentheses; the two previous examples are labeled
*T0 (1) and *It (1), respectively.11
Figure 1-1 shows five chords, labeled a-e, from Arnold Schoenberg’s Drei
Klavierstuecke Op. 11 n. 2, mm. 11-13.12
Straus’ method for describing voice
leadings may be applied to this passage by considering the harmonic structures of the
chords, and the pitch-class transformation(s) that most closely approximate “crisp”
transposition or inversion. Chords a and b are not members of the same set-class, but
both contain a member of [036]. As Figure 1-2a shows, the C# diminished triad in
chord a is mapped onto the B diminished triad in chord b by Tt, while the singleton
10
Ian Quinn, “Fuzzy Transposition of Pitch Sets” Paper presented at the Society for
Music Theory Conference, Baton Rouge (1996), cited in Straus 2003. 11
Straus 2003, 314-320. 12
David Lewin’s “A Tutorial on Klumpenhouwer Networks, Using the Chorale in
Schoenberg’s Op. 11, n. 2” in Journal of Music Theory, 38, n. 1 (1994): 79-101
features this passage and the preceding two measures. Edward Jurkowski describes
the voice leading of this passage using what he terms I-DIFF Networks; these
partition the tetrachords into dyads, and are linked by the difference between the
interval-class values of the dyads. See “A Theory of Harmonic Structure and Voice-
leading for Atonal Music,” (Ph.D Diss., Eastman School of Music, 1998): 93-99.
Lewin 1994 also focuses on an ic 4/5 interpretation of the passage, partitioning the
harmonies in many different ways to do so; see pp. 86-93.
7
F# moves to D# by T9; therefore, the voice leading between chords a and b may be
described by *Tt (1).
John Roeder’s work provides another way to view voice leading in post-tonal
music. Roeder uses vectors to describe individual motions among chord members.13
These vectors map registrally-corresponding chord members onto one another.
Unlike Straus, Roeder focuses on pitch voice leadings in his analyses, as he uses his
method to model voice leadings on the surface of post-tonal pieces.14
As Figure 1-2b
shows, the pitch voice leading between chords a and b in Figure 1-1, ordered from
lowest to highest, is <1,1,4,-3>; that is, the lowest chord members ascend by
semitone, the next by four, and the highest descends by three.
The present study combines the approaches of Straus and Roeder to develop a
theory of voice-leading sets.15
A voice-leading set (vlset) is an ordered set of
transpositions or inversions which operates upon a pitch-class set or multiset. Like
Lewin, Straus, and Roeder, voice leadings are understood as transformational
mappings from one set to another. As in Straus’ work, these mappings define an
underlying pitch-class counterpoint, that is, the individual pitch-class voices
interpreted from a succession of pitch-class sets. Unlike Straus, no emphasis is
placed upon either harmonic similarity or voice-leading smoothness. Vlsets employ
Roeder’s notation, but apply to pitch-class, rather than pitch, voice leadings.
13
Roeder 1984, 162. 14
“Voice Leading as Transformation,” in Musical Transformation and Musical
Intuition: Essays in Honor of David Lewin, ed. Raphael Atlas and Michael Cherlin
(Roxbury, MA: Ovenbird Press, 1994): 41-43. 15
I am grateful to Andrew Wilson for sharing an unpublished paper with me that
overlaps with some of the ideas presented in this chapter considerably.
8
Vlsets express ordered sets of pitch-class transpositions or inversions as
vectors. For example, if chords a and b of Figure 1-1 are placed in normal order,
vlset <tt9t> maps ordered pcset <1467> onto <e235>. Vlset <tt9t> is a vector of
transpositions; each maps an individual member of ordered pcset <1467> onto the
corresponding member of <e235>.16
This series of mappings defines the pitch-class
voices between the two chords, as shown in Figure 1-2c. If the vlset from pcset
<1467> to <e235> is interpreted as a series of inversional mappings, the vlset
spanning them is <0690>i. The lowercase i designates that the vlset is inversional.
This alternative is also shown in Figure 1-2c. Vlsets may also span multisets. For
example, in Figure 1-2c, chord d is a hexachordal multiset, <569tt1>, and chord e is a
hexachord, <578t01>; using this ordering, chord d is mapped onto chord e by vlset
<01e020>, forming the pc voices shown in the figure.
This brief demonstration has shown that vlsets explicitly define the pc voices
formed by transformational mappings. In doing so, post-tonal voice leading may be
studied on its own terms, without reference to harmonic similarity or semitonal offset.
As will be shown in chapter 3, vlsets may be used to generate voice-leading spaces in
which they form the unit distance. This defines voice-leading parsimony
contextually, which may be more appropriate for the representation of individual
post-tonal works than a semitonal model.
16
In order to avoid confusion between pcsets and vlsets, in this study vlsets will
always be expressed in bold using angle brackets.
9
Background
Post-tonal music has been interpreted in many ways which generally reduce to
two approaches: 1) forms of extended tonality, and 2) discussions of symmetry and
cycles, with properties stemming from the operations and transformations of set and
group theory.17
The first methodology is more or less compelling depending on the
piece, and the skill of the analyst.18
Post-tonal pieces may have a focal pitch or pitch-
class, but typically do not demonstrate the same relationships as tonal pieces.19
The second method above, the interpretation of post-tonal music in terms of
symmetries and group properties, has proven analytically fruitful. Many artifacts of
post-tonal study, from interval-class vectors and lists of M-related sets to
Klumpenhouwer networks, originate in group theory. This category includes serial
operations, which form a symmetrical Klein group among the four transformations of
17
See for example: Allen Forte “Pitch-class Set Genera and the Origin of Modern
Harmonic Species” in Journal of Music Theory, 32, no. 2 (1988): 187-270, Lewin
2007, and George Perle, Twelve-Tone Tonality 2nd
ed. (Berkeley: University of
California Press, 1996). 18
Felix Salzer applies Schenkerian methodology to post-tonal music in Structural
Hearing (New York: Dover, 1982). Heinrich Schenker himself provides an analysis
of a passage from Stravinsky’s Concerto for Piano and Winds in “Fortsetzung der
Urlinie-Betrachtungen,” in Das Meisterwerk in der Musik: Jahrbuch II (Munich: Drei
Masken Verlag, 1926); trans. by John Rothgeb as “Further Consideration of the
Urlinie: II,” in The Masterwork in Music: A Yearbook, vol. 2, ed. William Drabkin
(New York: Cambridge University Press, 1996), 1–22. Schenker’s work differs from
Salzer, in that his purpose is to show the ways in which Stravinsky’s piece is lacking;
that is, why it is poorly-composed. A number of more modern analysts have tackled
post-tonal music using Schenkerian techniques, including James Baker’s study of the
first of Schoenberg’s Six Little Piano Pieces, Op. 19 in Music Analysis, 9, n. 2 (1990):
177-200, and Olli Vaisala’s “Concepts of Harmony and Prolongation in Schoenberg’s
Op. 19/2” in Music Theory Spectrum, 21, n. 2 (1999): 230-259 and “Prolongation of
Harmonies Related to the Harmonic Series in Early Post-tonal Music” in Journal of
Music Theory, 46, n. 1/2 (2002): 207-283. 19
Joseph Straus critiques post-tonal prolongational analysis in Straus 1987: 1-21.
10
P, I, R(P), R(I). Stemming originally from Milton Babbitt, then through his student
David Lewin, group theory concepts continue to dominate the post-tonal theoretical
and analytical landscape.
The group structure of triadic space, as well as the space of tonal chords in
general, has been explored extensively in the Neo-Riemannian literature of David
Lewin, Brian Hyer, and Richard Cohn, among others.20
Drawing upon ideas and
vocabulary from 19th
-century theorists, Lewin uses group theory to develop a system
of triadic transformations. While Lewin’s approach is mathematical and uses the
language, if not always the categories, of set theory, he includes familiar tonal labels
such as dominant and mediant. Lewin’s goal appears to be to encompass as much
music as possible, tonal, post-tonal, or otherwise, using transformations.
The starting point for Neo-Riemannian theory is the voice leading by the
smallest intervals of 1 and 2 between inversionally-related triads, which is often
found as such in late 18th
and 19th
century representations of tonal space. Using the
triadic transformations P (Parallel) (C-E-G, C-Eb-G), L (Leittonwechsel) (C-E-G, B-
E-G), and R (Relative) (C-E-G, C-E-A), Brian Hyer, constructs the space of triadic
transformations and demonstrates their group structure, drawing on the theories of
Riemann, Oettingen, and others.21
As Hyer shows, P, L, and R transform triads by
maintaining two common tones, and moving one by step. Of course, not all triads
may be linked this way, but the distance between any two triads may be viewed as
20
See, for example, Cohn 1996 and 1997, Hyer’s ”Reimag(in)ing Riemann” Journal
of Music Theory, 39, n. 1 (1995): 101-138, and Lewin 1987. 21
Hyer 1995.
11
some combination of the three transformations. Richard Cohn also focuses on
parsimonious voice leading and demonstrates that triads are the only trichords in
which all three maximal-common-tone preserving transformations involve motion by
step (interval-class 1 or 2) in the moving voice. Figure 1-3 replicates Cohn’s Figure
4, showing the interval-class by which each voice of a trichord must move to produce
another member of the set-class that shares two common tones.22
The actual pitch-
class interval required is determined in part by the Tn form of the set-class in
question. For example, P transforms a C minor triad into a C major triad through
moving the third by interval 1, while it transforms a C major triad into a C minor triad
through the interval 11.
As shown in Figure 1-3, triads and parsimonious voice leading are
interdependent. While other trichords, such as [014], may be capable of
parsimonious voice leading in one particular transformation, [037] is the only one in
which all three transformations involve motion by step.23
As Cohn shows, this is
related to the triad’s internal intervallic structure; the triad’s largest interval, the fifth,
is the most nearly symmetrical division of the octave possible. Similarly, the third is
22
Cohn 1997, p. 6. Figure 1 differs slightly from Cohn’s Figure 4 in a few ways:
step-intervals are listed as ics instead of directed pitch-class intervals, and voice-
crossing is allowed. Interval-classes are used because the value is different when
moving from one Tn-class to another, {037} moves to {047} by ascending semitone,
but {047} moves to {037} by descending semitone, and the purpose of the table is to
show minimal motion; the non-distinct-invertibility of the tritone results in a voice-
leading possibility not included by Cohn’s Figure, that of transposition or inversion
with maximal common-tone retention where the moving voice crosses one of the
common tones. For example: {016}—{076}, where the “third” moves by tritone. 23
Joseph Straus generalizes PL and R and constructs tonnetze for all other trichordal
set-classes, then demonstrates their analytical usefulness in “Contextual-Inversion
Spaces,” in Journal of Music Theory 55, n. 1 (2011): 43-88.
12
the most nearly symmetrical division of the fifth possible. These features allow for
very smooth voice leading when the fifth is inverted within the octave, and when the
third changes quality within the fifth. Thus, smooth voice leading and maximal but
asymmetrical spacing depend upon one another.24
Because parsimonious voice
leading and triads are inextricably linked, an emphasis on one supports the other.25
Therefore, it is not necessarily appropriate to emphasize stepwise voice leading
among non-triadic set-classes.
Julian Hook’s work expands upon that of the previous authors by studying
group structure among triadic transformations without focusing on voice leading.
Hook defines a “Uniform Triadic Transformation” as an ordered triple, <x,y,z>,
where y and z indicate the pitch-class interval by which a triad’s root changes, with
each number acting upon major or minor triads, respectively, while x indicates
whether the mode is preserved or changed. The Neo-Riemannian operation P
corresponds to <-,0,0,>; the root stays the same, while the mode is changed. Hook
then shows that other trichordal set-classes may be substituted for triads using the
transformations.26
Although not restricted to triads, Hook’s transformations may still
only be used between members of the same set-class, or between the members of two
24
Cohn’s work intersects in interesting ways with that of John Clough and Jack
Douthett in “Maximally Even Sets” Journal of Music Theory, 35, 1/2, (1991): 93-173.
See especially pp. 169-172, where the authors show that major and minor triads are
maximally even with respect to the diatonic scale—what they call second-order
maximally even sets. 25
Cohn 1997 describes the triad as “overdetermined” due to the incidental
relationship between its voice-leading properties and acoustic properties, 5. 26
“Uniform Triadic Transformations” Journal of Music Theory, 46, n. 1-2 (2002):
57-126.
13
set-classes. The groups formed by voice-leading sets, as shown in chapter 2, may be
seen as expanding the use of group theory to encompass all pitch-class sets and
multisets in a given cardinality. In addition, the objects—pitch-class sets and
multisets—corresponding to the various subgroups formed by vlsets consist of
multiple set-classes.
Robert Morris’ work on voice-leading spaces is very influential on the
generation of voice-leading spaces proposed in chapter 3. Morris defines general
spaces with relative distances as “contour spaces” as well as modular and linear
spaces with defined spans, such as pitch and pitch-class spaces. More recently, his
work on voice-leading spaces defines properties of these spaces and the restrictions
and definitions of tonnetz spaces. He also shows Perle spaces as an outgrowth of
realignments of the cyclic spaces which constitute tonnetze.27
Morris’s definition of
voice-leading spaces can be interpreted as defining the transformations that relate
forms of the same set-class, as he shows in his “Riemann wreath” of trichords that
combine to create members of [013478].28
José Martins generates spaces built on scales from the medieval Enchiriadas
treatises, termed dasian and Guidonian spaces, respectively. He then shows that the
spaces may be useful in the analysis of post-tonal music, focusing chiefly on the
music of Bela Bartok. After showing that the spaces derive from interlocking
27
Morris 1987 and 1998. 28
Morris 1998, 198.
14
interval-cycles, Martins generalizes their structure in order to produce a large number
of new spaces which may be used to analyze post-tonal music.29
The most recent explorations of voice leading incorporate geometrical models.
Figures 1-4a and b display the trichordal orbifold as described by Tymoczko, Quinn,
and Callender.30
This work intersects with Perle spaces, as well as an earlier article
by Walter O’Connell where the geometry of set-class space, here the all-interval-class
tetrachords, is examined in modern terms for the first time.32
Figure 1-4b provides a
close-up of one-quarter of the orbifold, corresponding to three of the triangular
sections shown in Figure 1-4a. As Figure 1-4b shows, each trichord is located in
closest proximity to chords that have two common tones and minimal offset in the
intervals involved. A chord’s location also corresponds to its evenness, that is, how
evenly it divides the octave. For instance, [048] divides the octave into three equal
parts, and is located at the center. Triads, [037], are nearly symmetrical, and
therefore orbit the central [048]s. Given this feature, traditionally defined
“parsimonious” voice leading may take place within one particular region of the
29
See Dasian, Guidonian, and Affinity Spaces in Twentieth-Century Music, (PhD
diss., The University of Chicago, 2006), “Stravinsky’s Discontinuities, Harmonic
Practice and the Guidonian space in the ‘Hymne’ for the Serenade in A” in Theory
and Practice, 31, (2006): 39-64, and “Affinity Spaces and Their Host Set Classes” in
Mathematics and Computation in Music 38, (2009): 496-508. 30
Clifton Callender, Ian Quinn, and Dmitri Tymoczko “Generalized Voice-leading
Spaces” in Science, 320, (2008): 346-348, and Dmitri Tymoczko “The Geometry of
Musical Chords” in Science, 313 (2006): 72-74. The images were generated using
the program Matlab a scientific imaging and calculating program. 32
Dave Headlam “Introduction” in Theory and Practice, 33, (2008): 1-47, and Walter
O’Connell “Tone Spaces” in Die Reihe, 8, (1968): 35-67.
15
orbifold, such as among the triads at the center. This spatial display of chordal voice
leading and harmonic properties is a compelling feature of the orbifold.
Andrew Wilson also uses set theory to describe voice leadings in post-tonal
music. Wilson shows pitch voice-leadings as unordered sets, then demonstrates how
they may be grouped into equivalence classes using permutation, inversion, register,
transposition, and cardinality, describing the possible perceptual issues raised by each
equivalence.33
As mentioned above, the description of voice leading as an ordered set is also
indebted to John Roeder’s formalization of voice leadings as vectors. Roeder
explores some of the relationships between harmonic structure and voice leading, and
categorizes various types of voice leading. A T-class voice leading connects two
chords that have a relative transpositional level of zero, that is, their lowest pcs are
the same. A transpositional voice leading connects two members of a tn-class, and a
permutational voice leading connects two chords which contain the same pcs, but in a
different registral order.34
Roeder focuses on pitch voice leadings, rather than pitch-class, in his
analyses, in order to describe voice leadings as they occur on the surface of the
music.35
One of the challenges in analyzing pitch-class voice leadings is that there
are multiple possible interpretations of a passage. For example, any member of one
33
“Voice Leading as Set,” unpublished manuscript, (2011). Although we worked
independently, Wilson and I developed very similar ideas, extending even to terms
such as vlset and vlclass. 34
Roeder 1984, 162-165. 35
Roeder 1994, 41-43.
16
pc set may map onto any member of another pc set. This difficulty is faced by those
using set theory to analyze pc sets as well, as they must decide how the music should
be segmented and partitioned.
Roeder’s ordered interval spaces are similar in structure to the orbifold spaces
discussed above. One important difference is that Roeder interprets the coordinates
of his spaces as ordered pc intervals. For example, <1,2> represents the
transposition-class [013]. Using transpositional equivalence, Roeder is able to
represent pc sets in a lower dimension, so that the two-dimensional space, of which
<1,2> is a member, represents trichordal transposition-classes.36
Klumpenhouwer-networks, or K-nets, as defined by Henry Klumpenhouwer
and described by David Lewin, are networks of nodes and arrows, in which nodes are
populated by pitch-classes and arrows represent transposition or inversion operations.
Networks are related to one another through the similarity of their graphs, known as
isography. The arrows of strongly isographic K-nets have the same T and I values.
Positively isographic K-nets have the same T arrows, and the same difference
between their I arrows. The value of this difference, known as <T>, is used to
describe the relationship between two K-nets. Negatively isographic K-nets have
inverse T arrows, and their I arrows add up to the same sum. This sum is known as
<I>.37
36
“A Geometric Representation of Pitch-Class Series” in Perspectives of New Music,
25, ns. 1/2 (1987): 362-409. 37
See, for example, “A Tutorial on Klumpenhouwer Networks, Using the Chorale in
Schoenberg’s Op. 11, n. 2” in Journal of Music Theory, 38, n. 1 (1994): 79-101.
17
As Shaugn O’Donnell has shown, the <T> and <I> relationships among K-
nets may be seen as dual operations—two transpositions or two inversions,
respectively.38
The pcset a K-net represents may be divided into two subsets, which
are mapped onto the subsets of some other pcset. <T> and <I> values are equal to the
sum of the two operations. Therefore, to form isographic K-nets, two pcsets must be
divisible into the same two Tn-classes, or inversionally-related Tn-classes. Pcset
pairs that are subdivided into the same Tn-classes form strongly or positively
isographic K-nets, while those with inversionally-related Tn-class subsets form
negatively isographic K-nets. Therefore, using K-nets, one may describe
relationships among pcsets which are not members of the same set-class.
Dual operations provide relationships among the same pcsets as K-nets, but do
not form recursive network structures. Like K-nets, they are useful in post-tonal
repertoire for describing relationships among pcsets which do not belong to the same
set-class. One analytical advantage dual operations have over K-nets, is that they
explicitly describe pcset mappings. K-net <T> and <I> relationships, on the other
hand, describe operations upon network arrows. This is useful for building recursive
levels of networks, but does not describe the musical surface of a piece.39
38
Transformational Voice Leading in Atonal Music, (PhD diss., CUNY, 1997). In a
response to a critique of K-nets by Michael Buchler, Henry Klumpenhouwer outlines
the differences between dual transformations and K-nets, noting that the <T>/<I>
automorphisms of K-nets map the arrows of one network onto another, while dual
transformations map the contents of the nodes of one network onto another, among
other differences. 39
Headlam 2008 points out connections between Perle’s compositional materials and
K-nets; see pages 30-32.
18
K-nets and dual operations, are analytically useful in post-tonal music because
they may be applied to passages which employ multiple set-classes, and demonstrate
relationships involving combinations of transformations. Voice-leading sets take this
idea a step further; all voice-leadings are interpreted as the combination of individual
pitch-class transformations. Dyadic vlsets are essentially identical to dual operations,
except that they apply to individual pitch-classes, rather than pitch-class sets.
Joseph Straus has also written and presented geometrical visualizations of
“set-class” voice leading, where we may move systematically through some sequence
of set-classes by a direction in the geometrical space presented.40
Straus’s ideas are
described more fully below. Finally, Michael Callahan demonstrates connections
between Perle spaces and Knet transformations by creating a lattice in which adjacent
trichords are related by strong-isography and/or minimal pc offset; graphically
demonstrating consistent relationships among pc sums and differences.41
40
“Voice Leading in Set-Class Space” in Journal of Music Theory 49, no. 1 (2005):
45-108. The space will be discussed in detail below. 41
“Mapping Sum-and-Difference Space: Parallels Between Perle and Lewin” in
Theory and Practice 33, (2008): 181-217.
19
Chapter 2: Voice-leading Set Theory
As discussed in the previous chapter, the study of voice leading in post-tonal
music has focused mainly on harmonic similarity, using semitonal offset as a metric.
While this emphasis is appropriate for some musical contexts—especially triadic
contexts, as discussed in chapter 1—it is insufficient for post-tonal music. The
following discussion will demonstrate that voice-leading sets provide a way in which
to discuss post-tonal voice leading without relying on harmonic similarity or
semitonal offset; instead, voice-leading sets explicitly define pitch-class voices
produced by mapping one pcset onto another. Voice-leading sets will then be
compared to traditional pitch-class set transformations, such as transposition and
inversion, followed by a brief analytical example using Arnold Schoenberg’s Drei
Klavierstuecke, Op. 11, n. 1; the chapter will conclude with a discussion of the group
properties of voice-leading sets.
A voice-leading set (vlset) is an ordered set of transpositions or inversions
which operates upon a pitch-class set or multiset. Let us begin with pc dyads {01}
and {5t}. There are two possible voice leadings between them; either pc 0 maps onto
5 and 1 onto t, or 0 maps onto t and 1 onto 5. If the first pcset is ordered, <01>, and
the mappings are induced by transposition, they form vlsets <59> and <t4>. If the
mappings are induced by inversion, they form vlsets <5e>i and <t6>i.42
Because vlsets are ordered sets of transformations, they naturally order the
pcsets they operate upon. That is, vlsets do not operate consistently upon the various
42
Vlsets will always be shown in bold in angle brackets. Inversional vlsets include a
lower case i.
20
permutations of a pcset, unless the vlset or pcset has some symmetrical property
which produces order equivalence. This is a significant difference between vlsets and
other proposed methods for studying voice leadings. A transposition or inversion
maps the set of pcs onto itself consistently, that is, by the same interval or sum. For
example, T1 maps every pc by T1. Vlsets do not operate uniformly on the set of pcs,
but on individual pcs. Therefore, they may be seen as operating upon multiple sets of
pcs, equal to the cardinality of the vlset. For instance, vlset <356> operates upon
three distinct sets of pcs simultaneously.43
Imagine that ordered pcset <000> is a slot
machine and each reel contains pcs 0-e; vlset <356> directs the first reel to spin three
places, the second five places, and the third six, producing pcset <356>. The analogy
breaks down with inversional vlsets, as these direct the reels to flip, rather than spin.
While the ordered structure of vlsets requires strict ordering of pcsets, it also allows
for multisets; each doubling in a multiset represents an independent set of twelve pcs.
The number of vlsets in each cardinality n is equal to (12^n)*2. For example,
the dyadic vlsets can have any of the twelve pcs in either position, and there are two
types of each vlset, transpositional and inversional; Figure 2-1 shows the number of
vlsets in each cardinality. The vlsets in cardinality one correspond to the twenty-four
transposition and inversion operations, with the exception that they apply to single
pcs as vlsets. Because vlsets include multisets, they can be any cardinality, although
it is difficult to imagine what usefulness a ten-thousand-member vlset might have.
43
The group structure of vlsets is addressed at the end of this chapter.
21
Traditional transposition and inversion are preserved by vlsets as singleton
multisets; for instance, <000> is equivalent to T0, <44> is equivalent to T4, <3333>i
is equivalent to I3, etc. However, vlset notation does not necessarily demonstrate a
transpositional or inversional relationship between pcsets. Vlsets define individual
transpositional or inversional mappings. They are therefore independent of the
harmonic structure of the pcset mappings they produce. This is significantly different
than the transposition and inversion of pcsets, which preserves harmonic structure.
For example, pcsets {014} and {458} are both members of tn-class [014]. If placed
in normal order, <014> and <458>, the sets are spanned by vlset <444>, showing that
they are harmonically equivalent under transposition. In any other ordering, however,
the sets’ harmonic relationship is not shown by the vlset. Figure 2-2 shows the
possible mappings between pcsets {014} and {458}, including inversional vlsets.
Each of these mappings is a viable interpretation of the voice leading between the two
pcsets. Unlike transposition and inversion, the harmonic relationship between any two
sets is not addressed by vlsets, except for the way in which it limits the possible
mappings between them.44
Using vlsets, therefore, voice leading may be removed from its harmonic
context and examined more abstractly. The above examples begin with pcsets and
demonstrate how vlsets may map one onto another. As a transformation, with no a
44
Given two pcsets, no vlset spanning them is the “correct” interpretation. All
possible vlsets, such as those shown in Figure 2-3, are equally viable. An analyst,
however, may choose one particular voice-leading interpretation which conveys some
meaningful data about a piece. Analytical issues will be addressed further in chapter
four.
22
priori pcsets, a particular vlset spans many pcset pairs. For example, in addition to
mapping pcset <1467> onto <e235>, vlset <tt9t> maps pcset <0245> onto <t013>, or
<0167> onto <te35>, or <0022> onto <tte0>, etc.45
Contrapuntal Motion in Pitch-class Space
Pitch voice leading categories such as parallel, oblique, contrary, and similar
motion may be adapted for pitch-class voice leading, and articulated with vlsets.
These categories are altered by their transformation from linear pitch space to
modular pitch-class space. Pc space counterpoint also introduces interpretive
challenges; the individual motions between pcsets may be interpreted in a number of
ways, producing multiple contrapuntal interpretations. In addition, due to octave
equivalence pitch-class counterpoint may or may not be reflected in pitch-space on
the musical surface.
Parallel motion in pitch space is formed by multiple voices moving in the
same direction by the same distance.46
In pc space, parallel motion corresponds to
transposition and inversion, which move each pc the same distance relative to some
axis. This axis is rotational for transposition and reflective for inversion.47
Parallel
motion occurs whenever a vlset’s values are the same. For example, vlsets <222>
45
The first example is shown in Figures 1-3 and 1-4. 46
This distance may be measured in diatonic pitch space as well, as in the parallel
thirds of a major scale. 47
Parallel motion corresponds to crisp transposition or inversion; see Joseph Straus’
“Uniformity, Balance, and Smoothness in Atonal Voice Leading” in Music Theory
Spectrum, 25, n. 2 (2003): 318.
23
and <777>i create parallel motion, as they transform each pc voice by the same
amount; see Figure 2-3.48
Oblique motion in pitch-space is the movement of one voice while another
maintains the same pitch. Transpositional vlsets which include a zero, such as <05>,
show that one voice maintains the same pc, while another moves. Inversional vlsets
may also produce oblique motion, but are defined in part by the specific pcset they
operate upon. In order to map a pc onto itself, the index of inversion must be double
the pc value. See Figure 2-4 for some examples.
Contrary motion in pitch-space is defined as motion in opposite directions;
one voice ascends while the other descends. In modular pc space, this can be more
rigorously defined as motion by complementary intervals, that is, when one voice
moves by ordered pc interval X, the other voice moves by 12-X. This motion is
easily demonstrated by transpositional vlsets, such as <39>. Cardinalities greater
than two may produce sum-0 vlsets which are not made up of complementary pairs,
such as <et96>. Nevertheless, the total ordered pc distance invoked by the vlset is 0,
therefore the definition may be generalized to include these sets; a few examples are
shown on Figure 2-5.49
48
The lines on Figures 2-3 through 2-6 demonstrate the pc voices created by the
vlsets. Some of the examples demonstrate their contrapuntal motion in pitch space as
well as pitch-class. 49
Strongly isographic K-nets also involve the concept of contrary motion, as the T-
related nodes may be transposed by complementary transpositions from one to
another. Mike Callahan organizes these motions on a torus he terms sum-and-
difference space. One circle of this torus is formed by strongly isographic K-nets,
whose constituents, an ic 5 dyad and a singleton, move by T1 and Te, respectively;
the other circle is formed by positively isographic K-nets, where the ic 5 dyad is
24
Inversional vlsets may also create pc contrary motion. However, this motion
results from the relationship between the sum of the inversional index numbers and
the total sum of the pcset. For an inversional vlset to produce contrary pc motion, it
must be double the total sum of the pcset it operates upon. For example, the total sum
of pcset {24} is 6, therefore any inversional vlset of sum 0 will produce contrary
motion. This is demonstrated by Figure 2-5. Although the Figure shows wedge
motion, each individual mapping from the first set onto a later set forms contrary
motion.
Just as contrary-motion producing inversional vlsets must be double the sum
of the pcs they operate upon, the total sum of transpositional vlsets which produce
contrary motion must be 0. Due to these sums, the vlsets preserve the sums of the
pcsets they operate upon. If Q is the sum of a pcset, inversional vlset sums are 2*Q,
and 2*Q – Q = Q; for transpositional vlsets, 0 + Q = Q. The relationship between the
total sum of two pcsets also demonstrates this relationship. Any pcsets of the same
sum will demonstrate contrary pc voice leading, no matter how the voice leading is
interpreted. For example, all of the transpositional vlsets shown in Figure 2-2,
between {014} and {458}, are sum 0, and all of the inversional vlsets are sum 10,
double the sum-5 pcsets.50
maintained, and the singleton moves by T1. “Mapping Sum-and-Difference Space:
Parallels Between Perle and Lewin,” in Theory and Practice, 33 (2008): 181-217.
The torus is shown on page 198. 50
On the orbifold spaces developed in Tymoczko 2006 and Callender, Quinn, and
Tymoczko 2008, pcsets of the same sum are grouped together; they are aligned on the
dyadic space, and in the same plane on the trichordal space. Therefore, any
movement along a same-sum line or within a sum plane may be characterized as
25
Similar pitch motion is simply defined as two voices moving in the same
direction. It is distinguished from parallel motion in that the voices move by different
intervals; in horn fifths, for instance, one voice moves by step while the other moves
by arpeggio in the same direction. Pitch-class intervals make no distinction between
ascent and descent, of course, but do specify the total distance traversed in pitch-class
space. Therefore, similar pc motion may be defined by the difference between the
intervals of each voice. This is equivalent to Straus’ uniformity and balance,
discussed above.51
For example, vlset <12> moves one pc voice by T1, and the other
by T2, this voice leading is very similar, without being parallel or oblique. Similarity
among inversional vlsets may be defined in the same manner, as the difference of
inversional sums; <78>i would produce very similar motion.52
Pc voice-leading
similarity is a continuum, with a maximum dissimilarity of six in two voices, as in
<17>, and a midpoint of three, as in <t1>. For higher cardinalities, relative similarity
contrary. “Generalized Voice-leading Spaces” in Science, 320, (2008): 346-348, and
“The Geometry of Musical Chords” in Science, 313 (2006): 72-74. David Headlam
notes this organization of the orbifold and demonstrates that it corresponds with some
of the theories of George Perle, most notably his triadic arrays in his introduction to
Theory and Practice, 33 (2008): 32-39. This definition of contrary pc motion is also
related to Richard Cohn’s directed voice-leading sums, or DVLS, which are the total
ordered pitch-class sums produced by the voice-leading from one triad to another; see
“Square Dances With Cubes” in Journal of Music Theory, 42, n. 2 (1998): 283-296.
A DVLS value of 0 corresponds to the definition of contrary motion proposed here. 51
Straus 2003. The vlsets shown on Figure 2-7 correspond to fuzzy transposition or
inversion with very little offset. 52
Of course, it may also produce oblique motion, depending on the pcset.
26
is given by the range of values; as shown in Figure 2-6, vlsets <1122> and <011>i,
with a range of one, produce very similar pc voice leading.53
Voice-leading Set-classes
The vlsets in any cardinality may be organized into voice-leading set-classes
(vlclasses) through various equivalences, such as transposition, and inversion. For
example, vlsets <01>, <23>, <45>, <43>, and <87> are members of vlclass [01]. The
set of vlclasses formed through transposition and inversion is isomorphic to the set of
set-classes of ordered pcsets, including multisets. Vlclasses preserve the intervals
among the pc voices produced by their vlset members. For example, Vlset <025>
transposes the first pc by T0, the second by T2, and the third by T5; the second pc
moves two steps further than the first and the third pc moves five steps further.
Another member of vlclass [025], such as <247>, increases each pc transposition by
two; the first pc moves by T2, the second moves two steps further, and the third pc
moves five steps further. Vlclasses thus preserve the harmonic structure of voice
leadings, rather than pcsets.
Vlset transformations are linked to pcset transformations when operating upon
ordered pcsets. For example: pcset <02> is mapped onto <03> by vlset <01>. The
same pcset, <02>, is mapped by <23> onto <25>; the T2 transposition of the vlset is
matched by a T2 transposition of the resulting pcset. Similarly, vlset <45> produces
53
Above I suggested that contrary motion might be broadly defined to include sum-
zero transpositional vlsets which are not made up of complementary transposition
pairs, as the total voice-leading is balanced among the pc voices. In my opinion, pc
voice-leading similarity should be defined by the relationships among all of the
individual voices, so the range, not the total sum should be used. This differs from
Straus, who measures the total offset in a fuzzy transformation.
27
pcset <47> from <02>, a T4 transposition of <03>. When operating upon ordered
pcset <02>, the twelve transpositions of vlset <01> produce the twelve transpositions
of pcset tn-class [03], as Figure 2-7 shows. Therefore, vlclass [01], under
transposition, maps pcset tn-class [02] onto tn-class [03]. Inversions of vlset <01>,
such as <10>, reverse the direction of the voice leading. That is, they transform
ordered pcset <03> into some transposition of <02>. For example: ordered pcset
<03> is transformed by vlset <10> into pcset <13>.
As is evident from the above definition, the voice leading theory proposed
here uses an expanded form of the set theory currently in practice, but applies it to
voice leadings in addition to pc sets.54
A brief discussion of Arnold Schoenberg’s
Drei Klavierstuecke Op. 11 no. 1 will show some of the ways in which the ideas
presented thus far may be applied. The piece will be revisited in chapter 4. Figure 2-
8a shows the first eleven measures with ten trichords labeled a-i, and Figure 2-8b
demonstrates that voice-leading class [002] links many pairs of trichords; that is, the
vlsets are some transposition or inversion of one another. This preserves the
relationship among the intervals traversed by the individual moving voices. In the
case of vlclass [002], two voices move by the same interval, and one voice moves two
semitones further, or two semitones less. For example, the [04] dyad in chord a, {7e}
moves by Tt to the [04] dyad in chord b, {59}, and the G# in chord a moves by T8, a
difference of two semitones, to E in chord b. Similarly, two members of chord c,
{56}, move by T4 to {9t} in chord d, while the B in chord c moves by T2 to Db in
54
A focus on a painting’s negative space is a possible analogy.
28
chord d. The voice-leading consistency provided by [002] ensures that two voices of
each trichord move by the same interval, and the third voice moves by an interval that
differs by two semitones. This consistency unifies the passage despite the harmonic
diversity presented by the trichords, which represent six different Tn-classes—[012],
[014], [015], [016], [024], and [026].
The voice leadings shown in Figures 2-8a and b are not necessarily intended
to be heard in real time. Rather, they are analytical interpretations of the paths
between various pcsets. In other words, I am not suggesting that a listener attempt to
“hear” the individual pcs of chord a mapping onto chord c and chord b at the same
time. Instead, vlclass [002] may be understood as a motive, which is varied through
transposition, inversion, and rotation. This motivic interpretation changes the way in
which I understand the piece, whether or not I am concentrating on hearing individual
voice leadings in a performance.55
The unique properties of vlclass [002] also affect
the way in which I understand the piece’s succession of trichords. Because [002] is a
multiset, each trichordal pair must share one interval, such as the [04] dyads in chords
a and b. Further, [002] preserves the whole-tone content of a pcset; either each pc is
mapped onto another within the same whole-tone scale, or each is mapped onto the
other whole-tone scale. These properties explain why, for example, chord e cannot be
mapped onto chord f by a member of vlclass [002]; the three members of chord e
55
A number of theorists, such as those involved in the Society for Music Perception
and Cognition, study music perception in many contexts, including transformation
theory. In this dissertation I concentrate on the presentation of a theory of voice
leading with analytical applications, and do not engage with issues of perception at
length; I hope to pursue these issues in the future.
29
belong to the same whole-tone scale, while chord f contains two even whole-tone
members, and one odd.
Each vlset aligns the set of ordered pcsets with itself in specific pairs. As
mentioned above, any member of ordered vlclass [01] maps any member of ordered
pcset tn-class [02] onto some member of [03]. The remaining pairings are shown in
Figures 2-9a and b. As shown, each tn-class-pair in ordered vlclass [01] is reversed
by inversion to ordered vlclass [10], and vice versa. For instance, pcset <49> is
mapped onto <26> by vlset <t9>, and is mapped onto <17> by vlset <9t>. This
feature is similar to the relationship between a transposition operation and its
complement: T1({04}) = {15}, and Te({15}) = {04}, for example. The difference is
that the relationship among vlsets can be applied to pcset tn-classes, as there are
twelve transpositions and inversions of most vlsets. Like pcset transpositions, each
transpositional vlset has a complement when operating upon pcsets—the vlset which
is the specific sum-zero inversion: <34>(<16>) = {4t}, and <98>(<4t>) = {16}.
Figures 2-9c – l demonstrate the tn-class pairings produced by the remaining
dyadic ordered vlclasses. The vlclasses are aligned with their inversions to
demonstrate their reciprocal relationships. Among transpositions, T0 and T6 are
unique. T0 transposes every set onto itself, and T6 is its own inverse. These
properties are maintained to some extent by vlclasses [00] and [06]. While vlset
<00> has the same effect as T0—transposing any dyad onto itself—the remaining
members of vlclass [00] <11>, <22>, etc, preserve the tn-class of any pcset they
operate upon, as shown in Figure 2-9l. This is not surprising given that vlsets of
30
vlclass [00] behave like traditional transpositions. In addition, vlset <66> is its own
inverse, just as T6 is. Figure 2-9k demonstrates that vlclass [06] also behaves in
some ways like pcset transposition by T6. Each tn-class is paired with only one
other, that is, members of vlclass [06] are their own pcset tn-class complements. This
is true regardless of any transposition of the vlset. For example: pcset <38> is
mapped by <39> onto <65>, and <65> is mapped by <4t> onto <t3>; vlclass [06]
maps ordered tn-class [05] onto [10], and [10] onto [05]. In addition, specific vlsets
<06> and <60> are their own complements.
Because they are made up of multiple transpositions, transpositional vl-classes
produce ordered tn-class interval cycles. Figures 2-10a – j display the cycles created
by each vlclass.56
Each vlclass creates an ordered tn-class cycle which is isomorphic
to a pcset interval cycle. Like pc interval cycles, vlclasses create cycles according to
the way their individual transpositions divide twelve. For example, [04] creates a
three-member tn-class cycle, because four (T4) divides twelve into three. 1, 5, 7, and
e are prime to twelve, therefore the ordered vlclasses with these transpositions, [01],
[05], [50], and [10], cycle through all twelve ordered pc tn-classes.
The tn-class cycles shown in the figures simply tabulate the results of the tn-
class pairings shown earlier. As in the tn-class pairs, vlclasses are aligned by
inversion, such as [01] and [10]; this demonstrates that their cycles are also inversions
of one another. In addition, vlclasses related by multiplication, such as [01] and [05],
create multiplication-related cycles. As their name suggests, the ordered tn-class
56
[00] and [06], discussed in the previous paragraph, create one- and two-member
cycles, respectively, and are left off of the figure.
31
cycles are independent of the transposition of the vlset involved. For example, Figure
2-11 shows the ordered tn-class cycle created by three transpositions of vlclass [04]
applied to pcset <34>. As shown, the three transpositions create an ordered tn-class
cycle of ([01] – [05] – [30]).
Vlsets also create cycles of pcsets. Like the tn-class cycles just discussed,
these are directly linked to the individual transposition values in the particular vlset.
These pcset cycles operate independently of their ordered tn-class cycles. For
example, ordered vlclass [03] creates a four-member ordered tn-class cycle. Figure 2-
12 displays the pcset cycle and tn-class cycle created by vlset <14>. As the figure
shows, the tn-class cycle, ([01] – [04] – [50] – [20]) cycles through three times before
the pcset cycle is complete. The pcset cycle is twelve sets in length, because the first
member of the vlset, T1, cycles through all twelve pcs. The cycle can be seen as a
result of the combination of cycles formed by T1 and T4.57
Like transpositional vlsets, inversional vlsets align the set of ordered pcsets
with itself. For example, vlset <01>i maps pcset <34> onto <99>, and pcset <87>
onto <46>. Inversional vlset operations are involutions: vlset <01>i maps pcset <34>
onto <99>, and <99> onto <34>. Like transpositional vlsets, a transposition of an
inversional vlset results in a transposition of the pcset produced. Figures 2-13a and b
show the transpositions of vlset <01>i and <0e>i applied to pcset <04>. The pcsets
produced are members of the same tn-class, [30] and [50], respectively.
57
These cycles intersect with George Perle’s work in a number of ways, especially
his difference tables, which produce interval cycles through arrangements of multiple
cyclic sets. See Twelve-Tone Tonality, 2nd
edition, (Los Angeles: University of
California Press, 1996), 31-32.
32
Transpositional and inversional vlsets are not commutative, just like their
counterparts—pc transposition and inversion. For example, <01>(<36>) = {37}, and
<01>i(<37>) = {96}, while <01>i(<36>) = {97}, and <01>(<97>) = {98}, therefore
<01>i(<01>(<36>)) ≠ <01>(<01>i(<36>)).
As mentioned above, invesionally-related transpositional vlsets produce
inversonally-related tn-class pairings. No such relationship exists for inversional
vlsets, because they are their own inverses. Instead, inverting an inversional vlset
produces a new pairing, as Figures 2-14a and b show. The two pairings replicate the
vlsets’ inversional relationship; that is, the paired pcsets in each collection are
inversions of one another. For example, ordered vlclass [01]i pairs ordered tn-classes
[04] and [30], while [10]i pairs [40] and [03]. For another example, see Figures 2-15a
and b, which demonstrate the tn-class pairings produced by ordered vlclasses [04]i
and [40]i. Figures 2-16a – h show the remaining tn-class pairings produced by
inversional vlclasses.
The properties of dyadic vlsets just discussed are also true for trichordal
vlsets. The main difference, of course, is the greatly expanded number of both pcsets
and vlsets. There are 1728 ordered pcsets, transpositional vlsets, and inversional
vlsets. The pcsets reduce to 144 ordered tn-classes, which, like the dyads, are paired
by the individual vlsets. Figure 2-17 demonstrates the tn-class pairings produced by
ordered vlclass [025]. As with dyadic vlsets, these pairs are reversed by inverting the
vlset, producing ordered vlclass [0t7]. For example, vlset <247> maps pcset <016>
onto <251>, and vlset <t85> maps pcset <251> onto <016>. Triadic inversional
33
vlsets also share some of the properties with their dyadic counterparts. Each
individual inversional vlset is an involution: <278>i(<027>) = {251}, and
<278>i(<251>) = {027}.
As a result of the volume of trichordal vlsets, many distinct vlsets are similar.
For example, <046>, <460>, and <604> are rotations of one another—the same
vector of transpositions in the same order—but produce very different results when
applied to pcset <014>: {05t}, {474}, and {618}, respectively. Three more vlsets
share the same transpositions, but not the order: <064>, <640>, and <406>, which
produce {078}, {654}, and {417}, respectively. The same is true of ordered pcsets,
of course; <014>, <140>, <401>, <041>, <410>, and <104> all belong to different
ordered tn-classes. This permutational redundancy is eliminated by moving from
ordered to unordered pcsets.58
Depending on the analytical situation, it may be useful
to use either or both unordered or ordered vlsets, just as it is with pcsets. Analytical
issues will be addressed further in chapter four.
Voice-leading Group Theory
A group is a set of objects and transformations with a few special properties:
closure, associativity, identity, and inverse. A group must have closure, that is, the
result of every transformation must be a member of the group. If two transformations
are performed in succession, the result must be the same no matter which
transformation is performed first, for example: (a + b) + c = a + (b + c). There is an
58
This is the reduction used to move from 1728 ordered pcsets to 364 unordered
pcsets in the triadic orbifold.
34
identity element in every group, which leaves the other elements unchanged. Finally,
each member of the group must have an inverse.
The most familiar groups used in music theory are formed by transposition
and inversion. In group theory, these transformations are equivalent to the groups
formed by the rotation and reflection of symmetrical objects. The twelve
transpositions operating upon pitch-classes satisfy the requirements listed above; the
transposition of any pc results in another pc, (T1 + T3) + T4 = T8 = T1 + (T3 + T4),
T0 is the identity, and each transposition has an inverse mod 12. T6 is
its own inverse. The transpositions form a cyclic group which is equivalent to the
rotations of a dodecagon, C12. The combination of transpositions and inversions
forms a twenty-four-member dihedral group, D12. Each inversion is its own inverse,
and T0 remains the identity.
The T/I groups contain a number of subgroups. For example, the rotations
and reflections of a triangle form a dihedral group (D3) with six members: T0, T4,
T8, In, In+4, and In+8. These subgroups have fewer elements than the larger group,
but must fulfill the same requirements—closure, associativity, identity, and inverse.
As mentioned above, the 144 dyadic transpositional vlsets behave much like
traditional pcset transpositions. The group formed by the transpositional vlsets is
simply the direct product of this group and itself, C12 C12, essentially squaring the
group of transpositions. This produces a simply transitive Abelian group of order
144, with <00> as the identity. Although it is the product of two cyclic groups, the
group is not cyclic, as it must be generated by more than one element: {<01>, <10>}.
35
There are 90 subgroups of C12 C12, including the full group, and trivial
identity group. Table 2-1 lists the subgroups by order, and provides a generating set
for each. The subgroups generated by singleton multisets are isomorphic to the
familiar groups of PC transpositions found in transformation theory: C2 {<66>}, C3
{<44>}, C4 {<33>}, C6 {<22>}, and C12 {<11>}. In most cases, the other
subgroups share features with the transposition subgroups of the same order. For
example, the remaining subgroups of order 4 are generated by <03>, <30>, <36>,
<39>, <63>, and {<06>, <60>}. The latter subgroup is the direct product of 2
subgroups of order 2. The subgroups with more than 12 elements are less familiar, as
they are all formed by multiplying smaller subgroups.
Vlsets, and the groups they form, operate upon any pcsets of the same
cardinality. Therefore, the subgroups create networks of pcsets. Figure 2-18 shows
the subgroup generated by vlset <17> with three sets of dyadic pcsets as objects. As
the figure shows, the group of vlsets operates independently of the specific pcsets
used as objects; the first set of pcsets is made up of the even whole-tone singletons
and the odd whole-tone tritones, while the second is made up of the even whole-tone
[02] sets, and the odd whole-tone [08] sets. The third set of pcsets begins with <01>,
but is similarly constructed; it contains the even whole-tone [01] dyads, and the odd
whole-tone [50] dyads. The group table is shown in Table 2-2.
Each corresponding collection of ordered pcsets forms a Generalized Interval
System along with the <17> subgroup. The pcsets form the space, and the subgroup
is the set of intervals between each ordered pair of sets in the space. Each subgroup
36
and corresponding set of ordered pcsets forms a GIS in this fashion, including the full
group of 144 transpositional vlsets.61
The whole-tone features of the sets in Figure 2-18 result from the properties of
vlset <17> and the group it generates. The cycles created by intervals 1 and 7
alternate odd and even intervals, and overlap on the evens. This partitions the vlset
operations into even singletons and odd tritones—in this case, the tritone is the
difference between the intervals. The even singletons always maintain the whole-
tone membership of the pcsets they operate on, while the odd tritones always change
the whole-tone membership. As mentioned above, vlclass [06] is its own tn-class
involution; therefore, the specific pcsets the <17> group operates on belong to only
two tn-classes.
There are 288 dyadic voice-leading sets; these correspond to the ordered
dyadic pcsets, including multisets, which may be organized into vlclasses. The full
set of 288 vlsets form a semidirect product group, [C12 C12] C2. Vlset <00> is
the identity, each transpositional vlset has an inverse, and each inversional vlset is its
own inverse. The C2 group acts upon C12 C12 through the inverse automorphism.
This could be shown using 0 and 1; all elements of the group would be of the form
{a, b, 0} and {c, d, 1}, where a, b, c, and d are members of 12. Group operations
combine differently depending on whether they have a 0 or 1; {a, b, 0} + {c, d, 0} =
{a + c, b + d, 0}, {a, b, 0} + {c, d, 1} = {-a + c, - b + d, 1}, {a, b, 1} + {c, d, 0} = {a +
c, b + d, 1}, and {a, b, 1} + {c, d, 1} = {-a + c, -b + d, 0}. This is the same way in
61
Lewin 2007.
37
which transposition and inversion operations combine. The vlset group
representation simply substitutes the lowercase “i” for “1”: <12> + <48> = <5t>,
<12> + <48>i = <36>i, <12>i + <48> = <5t>i, and <12>i + <48>i = <36>. This may
be easily verified using any ordered pc dyad, for example: <12> (<04>) = {16},
<48>i (<16>) = {32}, and <36>i (<04>) = {32}.
There are many subgroups of [C12 C12] C2. The 90 subgroups of C12
C12 are subgroups of the full group as well, of course, and also combine with
inversional vlsets to form other subgroups. The set of inversional vlsets in any
subgroup is calculated by adding any individual vlset to the cycle of transpositional
vlsets formed by the subgroup’s generator. The generator of each subgroup
essentially acts as a pair of interval cycles. For example, the transpositional vlsets in
the subgroup generated by <02> all contain a member of the 0-cycle—which is
always 0, of course—followed by a member of the even 2-cycle, 0, 2, 4, 6, 8, and t.
A set of corresponding inversional vlsets is generated by adding any particular vlset
to this transposition cycle. For n = , and vlsets <x,y> and <a,b>i, the
transpositional vlsets = <x*n, y*n>, and the inversional vlsets = <x*n+a, y*n+b>i.
Therefore, the inversional vlsets for the <02> subgroup are <a+(n*0), b+(n*2)>i, or
<a, b+2n>i, as addition by 0 has no effect. This produces the full set of inversional
vlsets for this subgroup: <a, b+(2*0)>i, <a, b+(2*1)>i, <a, b+(2*2)>i, <a, b+(2*3)>i,
<a, b+(2*4)>i, <a, b+(2*5)>i, <a, b+(2*6)>i, <a, b+(2*7)>i, <a, b+(2*8)>i, <a,
b+(2*9)>i, <a, b+(2*t)>i, and <a, b+(2*e)>i. If <25>i is arbitrarily chosen to serve
as an example, the remaining vlsets are <27>i, <29>i, <2e>i, <21>i, and <23>i along
38
with <02>, <04>, <06>, <08>, <0t>, and <00>. The group table is shown in Table 2-
3.
For another example, the three two-member subgroups, generated by
<06> <̧60>, and <66>, respectively, each combine with inversional vlset pairs to
form many subgroups. <06> combines with any <a, b+6n>i to form a four-member
subgroup with <00> as the identity. See Table 2-4 for some example group tables.
Because addition by six is an involution, each generator partitions the 144 inversional
vlsets into 72 pairs; <a, b>i and <a, b+6>i, <a, b>i and <a+6, b>i, and <a, b>i and
<a+6, b+6>i. These produce 216 four-member subgroups.
The procedure used to generate subgroups above may be extended for any of
the remaining subgroups of C12 C12. Because inversional vlsets are their own
inverses, all 144 combine with the trivial subgroup consisting solely of <00>, to
produce two-member subgroups; an example is shown in Table 2-5. Table 2-6 lists
the subgroups formed by combining inversional vlsets with the subgroups of C12
C12; the order is listed first, then the transpositional generator and inversional vlsets,
and finally the number of subgroups. In order to save space, inversional vlsets are
shown as multiples of n, where n = .
Some of the subgroups shown below cannot be generated by a single element.
For these subgroups, the method for calculating inversional vlsets shown above is
insufficient. The two transpositional generators form the subgroup through all
combinations of their cycles, and the inversional vlsets must match all of these
combinations. For example, the subgroup generated by {<04>, <40>} includes the
39
two cycles formed by these vlsets, <04> - <08> - <00> and <40> - <80> - <00>, as
well as all of their combinations: <44>, <84>, <48>, <88>. Each of these vlsets must
have a corresponding inversional vlset. On Table 2-6 below, these subgroups are
shown as simply adding some inversion <a,b>i to the full spectrum of vlsets
generated by the transpositional vlset cycles, for example: {<04>, <40>} + <a,b>i
should be understood as showing that the values of each transpositional vlset
generated are added component-wise to some inversional vlset. Table 2-7 shows the
group table for the {<04>, <40>} + <61>i subgroup.
The groups formed by vlsets of larger cardinalities are similar to those
discussed above. The transpositional trichordal vlsets form the direct product group
C12 C12 C12, of order 1728, with <000> as the identity; the group is generated
by {<001>, <010>, <100>}. Like the dyadic direct product group, this group is
abelian and simply transitive; that is, it is commutative, and any pair of pcsets is
spanned by a unique member of the group. Similarly, the full group of transpositional
and inversional vlsets is the semidirect product [C12 C12 C12] C2 of order
2456. Larger vlset cardinalities result in larger groups, of course.
Group Representations
Groups formed by traditional transpositions are cyclic groups—C2, C3, C4,
C6, and C12. These groups correspond to the rotational symmetries of regular
polygons. The order of the group is equal to the number of sides; C2 is the rotations
of a line, C3 the triangle, C4 the square, C6 the hexagon, and C12 the dodecagon.
Figure 2-19 shows the symmetries of the triangle and the accompanying C3 group.
40
T4 is a single clockwise rotation, moving A to B, B to C, and C to A, T8 is a single
counterclockwise rotation, moving A to C, C to B, and B to A, and T0 is no rotation.
While these rotations change the positions of the vertices, they preserve the structure
of the equilateral triangle. If a rotation between T4 and T8 were added, such as T6,
the rotation would result in an upside-down triangle. Because this operation does not
preserve the symmetry of the triangle, it is not a member of C3.
Dihedral groups are made up of transpositions and inversions. These groups
also correspond to the symmetries of regular polygons, but include reflections in
addition to rotations. Figure 2-20 shows the three axes of reflection included in D3—
the symmetries of the triangle. As shown, each inversion maps one vertex into itself,
and the other two onto one another. The inversions are named for the vertex they
preserve, therefore IA maps B and C onto one another, IB maps C and A onto one
another, and IC maps B and A onto one another. When operating upon pcs or pcsets,
these inversions follow the cyclic structure of the transpositions, In, In+4, and In+8,
for any n in . The D12 group works the same way, but corresponds to the
rotations and reflections of a dodecagon. Figure 2-21 shows a dodecagon with pcs at
the vertices. A transposition Tn rotates the figure n*30 degrees clockwise, and an
inversion In reflects the figure about some axis, where each pc a maps onto the pc n –
a. For instance, I0 reflects the figure about a vertical axis, and each pc maps onto its
mod 12 inverse 0 – 0, 1 – e, etc.
The direct product group C12 C12 essentially multiplies the rotational
symmetries of the dodecagon with itself. This produces a dodecagonal torus—a
41
dodecagon that traces the path of a larger dodecagon; see Figure 2-22. Each member
of a transpositional dyadic vlset operates upon either the vertical or horizontal
dodecagon of the torus. For example, vlset <23> describes a vertical rotation of two
vertices and a concurrent rotation of three. This maps each pcset onto the location
determined by the vlset.
The semidirect product group [C12 C12 C12] C2 includes reflectional
symmetries as well as rotational symmetries. Like the rotational symmetries, each
inversional vlset may be understood as inverting the vertical and horizontal
dodecagons at the same time. For example, vlset <00>i creates an axis of symmetry
that runs straight through the torus through pcsets {00}, {06}, {66}, and {60}, all of
which map onto themselves. The torus is then flipped about this axis, mapping each
pcset onto its <00>i—related pair. Other axes of symmetry are not as simple, as they
do not form straight lines through the torus. For example, <11>i produces different
axes of symmetry at different parts of the torus. It may be helpful to decompose the
inversion operation in order to make the representation simpler. Just as T5I produces
the same result as I5, <11>i may be interpreted as <11><00>i, where the torus is
flipped about the <00>i axis, then rotated by the transposition <11>.
The toroidal representation of vlset groups is somewhat unwieldy, and
virtually impossible to conceive of for higher cardinalities; the transpositional triadic
vlset group, C12 C12 C12, for example, corresponds to a dodecagonal 3-torus, a
figure which is created by gluing each of the opposing faces of a cube to one another.
42
This requires four dimensions.62
As an alternative to the higher-dimensional tori, the
vlset groups may be represented by the combination of dodecagons rotating and
flipping next to one another. This is what the groups essentially represent—multiple
C12 and/or D12 groups operating at the same time.
Figures 2-23a – c show three dodecagons aligned with one another.
Transpositional vlsets slide each dodecagon by the specified amount, while
inversional vlsets flip each one. Figure 2-23b demonstrates that vlset <216> maps
pcset <000> onto {216}. The vlset is applied to the dodecagons from outside in; the
outer dodecagon rotates by T2, the middle by T1, and the innermost by T6. In this
particular representation, the pcset is always determined from the vertical box which
the vertices of the dodecagons rotate through. Transpositions rotate the dodecagons
counterclockwise to move their vertices through the box in ascending order. Figure
2-23c shows that vlset <358>i maps pcset <000> onto {358}; like the transpositional
vlset, the inversional vlset begins with the outermost dodecagon and works inward.
As the figure shows, each dodecagon is flipped about a different axis, reversing the
order of the pcs. This group representation could be created for any vlset cardinality,
as each vlset member operates upon a different dodecagon.
Although the natural toroidal group representations are unwieldy, and even
unrepresentable in higher dimensions, one advantage they have over the interlocking
62
The toroidal objects discussed in this section are the same as the orbifolds in the
work of Tymozcko 2006, and Callender, Quinn, and Tymoczko 2008. In their work,
the authors use permutational equivalence to reduce the dyadic torus to a Moebius
strip, and the four-dimensional triadic torus to a torus in the shape of a triangular
prism with a half twist.
43
dodecagons is that each pcset is represented by some point on the figure. The
interlocking dodecagons must be arranged so that their alignment produces a
particular pcset. Figure 2-24 demonstrates the mapping <238> - <t27> - {053}; as
shown, the dodecagons must be arranged so that pcset 238} is aligned, before the
rotations of the vlset may be applied. One advantage of the toroidal representations is
that they show the vlset groups operating upon one figure, much like the cyclic and
dihedral groups they are products of.
44
Chapter 3: Voice-leading Spaces
Voice-leading sets provide a way to describe voice leadings on their own
terms, as they do not privilege any specific voice leading characteristic, such as
uniformity, balance, efficiency, contrary or similar motion, etc. This chapter
demonstrates some of the ways in which voice-leading spaces may be used to model
and represent these disparate voice-leading types. By generalizing the concept of
voice-leading distance, it becomes possible to create spaces for individual analytical
or compositional contexts.
The voice-leading spaces produced here may be used as representational,
analytical, or compositional resources. Pitch-class voices may be seen as “surfing”
the pcsets of a voice-leading space.63
The paths these voices travel will change if the
voice-leading space is changed, demonstrating the use of different metrics of voice-
leading distance. As an analytical tool, a voice-leading space may aid in the voice-
leading interpretation of musical surfaces, as will be discussed in the next chapter.
Finally, the spaces may be useful for composition in a variety of ways. For example,
if a composer wishes to maintain the same voice leading at various locations within a
piece, the space shows all of the possible pcset pairs that correspond to that voice
leading. Alternatively, a composer could maintain the same path through multiple
spaces, transforming the individual voice leadings, but preserving the relationships
among them; this creates a type of voice-leading contour transformation. Just as
63
David Lewin suggests that members of set-class [013] surf an [013] tonnetz in the
subject of Bach’s fugue in F# minor, WTC I in “Notes on the Opening of the F#
Minor Fugue from WTC I,” Journal of Music Theory, 42, n. 2 (1998): 235-239.
45
contour theory reduces individual melodies to their relative intervals, a voice-leading
contour reduction would reduce a passage to the transformations of its voice-leadings.
For example, imagine a three-chord passage which consists of two vlsets: X and
T2(X). If a given vlset Y substitutes for X, the chords will be transformed, but the
transpositional relationship between the voice-leadings—Y and T2(Y)—will be
preserved. Some ideas for future work in voice-leading contour will be suggested in
chapter 5.
Edward Gollin uses chess to describe the various conceptions of distance in
transformational music theories. Gollin distinguishes between Kingspace, which
involves motion between adjacent squares, and Knightspace, which is produced by
the Knight’s L-shaped motion.65
Possible musical analogs of these spaces are the
chromatic scale and circle of fifths, where the King moves by semitone, and the
Knight by fifth.66
The following will demonstrate some of the other chessboards
created by the remaining players—the vlsets.
The voice-leading spaces discussed in chapter 1, Tymoczko, Quinn, and
Callender’s orbifolds, Straus’ set-class space, and Roeder’s ordered interval spaces,
focus on harmonic objects organized in space. In the orbifolds, as well as Straus’
spaces, "voice leading" itself is understood as a means to connect these objects, and
voice-leading distances are measured by their total semitonal content. Although his
65
“Representations of Space and Conceptions of Distance in Transformational Music
Theories” (Ph.D. Diss., Harvard University, 2000): xii-xv. 66
Gerald Balzano compares the chromatic scale and circle of fifths to the motions of
the Queen and Knight, respectively, in “The Group-theoretic Description of 12-fold
and Microtonal Pitch Systems” in Computer Music Journal, 4, 4 (1980): 66-84.
46
presentation of ideas is significantly different, Straus’ space is geometrically identical
to the orbifold spaces presented by Tymoczko, Quinn, and Callender.67
As the latter
authors show, the set of ordered pcsets in any cardinality, and corresponding multi-
dimensional space, may be reduced through various equivalences, including
permutation (pc order), transposition, and inversion. The dyadic Moebius strip,
triadic toroidal prism, tetrachordal tetrahedral torus, and higher-dimensional
structures are all the result of permutational reduction.68
In Joseph Straus' trichordal set-class space, shown in Figure 3-1, each set-
class is connected to the set-classes it may move to by a fuzzy transposition or
inversion with a single semitonal offset.69
Alternatively, these operations may be
thought of as employing voice-leading class <01>, <001>, <0001>, etc. under
rotation, transposition, and inversion. Thus, Straus uses a type of parsimony to
generate his space, but generalizes that parsimony; each line segment represents
vlclass [001], whose constituents are organized through T, I, and rotational
equivalence. For example, the voice leadings <223>, <232>, and <9tt>, when
applied to ordered pcset <026>, produce {249}, {258}, and {904}, members of set-
classes [027], [036], and [037], respectively; these relationships are demonstrated by
67
“Generalized Voice-leading Spaces” in Science, 320 (2008): 346-348. As
discussed above, Roeder’s geometrical models are identical in structure, but
interpreted differently. Rather than modeling voice-leadings, Roeder interprets the
coordinates of his spaces as ordered intervals, uses them to generate pcsets, tn-classes,
and set-classes, and discusses the properties of these harmonic materials. 68
For a discussion of this reduction, as well as a comparison with the theories of
George Perle, see Dave Headlam’s introduction to Theory and Practice, 33 (2008):
32-39. 69
Straus 2005, 45-108.
47
the scs’ locations in the space: [026] is connected to [027], [036], and [037] (among
others). Straus then examines this space, as well as others in higher dimensions, and
explores the properties of the set-classes that it demonstrates.
Dyadic Spaces
The orbifold spaces proposed by Tymoczko, Quinn, and Callender provide a
starting point for the creation of voice-leading spaces. Figure 3-2a shows the full,
unreduced dyadic orbifold. As shown, each pair is an ordered pcset, and they are
connected by semitonal voice leading—<01> along the x-axis, and <10> along the y-
axis. The sum 0 inversions of the vlsets, <0e> and <e0> respectively, describe the
motions along these axes in the opposite direction. This space wraps around
vertically and horizontally to form a torus, discussed in the previous chapter as the
symmetrical object rotated and flipped by the semidirect product group of dyadic
vlsets [C12 C12] C2.
The torus is reduced to a Moebius strip through permutational equivalence; in
other words, (x, y) = (y, x). The space may be folded over along the diagonal, which
is made up of the singleton multisets, as shown on Figure 3-2b. Figure 3-2c
rearranges the pcsets to demonstrate the way in which the edges twist together. As
shown, the pcsets on the upper right side, {66}, {57}, {48}, {39}, {2t}, {1e}, and
{00}, map onto the pcsets on the lower left side with a twist. The other two sides are
the singleton multisets, which trace a circular path once around the torus, and once
through the center. This line becomes the edge of the Moebius strip when reduced
through permutational equivalence. Imagine that the line of multisets is a rubber
48
band. When the torus is reduced, this rubber band is twisted and realigned with itself,
so that tritone-related multisets are directly across from one another, as shown in
Figure 3-3. The line of multisets is a torus knot, a continuous line which wraps
around the outside of a torus. The Moebius strip lies in the plane formed between
each tritone-related pair as shown on Figure 3-4a.70
As Figure 3-4a shows, twelve lines of pcsets stretch across the Moebius strip.
The pcsets on these lines all add up to the same sum. Each line is a diagonal on the
toroidal space, corresponding to y = -x + n, where n = . For example, the line
between {00} and {66} is the diagonal y = -x + 0, and contains the zero-sum dyads
{1e}, {2t}, {39}, {48}, and {57}. The five other even-sum lines are transpositions of
this line, just as the six odd-sum lines are transpositions of one another. The
singletons form the edge of the strip, because they can only map onto two other
unordered pcsets by either rotation of the spaces’ generating voice leading. For
example, pcset {33} moves by <01>, <10>, <0e>, and <e0> to either {34} or {23}.
Non-singleton pcsets move to one of four unordered pcsets; for example, pcset {34}
may move to {35}, {44}, {33}, or {24}.71
70
Callender, Quinn, and Tymoczko 2008, and Tymoczko 2006 and 2011 discuss this
space, but only show the Moebius strip in its unfolded form. See “The Geometry of
Musical Chords,” in Science, 313 (2006): 72-74 and A Geometry of Music: Harmony
and Counterpoint in the Extended Common Practice (New York: Oxford University
Press, 2011). Rachel Hall states that the Moebius strip is twisted three times,
therefore the torus knot is a trefoil in “Geometrical Music Theory,” Science, 320
(2008): 328-329. 71
This space intersects with some of the theories of George Perle. Each of the same-
sum-dyad diagonals is equivalent to what Perle calls an inversionally complementary
cycle; Perle first shows these cycles as the combination of chromatic scales, as in the
orbifold, then demonstrates that they are reordered in the order-position pairs formed
49
The dyadic Moebius strip may be further reduced to a set-class line by
inversion, as shown on Figure 3-4b. This is equivalent to Straus’ set-class space, as
well as the tn-class cycle shown in the previous chapter, Figure 2-10a. The original
torus may also be reduced by transpositional equivalence, which results in a circle.
As mentioned above, the orbifolds locate pcsets by maximal harmonic
similarity. Voice leadings may be traced from one pcset to another as pitch-class
intervals. These intervals are easily understood as transpositions, but are more
difficult to translate to inversion, as the indices of inversion are not consistent among
adjacent pcsets. As Straus shows, all adjacent pcsets may be inverted onto one
another by some index with minimal offset. This is logical, as adjacent pcsets only
differ by one semitone. In Straus’ fully reduced set-class space, each set-class is
connected to those which it transposes or inverts onto with minimal offset. In terms
of vlsets, the dyadic orbifold can be seen as the space of <01>, and vl-class [01]i. All
inversions with minimal offset are simply some transposition or inversion of this vl-
class, such as <23>i, <65>i, etc.
The dyadic orbifold provides a model by which voice-leading spaces may be
created for other vlsets. These spaces differ from the orbifolds in the way in which
voice-leading distance and spatial distance are aligned; most of the spaces do not
equate voice-leading distance and semitonal offset. Rather, voice-leading parsimony
is defined by the vlset which generates the space. In other words, the space’s “unit”
distance is the generating vlset.
by two inversionally-related twelve-tone rows; Twelve-Tone Tonality 2nd
ed.
(Berkeley: University of California Press, 1996) 7-11.
50
The toroidal representation of dyadic pcsets, unrolled in Figure 3-2, can be
seen as generated by vlset <01> and pcset <00>. The x-axis is determined by vlset
<01>, the y-axis by <10>, and pcset <00> is placed at the origin. All other pcsets in
the space are generated by these elements, as all adjacent pcsets are related by single
semitone voice leading. In addition, adjacent pcsets map onto one another by some
member of vlclass [01]i. As Figure 3-2 shows, the transposition-class cycle shown in
chapter 2, Figure 2-10 forms a line along the x-axis. In fact, all lines parallel to the x-
axis contain some rotation of this cycle, and as well as all lines parallel to the y-axis.
Other vlsets may be substitued for <01> to generate their own voice-leading
spaces. To return to the chess analogy, if <01> produces Gollin’s Kingspace, <05>
could be used to produce Knightspace.72
For example, if vlset <05> is used with
pcset <00> at the origin, the resulting space locates pcsets by single-voice fifth voice
leading, rather than semitone. All adjacent pcset pairs are mapped onto one another
by some member of vlclass [05]i. This particular space, shown in Figure 3-5a, is an
M5 transformation of the original space. Just as the <01> space is reduced by
permutational equivalence to a Moebius strip, the <05> space may be folded along
the x=y diagonal and wound into a Moebius strip, as shown in Figure 3-5b. As on the
full space, adjacent pcsets are mapped onto one another by single-voice motion by
fifth, which corresponds to the rotations of vlsets <05> and <07>. Figures 3-6a and b
72
This is for conceptual comparison only—Gollin’s Kingspace is a two-dimensional
5x5 chessboard, but Knightspace is a four-dimensional object. The <01> and <05>
spaces defined here are similar in structure.
51
show the tn-class circle and set-class line produced by vlset <05>. Like the toroidal
space and Moebius strip, the figures are simply M5 of the <01> spaces.73
Although the space in Figure 3-5 is generated by vlset <05>, this is by no
means the only voice leading the space represents. Moving west parallel to the x-axis
inverts the vlset—<07>. Moving south parallel to the y-axis similarly produces
<70>. <05> is simply the space’s unit distance. The voice leading between any two
pcsets on the space can be measured in terms of this vlset. For example, two unit
distances (<05> + <05>) along the x-axis produce the vlset <0t>. This is also true for
inversional voice leadings; the unit distance is a member of vlclass [05]i, and a
straight line of four unit distances is a member of vlclass [04]i. Where n = ,
moving along any y = x + n diagonal is equivalent to transposition by the total sum of
the vlset, and motion along any y = -x + n diagonal is equivalent to wedge motion by
the difference between the members of the vlset.
The unit distance on a voice-leading space demonstrates the contextualization
of voice-leading parsimony. Within the <05> space, single-voice motion by fifth is
parsimonious, and single-voice motion by semitone is not. As Figures 3-5a and b
show, semitonal offset requires five moves; only single-voice motion by tritone
requires more steps. This is summarized nicely by the tn-class circle and set-class
73
This equivalence is also shown by Walter O’Connell. Using the all-interval-class
tetrachords as a model, O’Connell creates two M-related six-dimensional interval
spaces, which he calls tone-lattices; he then demonstrates the relationships among a
number of geometrical figures—symmetrical pitch-sets—on the spaces, as well as an
all-interval twelve-tone row as a bent-wire figure. He then describes the intervallic
transformations that result from rotations of the figures within the lattices. See “Tone
Spaces,” Die Reihe, 8 (1968): 53-64. Originally published in German in 1962.
52
lines shown in Figures 3-6a and b. The minimum distance from ordered tn-class [00]
to [01] or [10], for example, is five. Figure 3-6c shows a dyadic passage among
adjacent pcsets in the space shown in 3-5b; to produce a smooth motion on the space,
one voice moves by ic 5 while the other maintains its pc. Figure 3-6d shows another
passage which mixes smooth and compound moves in the space.
Toroidal voice-leading spaces may be produced by the combination of any
vlset <ab> and any pcset <cd>. The x-axis is determined by n*<ab>(<cd>), where n
= , and the y-axis by n*<ba>(<cd>). Adjacent pcset pairs may be mapped onto
one another by some member of vlclass [ab]i. Any two pcsets on the space are
related by some member of the subgroup generated by {<ab>, <ba>}, and combined
with their inversional equivalents.
Although any transpositional vlset may be used to generate a voice-leading
space, not all spaces include the full collection of pcsets.74
Complete spaces may
only be generated by vlsets whose sum and difference is prime to twelve—1, 5, 7, or
e. All others produce incomplete spaces. For example, the sum of vlset <2e> is
prime to twelve—1, but the difference is not—3 or 9. As a result the y = -x + n
diagonals of the space include some redundancy. As Figure 3-7a shows, these
diagonals are formed by <2e> - <e2> = <39>. The cycle produced by vlset <39> has
only four members, therefore the diagonals only contain four distinct pcsets each.
74
In a discussion of K-nets and Perle-Lansky cycles, David Lewin describes a
method of moving from one space to another as “hyperwarping,” then demonstrates
some ways in which a space may be transformed into another in “Thoughts on
Klumpenhouwer Networks and Perle-Lansky Cycles,” in Music Theory Spectrum, 24,
n. 2 (2002): 228-230.
53
The interaction of the two cycles of the vlset along the x-axis also shows redundancy;
the fourth member of the 2-cycle is 8, and the fourth member of the e-cycle is 8. If
{00} is at the origin, the fifth pcset along the x-axis is {88}; this pcset is replicated
along the y = x diagonal.
The <2e>, <00> voice-leading space shown in Figure 3-7a is different than
the spaces discussed thus far. A 4 x 12 box has been placed on the figure, which
shows the 48 discrete pcsets of the space.75
When forming a torus, the 12 pcsets of
the y axis wrap around vertically. The x axis, however, requires more maneuvering.
As the figure shows, the fourth pcset of the x axis, <69>, proceeds to pcset <88>, not
<00>. <88> is the fourth pcset along the y axis, therefore, the cycle begun along the
x axis overlaps with the cycle along the y axis. Four pcsets later in the cycle involves
another shift up four pcsets along the y axis, to {44}. Another four pcsets later, the
cycle returns to <00> at the origin. As discussed above, the <01> torus is made up of
twelve vertical and twelve horizontal dodecagons; pcsets are located at their vertices,
which intersect. In the <2e>, <00> space, the dodecagons overlap. Figure 3-7b
shows the torus formed by the space. The four y axis cycles are shown as vertical
dodecagons, which must be rotated around the torus, so that the x axis cycles are
aligned. These cycles are dodecagons that have been folded twice, as their twelve
vertices are now grouped in the four vertical planes of the y axis cycles.
75
This box could be drawn in other ways as well, such as 12 x 4. As long as the box
has these dimensions, it will produce 48 unique pcsets no matter where it is located.
All of these are equivalent.
54
Table 3-1 lists the dyadic spaces by vlset, demonstrating which are complete
by including the number of distinct ordered pcsets included in the toroidal space
generated by each vlset. Vlsets which are sum-zero inversions of one another are
aligned horizontally. These spaces are simply reflections of one another, provided
inversionally-related sets of pcsets are used as objects. Complete spaces are
unchanged no matter which pcset is placed at the origin, and their inversional
equivalents can always be mapped onto one another through flips and rotations.
Table 3-1 also demonstrates that vlsets related by multiplication generate spaces with
the same properties.
Many vlsets partition the pcsets into multiple collections, which require
multiple spaces to represent. For example, Figure 3-8 shows the toroidal space
generated by vlset <02> with pcset <00> at the origin. As the figure shows, the space
only includes even pcsets, that is, only those formed by the even whole-tone scale.
This results in a space which is six by six, rather than 12 by 12, one-quarter of the
size of the <01> space. Unlike the <2e>, <00> space shown above, the x and y
cycles of the <02> space do not overlap. As a result, the spaces fit onto a torus
formed by six vertical and six horizontal hexagons.
The small size of the <02>, <00> voice-leading space is a result of the
symmetries of the whole-tone scale. Although adjacent pcsets must be related by
either <02>, <20>, <0t>, or <t0>, any pcset pairs in the space are related by some
member of the subgroup generated by {<02> <20>} and combined with inversion.
This subgroup has 72 members—36 transpositional vlsets and their corresponding
55
inversional vlsets—which results in four distinct voice-leading spaces of 36 pcsets
each. The four spaces are determined by the whole-tone content of their pcsets. Two
of the spaces contain pcsets made up of one whole-tone scale; in addition to the all-
even space shown in Figure 3-8, the all-odd space, shown in Figure 3-9, may be
generated with pcset <11> at the origin. The other two spaces contain all pcsets made
up of both whole-tone scales, one space is even-odd, and the other odd-even—<01>
and <10> are representative pcsets (see Figures 3-10 and 3-11). Due to the group
structure of the spaces, any pcset member of a space may be placed at the origin, and
the full space will result. For example, if pcset <76> is at the origin, a rotated version
of the full odd-even space is generated. As these spaces are toroidal, the rotations
show that they are equivalent.
The toroidal <02> spaces shown in Figures 3-8 and 3-9 may also be reduced
to Moebius strips. Like the spaces they are reduced from, they contain fewer pcsets
than the <01> version. Figure 3-12 demonstrates the Moebius strip formed by <02>
with <11> at the origin. As the figure shows, there are only six lines of pcsets on this
space. Like the <01> Moebius strip, the singletons form a trefoil knot, and each line
of pcsets between them adds to the same sum. Each blue line on the space represents
the rotations and sum-zero inversions of vlset <02>: <02>, <20>, <0t>, and <t0>.
Not all of the <02> spaces may be reduced to a Moebius strip. Figures 3-10
and 3-11 show the toroidal voice-leading spaces formed by <02> with pcset <01> and
<10>, respectively. As mentioned above, vlset <02> preserves the whole-tone
contents of any pcset it operates upon. Therefore, each pcset in the spaces is made up
56
of one odd and one even whole-tone member. All pcsets in the <01> space are even-
odd, while all the pcsets in the <10> space are odd-even. Therefore, no pcsets may be
reduced by permutational equivalence. Instead, the two spaces are the permutational
equivalents of one another. Of course, either space may be reduced by transpositional
and/or inversional equivalence, but these form a circle (or hexagon), and line,
respectively, as shown in Figures 3-13a and b. Like the toroidal spaces, vlset <02>
produces multiple transpositional and inversional spaces, and pcset members of one
space cannot be linked to the other by any rotation or inversion of vlset <02>. In fact,
they cannot be linked by any transposition of <02> either; as mentioned in chapter 2,
vlclass [02] produces the transposition-class cycles shown in Figure 3-13a. Although
the toroidal spaces produced by vlsets <02> and <24>, for example, are different,
they both reduce to the transposition-class and set-class spaces shown below.
It is useful to describe transformations between voice-leading spaces, as a
given piece may “modulate” from one to another, or back and forth between a set of
spaces. As discussed above, inversion and multiplication are replicated throughout a
voice-leading space. This is only true for sum-zero inversion, and multiplication
without transposition. Other indices of inversion, and multiplication with
transposition change the sum of a vlset, therefore the changes to the voice-leading
space produced are more drastic.
As discussed in chapter 2, the transposition of a vlset—excluding T0—
produces a new vlset which maps the set of transposition-classes onto itself in the
same way. For example, <02> maps pcset <23> onto <25>, and <68> maps <23>
57
onto <8e>; both vlsets map the members of tn-class [01] onto [03], and the difference
between the mappings is equal to the transposition of the vlset. That is, <8e> – <25>
= <66>, and <02> + <66> (or T6) = <68>. This is replicated in the voice-leading
spaces produced by any vlsets related by transposition. T6 is a unique
transformation, in that it preserves the sum of the dyadic set it operates upon, as 6 + 6
= 12, and preserves the difference between the members of the set by adding six to
both of them. This produces pairs of voice-leading spaces in each sum which are
related to one another by <66>.
As Figure 3-14 shows, the toroidal space generated by <68>, <00> is similar
in structure to that formed by <02>, <00> in Figure 3-8. Although some of the pcsets
have been relocated, the transposition-class structure is preserved. Motion along a
diagonal in either space is the same, because <02> + <20> = <68> + <86> = <22>.
Motion parallel to the x or y axis alternates transposition by <66> with <00>. The
cause of this feature is easily shown if vlset <68> is split into (<66> + <02>): <02> +
<02> = <04>; <66> + <02> + <66> + <02> = <04>. Any even number of vlset
<66>s sums to <00>. The result of these relationships is a space in which the even
diagonals of the <02> space are preserved, and the odd diagonals are rotated by <66>.
Figure 3-15 shows the space generated by <67>, <00>, for comparison with
Figure 3-2a, the <01>, <00> space. Like the <02> and <68> spaces, every other
diagonal is transposed by <66>. The <67>, <00> space may be reduced to a Moebius
strip as shown in Figure 3-16. As mentioned above, the pcset lines on the space are
58
the diagonals y = -x + n. Therefore, the odd-sum lines are <66> transpositions of the
corresponding lines in the <01> Moebius strip, as shown in Figure 3-4a.
Nearly all vlsets paired by <66> generate similar spaces. Members of vlclass
[06] are exceptions, as <66> simply transforms them by rotation; this swaps the x and
y axes, producing a space where each pcset is a rotation of the corresponding set on
the other space. Some vlsets whose sums are 4, 8, and 0 also form exceptions,
because transposition by tritone drastically changes the pc cycles they form. For
example, vlset <04> produces a space which includes nine pcsets, while <6t>
includes eighteen. The other unequal sum-four pair is <22> and <88>, for the same
reason—the 2-cycle has six members, while the 8-cycle has 3. The sum-eight
unequal pairs are the inversions of these pairs: <08> and <26>, and <44> and <tt>.
The final exceptions are the sum-zero vlsets <00> and <66>, and <2t> and <48>.
Transpositions which change the sum of a vlset may change the size of the
voice-leading space that set produces. However, the spaces will still have the same
transposition-class structure. As mentioned above, the tn-class cycles of a vlclass
form the lines parallel to the x and y axes of the toroidal space. As shown in chapter
2, only ordered vlclasses [01], [05], [50], and [10] form complete tn-cycles. Because
the cycles are complete, any transposition of the vlclass combined with any pcset at
the origin will produce rotations of the vlclass’s cycle in lines parallel to the x and y
axes. This is true even for spaces which produce pcset redundancies, such as the
voice-leading space <12>, <47>, as shown in Figure 3-17. As the figure shows, the
vlclass [01] tn-class cycle along the x-axis begins with tn-class [03], and continues
59
through the remainder of the cycle to [02]. Just as in the <01> space, each line
parallel to the x axis is a rotation of this cycle, and each line parallel to the y axis is a
retrograde rotation.
The <12>, <47> toroidal space in Figure 3-17 contains 48 unique pcsets,
which have been boxed. This space is similar to the <2e>, <00> space shown above,
in that the x and y cycles overlap. The two spaces’ tori can both be represented using
four dodecagons with a twist. In this case, the dodecagons are twisted downward, as
the cycle along the x axis leaps to the eighth pcset of the y axis, and then the fourth
before returning. Because x + y = <33>, y = x diagonals begin to repeat after four
pcsets.
As shown in chapter 2, the remaining vlclasses form multiple tn-class cycles.
As mentioned above, these vlclasses also form multiple voice-leading spaces. The tn-
cycles these spaces produce is dependent upon the pcset used to generate the space.
For example, vlclass [02] produces two tn-class cycles, one which contains the even
tn-classes, and one odd. As Figures 3-8 through 3-11 demonstrate, if an even tn-class
is placed at the origin, such as <00> or <11>, the space produces the even cycle.76
If
an odd tn-class is at the origin, such as <01> or <10>, the space produces the odd
cycle. These two tn-class cycles are produced by any transposition of vlclass [02],
and the voice-leading space may be more or less complete depending on the sum of
the vlset. As table 3-1 shows, the odd sums contain all of the odd set-classes, while
the even sums contain all even set-classes. Therefore, each odd sum contains two
76
Although {11} is an odd pcset in terms of whole-tone membership, it is a member
of the even tn-class [00].
60
members of vlclasses (under transposition and inversion) [01], [03], and [05]; each
even sum contains two members of [00], [02], and [04], and one member of [06].
The two members of each vlclass in each sum are paired by <66>, as mentioned
above, and vlclass [06] maps onto itself under <66>.
As mentioned above, the sum-zero inversion of a vlset, <01> and <0e> for
example, simply retrogrades the x and y axes of a voice-leading space. In terms of
the torus, this is a reflection about the sum-zero axis. Inversion by other sums is
equivalent to this reflection, plus some rotation produced by transposition. As in the
discussion of transposition immediately above, this may change the size of the space,
but preserves the tn-class cycles it is made up of. The inversion, or reflection,
retrogrades the tn-cycles. For example, vlset <02> is transformed by <44>i into
<42>; Figure 3-18a shows the toroidal space <42>, <11>. A comparison of this
space with that of <02>, <11>, shown in Figure 3-9, demonstrates that the spaces
contain the same tn-class cycles in opposite order. Of course, the <42> space only
contains 12 pcsets, which are repeated throughout the 12 x 12 grid. The <02> space
has been reduced, but also would repeat itself in the full 12 x 12 space.
The <42>, <11> space is another in which the x and y cycles overlap. Due to
the small size of the space, the cycles form hexagons, rather than dodecagons. This is
not easily represented as a torus. As an alternative, the y hexagons may be aligned,
and the two x hexagonal paths traced between them, as shown in Figure 3-18b. If the
adjacent vertices of the hexagons were connected, this would form a hexagonal prism.
61
As mentioned above, multiplication is another way to navigate among voice-
leading spaces. Multiplication operates very much like inversion. When it is not
combined with transposition, multiplication preserves the structure of a voice-leading
space, and can be seen as operating upon each individual pcset; this is true whether
the multiplication operation maps the chromatic scale to the circle of fourths or fifths.
When combined with transposition, however, multiplication may change the sum of
the vlset it operates upon, and thus change the size of the voice-leading space it
produces.
In the following discussion, TnM5 will denote the mapping of the chromatic
scale to the cycle of fourths, and vice versa, followed by Tn; TnM7 will denote the
mapping of the chromatic scale to the cycle of fifths, and vice versa, followed by Tn.
For example, T1M5 maps vlset <45> onto <92>, as 5 * <45> = <81>, and <81> +
<11> = <92>; TtM7 maps vlset <45> onto <29>, as 7 * <45> = <4e>, and <4e> +
<tt> = <29>. Using TnM5 and TnM7, any member of vlclass [01] or [10] may be
mapped onto any member of [05] or [50], and vice versa. Although multiplication
may be used for other vlclasses, it is equivalent to either transposition or inversion.
For example, T3M5 maps <48> onto <e7>, which is equivalent to <33>i.
It is unnecessary to show all of the dyadic voice-leading spaces here.
However, it is helpful to show some representatives of the different sizes of possible
spaces. Figure 3-19 shows the Moebius strip reduction of Figure 3-17. The four
singletons—<11>, <44>, <77>, and <tt>—do not reduce by permutational
equivalence, but the remaining 44 reduce to 22, forming a Moebius strip of 26 pcsets.
62
The strip contains four lines of pcsets, which correspond to the y = -x diagonals of the
torus; each line contains all pcsets of the same sum. The sums follow the cycle given
by the vlset’s sum, 3. The generating pcset <47> sums to e, so the remaining sums
are 2, 5, and 8.
A small space—only 12 pcsets—is the <42>, <11> torus shown in Figure 3-
18. Like the previous space, this space may also form a Moebius strip, but only
contains 7 pcsets when reduced through permutational equivalence. This space has
only two lines of pcsets, which correspond to sums 2 and 8. Figure 3-20 shows the
reduced space. The two lines shown in the figure would be on opposite sides of the
Moebius strip; for ease of reading, they are shown as parallel.
Many of the small voice-leading spaces do not translate to a Moebius strip as
well as other, simpler shapes. Generally, spaces with less than twelve pcsets are
better represented some other way. For example, vlset <00> produces spaces with
single pcsets, which translates into a single point. Spaces with two pcsets, such as
those formed by vlset <66>, reduce either to a line, or a point, depending on the
pcsets involved. If a member of set-class [06] is a generating pcset, <66> will map
the pcset into itself under permutational equivalence. Toroidal spaces with three
pcsets may be represented by triangle, or a line if permutational equivalence is
involved. For example, the toroidal space <48>, <00> includes three pcsets, <00>,
<48>, and <84>, which reduces to <00> and <48> by permutation. The <48>, <01>
space—<01>, <49>, <85>—does not reduce further.
63
Some three-member spaces are not best expressed by a triangle, but a line,
such as the space formed by <06>, <11>, which includes pcsets <11>, <17>, <71>,
and <77> (see Figure 3-21). The space could be further reduced through
permutational equivalence. Both <11> and <77> are connected to <17> by the
generating vlset, but require two statements of the vlset to map onto one another. Of
course, vlset <06> also generates spaces that are not reducible, such as <06>, <01>,
which is made up of pcset <01>, <07>, <61>, and <67>. Figure 3-22 shows the
space as a square. This is the toroidal space—it cannot be reduced by permutation.
Because there are only four members, the circles which connect the pcsets can be
simplified to lines.
The voice-leading spaces formed by singletons may be represented by the
familiar geometric figures of their traditional transpositional counterparts; <11>,
<00> forms a dodecagon, <22>, <00> a hexagon, etc. These spaces are generally not
reducible by permutation, except for the special case of set-class [06]. For example,
<33>, <28> has four members, <28>, <5e>, <82>, <e5>, which reduce to two under
permutation equivalence. This changes the space from a square to a line. Other
twelve-member toroidal spaces generally reduce to one or two lines of pcs, if they are
reducible. As in the other spaces, the lines are made up of the pcsets which belong to
the same sum. Sum-zero vlsets, such as <1e> and <57>, produce pcsets in a single
sum, and therefore may reduce from a circle to a line, as shown in Figure 3-23.
Two vlsets form toroidal spaces with 18 members, <26> and <6t>; as
inversions, these vlsets form spaces which are reflections of one another. Because the
64
vlsets are members of vlclass [04], they produce four distinct tn-class cycles, as
discussed in chapter 2. Therefore, the vlsets produce four types of voice-leading
spaces. As a result of the 2-cycle in the vlset, each space preserves the whole-tone
membership of the generating pcset. This produces two versions of each basic space,
just as in the <02> spaces discussed above. Therefore, there are eight spaces of
eighteen pcsets each, adding up to the total of one hundred and forty-four.
Figure 3-24 shows the <26>, <11> toroidal voice-leading space. The boxed
portion of the space shows the 18 unique pcsets repeated throughout the figure. Like
the previous spaces, the x and y cycles overlap; in this case, they form six hexagons.
The six singletons do not reduce by permutational equivalence, but the remaining
twelve pcsets reduce to six. The space’s unusual structure results in a different
Moebius strip organization than shown by the other spaces. As Figure 3-24 shows,
both the y = x and y = -x diagonals contain only three distinct pcsets. The y = -x
diagonals are reduced to 2 through permutational equivalence. In the previous
spaces, all the pcsets of a particular sum are contained in a single y = -x diagonal.
These diagonals then become lines of pcsets across the Moebius strip. In the <26>,
<11> space, however, the pcsets of a particular sum are split into multiple diagonals.
For example, the diagonal containing pcset <77> contains the other sum-two dyads
<e3> and <3e>, but not <11>, <59>, or <95>. Although there are only three sums
represented by the pcsets of the space—2, 6, and t—they are divided into six distinct
diagonals. This is due to the fact that the voice leading represented by descending
along the diagonal is x – y, <26> - <62>, or <84>. <84> is a highly symmetrical
65
vlset, which maps the singletons onto [04] dyads, and the [04] dyads onto themselves
(under permutation) or back onto the singletons: <84> + <11> = <95>, <84> + <95>
= <59>, and <84> + <59> = <11>. Therefore, the two pairs of the same sum cannot
be mapped onto one another by this voice leading. The voice leading x – y forms the
lines of the Moebius strip, so as a result, the reduced space contains six pairs of
equally spaced pcsets. The two pairs of the same sum are located across from one
another.
Trichordal Spaces
Trichordal voice-leading spaces share many properties with the dyadic spaces.
It is more difficult to represent them, however, given that the toroidal spaces are four-
dimensional hypercubes. Through various equivalences, such as rotation,
transposition, inversion, and permutation, many of these spaces may be reduced to
smaller structures. Figure 3-25 shows the hypercube formed by <001>, <000>.77
The x, y, and z axes are generated by the three rotations of the vlset—<001>, <010>,
and <100>. As discussed above, this space may be reduced through permutational
equivalence, from 1728 pcsets to 364. Permutational equivalence includes rotation;
for example, the permutations of a, b, and c in the set <abc> include all six orderings.
Rotational symmetry alone differentiates between orderings of the set, therefore
<abc> is equivalent to <bca> and <cab>, but distinct from <acb>, <cba>, and <bac>.
Three-member trichordal sets thus reduce 6:1 through permutation, and 3:1 through
rotation alone.
77
This hypercube is equivalent to the unreduced trichordal orbifold.
66
The diagonal running through the origin contains the singleton multisets,
<000>, <111>, <222> etc., reflecting the sum of the voice leadings along each axis: x
+ y + z = <111>. This line, as in the dyadic Moebius strips, becomes a torus knot on
the edges of the triangular prism. The pcsets are sorted by their sums and placed in
twelve triangular planes. The least symmetrical pcsets, are placed at the edges, and
the most symmetrical sets at the center. The reduced trichordal orbifold is shown in
Figure 3-26.78
As in the dyadic spaces above, one may use alternative voice leadings to
construct spaces that are similar in structure to the orbifold. Like the dyadic spaces,
many of the trichordal vlsets produce incomplete spaces through pcset redundancy.
Table 3-2 lists the vlsets by sum and the size of the spaces they create. Vlsets are
grouped by T4, as well as inversional equivalence. Transposition by T4 and T8
produces the members of a vlclass who share the same sum. This creates networks of
T4-related vlsets whose spaces have the same properties. Like the dyadic spaces
paired by T6 above, these spaces share the same tn-class locations, and transform the
pcsets according to the relationship between the vlsets. T4 cycles the diagonal planes
of a space by T4, while T8 cycles them by T8. For example, diagonal planes 1, 4, 7,
and 10 in the <445> space are identical to the same planes in the <001> space
(assuming the same pcset is placed at the origin), diagonal planes 2, 5, 8, and 11 are
transformed by T4, and 3, 6, 9, and 12 are transformed by T8.
78
Callender et al. 2008.
67
As discussed above, sum-zero inversions simply flip a voice-leading space.
Alternatively, the sum-zero inversion of a vlset describes motion within a space in the
opposite direction. For example, vlset <001> maps pcset {024} onto {025}, while
vlset <00e> maps {025} onto {024}. In addition, inversions I4 and I8 preserve a
vlset’s sum. These vlsets are equivalent to the sum-zero inversion of the T4 or T8
transformation of the vlset. Therefore, the vlsets may be grouped by the Dihedral
group D3, to form networks of six spaces that have similar properties. These
networks are listed in each line of Table 3-2. As the table shows, the only vlsets that
do not participate in such a network are those of vlclass [048], as they map onto
themselves under transposition or inversion by multiples of index number 4.
Equivalent permutations have been left off the table, such as <015> and
<051>. The rotations of a vlset are expressed in its space, so <015> represents both
<150> and <501>. The permutations are left off of the table because they produce
spaces in which the pcsets are also permutations of the other space; that is, pcset
{02t} in the first space is replaced with {0t2} in the second.
As table 3-2 shows, each sum contains ten sets of three vlsets related by
T4/T8. Sums 0, 3, 6, and 9 also contain the augmented vlsets—<048>, <159>,
<26t>, and <37e>—for a total of thirty-one. Like the dyadic vlsets, complete spaces,
that is, spaces with all 1728 ordered pcsets, are only formed by vlsets whose sums are
1, 5, 7 or e; they must also be a member of vlclass [001], [013], [025], [037], [016],
[034], or [005], under transposition and inversion. Sums 1, 5, 7, and e produce three
vlclasses which do not generate the full collection of pcsets due to the interaction of
68
their cycles: [002], [004], and [026]. These vlclasses are all-even, meaning they
preserve a pcset’s whole-tone membership. [002] and [026] create two distinct voice-
leading spaces, much like dyadic vlclass [02] above. Vlclass [004] is even more
limiting, preserving the augmented triad content of a pcset. Let a, b, c, and d
represent the four augmented triads, where b = T1(a), c = T2(a), and d= T3(a). The
five distinct spaces of tn-classes may be represented by their augmented triad make-
up; 1) [aaa], such as [000], [004], and [048], 2) [aab], such as [045], [001], and [037],
3) [aac], such as [002], [026], and [024], 4) [aad], such as [003], [034], and [015], and
5) [abc], such as [012], [016], and [036].
Transposition increases the sum of a vlset by some multiple of three, as it adds
the same number to each member of the vlset. For example, a sum-one vlset such as
<02e> is mapped by T1 (or <111>) onto a sum-four vlset, <130>. Therefore, vlset
sums linked by three-cycles have the same vlclass content. As the table shows,
vlclasses of sum 1, 4, 7, and t are the same, as well as their inversions, sums 2, 5, 8,
and e. The remaining three-cycle—sums 0, 3, 6, and 9—are made up of vlclasses
[000], [003], [006], [012], [015], [024], [027], [036], and [048] under transposition
and inversion.
None of the vlclasses within sums 0, 3, 6, or 9 produce spaces with all 1728
pcsets, as their sums are factors of twelve. They do, however, differ in the number of
discrete tn-class spaces produced. Vlclass [000] is equivalent to traditional
transposition, and preserves the tn-class of any pcset it operates upon. [006]
preserves the tritone content of a pcset, and therefore links tn-classes whose
69
constituents belong to the same tritones. For example, let a, b, c, d, e, and f stand for
tritones, where b = T1(a), c = T2(a), d = T3(a), e = T4(a), and f = T5(a). The non-
intersecting tn-class spaces created by [006] may be represented by 1) [aaa], 2) [aab],
3) [aac], 4) [aad], 5) [aae], 6) [aaf], 7) [abc], 8) [abd], 9) [abe], and 10) [ace]. [003]
and [036] both preserve the diminished-seventh membership of a pcset. Let a, b, and
c represent diminished-seventh chords, where b = T1(a), and c = T2(a). The distinct
tn-class spaces are given by 1) [aaa], 2) [aab], 3) [aac], and 4 [abc].
The remaining tn-class spaces are more unusual than those discussed above.
[024], like [002], preserves the whole-tone content of a space; however, due to the
interaction of its cycles, it generates six distinct tn-class spaces—three each of the
whole-tone tn-classes and mixed whole-tone tn-classes. Although it contains three
distinct cycles, [012] behaves very much like [003], in that the tn-class spaces it
produces are partitioned in the same way as the 3-cycles of sums listed in Table 3-2.
That is, [012] generates three separate tn-class spaces, one with the sum-0, 3, 6, and 9
tn-classes, one with the sum-1, 4, 7, and t tn-classes, and one with the sum-2, 5, 8,
and e tn-classes. [015] produces the same tn-class partitioning as [012], although the
tn-classes are not in the same locations in the spaces. [027] generates similar spaces
to [012], as they are related by multiplication, and each of the spaces it generates are
related by multiplication to [012]’s spaces. [048] is extremely restrictive, given the
highly symmetrical structure of the vlset. When combined with itself under rotation,
[048] either produces the identity, [000], or maps onto itself. Therefore, the tn-class
spaces created by [048] are all pairs of tn-classes.
70
As shown above, the trichordal orbifold is reduced from a hypercube through
permutational equivalence. While any voice-leading set can produce a hypercube,
many of these cannot be reduced through permutational equivalence. Multiset vlsets,
such as <001>, have only three distinct permutations, <001>, <010>, and <100>,
which are rotations of one another. Along with their sum-zero inversions, <00e>,
<0e0>, and <e00>, these orderings transform a three-member pcset in at most six
ways, no matter the permutation of the vlset. For example, if the six vlsets are
applied to pcset {014}, they produce pcsets {015}, {024}, {114}, {013}, {004}, and
{e14}. If the pcset is re-ordered, {104}, the vlsets produce {105}, {114}, {204},
{103}, {1e4}, and {004}. If the rotations and inversions of a three-member vlset,
such as <025>, <250>, <502>, <0t7>, <t70>, and <70t> are applied to {014} and
{104}, the results are not equivalent under permutation. {014} is mapped onto
{039}, {264}, {516}, {0ee}, {t84}, and {712}, while {104} is mapped onto {129},
{354}, {606}, {1te}, {e74}, and {800}. As a result, the voice-leading hypercube
produced by vlset <025> cannot be reduced to a toroidal prism like the <001> space,
because the three-member pcsets could be mapped onto up to twelve other pcsets.
While three-member vlset spaces cannot be reduced by permutational
equivalence, they may still be reduced by rotation, transposition, and inversion. All
two and three-member pcsets in a space have three distinct rotations, such as {001},
{010}, and {100}, or {258}, {582}, and {825}. The singleton multisets, such as
{444} have one rotation. Therefore, under rotational equivalence alone, the 1728
71
trichordal pcsets reduce to 584; the 12 singleton multisets do not reduce, and the
remaining 1716 sets reduce 3:1 to 572.
The complete spaces, as shown on Table 3-2, reduce from four-dimensional
hypercubes to cones with two points at one end by rotation and transpositional
equivalence. These are similar to the set-class cone shown by Callender, Quinn, and
Tymoczko, but differ because the authors’ spaces are reduced by permutational
equivalence. In their trichordal set-class space, [000] and [048] are at opposite ends
of the cone, as they represent the least and most even distribution of pcs within the
octave, respectively. The spaces reduced only by rotation and transposition form
cones with two points at one end, as [000] is opposite both [048] and [084]. Figure 3-
27a demonstrates the vlclass [047] ordered tn-class space. Figure 3-27b shows a
trichordal passage among adjacencies within the space, the specific vlsets and
mappings are shown below the score.
As Figure 3-27a shows, there are 50 ordered tn-classes under rotational
equivalence. These are made up of 12 multiset tn-classes, and 38 trichordal tn-
classes. The 19 unordered tn-classes each have two forms as ordered classes, such as
[013], [031], [023], and [032]. The 12 multiset tn-classes are the dyadic intervals and
singleton sets: [000], [001], [002]... [011]. Ordered voice-leading tn-classes [001],
[005], [055], [011], [013], [023], [031], [032], [014], [034], [041], [043], [016], [056],
[061], [065], [025], [035], [052], [053], [037], [047], [073], [074] produce complete
ordered tn-class spaces.
72
Each of the complete ordered tn-spaces may be reduced further by ordered
inversion. Figure 3-28 shows the ordered set-class space for vlclass [047]. For the
symmetrical tn-classes listed above, each ordering is an inversion of the other. For
instance, ordered tn-classes [012] and [021] are inversions of one another under
rotational equivalence. They are not transpositions, because they may not be
reordered. Non-symmetrical tn-classes reduce from four forms to two; [025] and
[053] are inversionally-related pairs, as are [035] and [052]. Inversional equivalence
reduces the space to a more manageable 26 ordered set-classes. [000] and [006] are
already in their most reduced form, and the remaining 48 ordered tn-classes reduce by
half. As Figure 3-28 shows, when the spaces are reduced by inversion, the open end
of the cone is stitched together, and the two augmented triad points are merged.
Figures 3-29a-e show some representative ordered set-class spaces. As the Figures
show, these spaces are identical in structure, but their ordered set-classes are
redistributed to reflect their differences in voice leading. Highly symmetrical set-
classes, such as [000], [006], and [048] are located in the same positions in each
space.
Many ordered vlclasses produce multiple ordered tn-class and set-class
spaces. These spaces are disjunct due to the interaction of their generating vlclass’s
interval cycles. For example, all spaces produced by members of vlclass [002] reduce
by rotation, transposition and inversion to one of two ordered set-class spaces, shown
in Figures 3-30a and b, respectively. These spaces partition the trichordal tn-classes
by their whole-tone contents, Figure 3-30a includes ordered set-classes made up of
73
members of one whole-tone scale, while the ordered set-classes in 3-32b are made up
of both whole-tone scales. Both spaces are three-dimensional; the two sides of Figure
3-30a fold onto one another, and Figure 3-30b wraps around itself to form an
irregular hexagonal prism, as shown.
The scope of the present study does not allow a full examination of the spaces
produced by the remaining trichordal vlclasses. As shown on Table 3-2, the
remaining vlclasses partition the pcsets, ordered tn-classes, and ordered set-classes
into multiple spaces. The previous examples have shown the construction of voice-
leading spaces for many distinct types of voice leading. These form alternatives to
the spaces based on harmonic similarity found elsewhere in contemporary theory.
The complete dyadic and trichordal spaces are especially useful, as they may be used
to model the voice leading between any pcsets or set-classes an analyst chooses.
74
Chapter 4: Analysis
Post-tonal music is, in general, characterized by surface diversity. As the
following discussion will show, many analytical methods for post-tonal music
attempt to reduce the amount of information presented in each piece to a small
number of unifying elements, or principles governing behavior. Voice-leading sets fit
within this analytical tradition by demonstrating underlying unity through
transformational voice leading. After the opening of Arnold Schoenberg’s Drei
Klavierstucke Op. 11, n. 1 is briefly revisited, this chapter presents analyses of
movements 2 and 3 from Anton Webern’s Five Movements for String Quartet Op. 5
and movements 1 and 2 of Alban Berg’s Four Pieces for Clarinet and Piano, Op. 5.
Set theory is often used for the analysis of non-serial post-tonal music. Set
theory is especially useful for demonstrating harmonic relationships in various
musical contexts. In Forte’s formulation, an analyst shows how much of the
harmonic material of a piece is related to a central nexus set, most often through
inclusion relations.79
Set theory is also useful simply as a labeling system for the
harmonic materials of post-tonal music, and therefore may be used in conjunction
with other analytical systems, such as developing variation, basic cell analysis, and
transformation theory. In the following analyses, instead of a pitch-class set-class
nexus, the voice-leading set-class presented by prominent motivic material is treated
as a type of nexus set. Because any two pcsets have multiple vlset interpretations,
many diverse pcset pairs may provide voice leadings which belong to the same
79
These inclusion relations may be abstract. See The Structure of Atonal Music (New
Haven: Yale University Press, 1977).
75
vlclass; the choice of which vlset “path” to follow is partly a statistical choice and
partly an interpretive choice. That is, it is based on the extent to which the path
accounts for an appreciable amount of the piece, how convincing the path is, and
whether it accounts for the most salient materials.
The voice leadings interpreted through my analyses are presented in a wide
variety of ways, with a wide range of salience. The goal of these analyses is not to
“hear” vlsets per se, although it is possible to derive a perceptual path based on the
intervals outlined. In the analytical context proposed here, voice leading is treated
motivically, through the “variation” (as in motivic variation) of one vlset, or a small
group of sets. The pitch information in the rest of the piece may then be thought of as
filtered through the vlclass, establishing relationships among a wide variety of
contexts within a piece.80
Salience does factor into the decision of which vlsets may
be primary in a piece, but this decision has multiple inputs.81
Developing variation and grundgestalt, or basic shape, are two important
concepts from Arnold Schoenberg’s analytical practice. Both concepts provide ways
in which to interpret a composition as a process of organic growth. In the most
80
This is not unlike Allen Forte’s octatonic filter employed in The Atonal Music Of
Anton Webern (New Haven: Yale University Press, 1998). 81
Common criticisms of pcset analysis are based on issues of segmentation, and the
analytical method proposed here does not offer a solution. My analysis admits a wide
variety of segmentation possibilities; as in pcset analysis, the analyst is responsible
for making meaningful choices. Other analysts could, of course, employ more
rigorous segmentation methods if they wish. See Ethan Haimo’s criticism of Allen
Forte in “Atonality, Analysis, and the Intentional Fallacy” in Music Theory Spectrum,
18, n. 2 (1996): 167-199. For an earlier discussion of segmentation, see Christopher
Hasty’s “Segmentation and Process in Post-tonal Music,” in Music Theory Spectrum,
3, (1981): 54-73.
76
conventional interpretation, developing variation refers to a method of motivic
variation, while the basic shape is a musical figure presented near the outset of a
composition, whose “endless reshaping” forms the remainder of the piece.82
The size
and musical parameters of a basic shape may vary from piece to piece, while
developing variation typically refers to the transformations of a small collection of
pitches or intervals.83
In post-tonal music, Arnold Schoenberg uses his Op. 22 songs
to demonstrate ways in which a small motivic idea may be altered and transformed to
create new motives. There are many possibilities for variation of a motive, including
re-ordering, expansion, contraction, rearrangement, etc.84
This type of variation often
changes the harmonic content of a motive. For example, in the course of his talk,
Schoenberg shows two trichords from the second and third phrase of the piano
accompaniment in his song Seraphita, Op. 22 n. 1; ordered pcsets <032> and <478>.
As Schoenberg shows, the second trichord employs the same interval succession
(labeled in tonal terms) —minor 3rd
followed by minor 2nd
—as the first. In the
second trichord, however, the motive is varied by reversing the direction of the minor
2nd
.85
The two motives thereby belong to different set-classes, [013] and [014]
respectively, and cannot be related by transposition and inversion. Voice-leading
sets, however, provide a method for describing the systematic transformations of the
82
“Linear Counterpoint” in Style and Idea edited by Roy Carter with translations by
Leo Black (New York: St. Martins Press, 1975, reprint Los Angeles: University of
California Press, 1984). 83
See for example, Schoenberg’s analysis of Brahms’ F-major Cello Sonata, Op. 99
in “The Orchestral Variations, Op. 31: A Radio Talk,” The Score, 27 (1960): 28. 84
Arnold Schoenberg “Analysis of the Four Orchestral Songs Op. 22” translated by
Claudio Spies in Perspectives of New Music, 3, n.2 (1965): 1-21. 85
ibid., 5.
77
motive. Vlset <446> maps pcset <032> onto <478>. The double T4s in the set show
that the first interval is unaltered, as both pcs are transposed by the same amount, and
the six shows the difference between them.
George Perle’s basic cell, related to developing variation, is essentially a pitch
or pitch-class set which provides unity through transposition, inversion, and
permutation. Perle focuses on small units and demonstrates their replication in
various harmonic and melodic guises. For example, in his discussion of
Schoenberg’s Op. 11 n. 1, Perle shows many ways in which the opening [014] can be
found in the opening three measures, as well as at important formal junctures
throughout the work. Perle’s analysis is not limited to one cell, but he discusses
multiple important cells in the piece.86
Although Perle does not employ the language
of set theory, it is easy to interpret his work as focusing on a piece’s primary sets; that
is, Perle demonstrates how the transformations of a piece’s most important pcsets
shape the work. The analyses shown below use Perle’s method, but apply it to voice
leading. That is, a small number of specific vlclasses form a piece’s basic cells.
As David Lewin has shown, transformation theory may be used in conjunction
with set theory, where pitch-class sets are mapped onto one another using various
transformations. Some of the transformations Lewin demonstrates are operations,
meaning they are one-to-one and onto, such as transposition, inversion, and
multiplication; others are more exotic, such as wedging-to-E, which moves each
86
Serial Composition and Atonality 6th
ed. (Berkeley: University of California Press,
2001) 9-15.
78
pitch-class a semitone closer to E while preserving E and Bb.87
The latter is a many-
to-one transformation, and therefore not an operation, as it maps three different pitch-
classes, F, E, and D#, onto E. Lewin also describes a number of contextual
transformations, whose mappings change depending on the object they operate upon;
for example, the neo-Riemannian operation L, or Leitonwechsel maps major triads
onto minor triads a major third above, and minor triads onto major triads a major third
below, such as C major – E minor, and Eb minor – B major.88
Because they describe
individual pitch-class mappings, vlsets eliminate the distinction between interval-
preserving operations such as transposition and inversion, many-to-one or one-to-
many mappings, and contextual transformations, and they can model essentially any
pcset relationship.
The following analyses demonstrate ways in which voice-leading sets unify
pitch and interval diversity in post-tonal music. In each piece, much of the musical
surface can be viewed as expressing a small number of voice-leading classes, that is,
voice-leading transformations related by transposition or inversion. These vlclasses
are determined by the ordered voice-leading transformations presented by prominent
motivic material; in three of the pieces, this is the ordered voice leading between
trichordal members of the opening melodic gesture. The vlclass thus forms a type of
basic cell, or basic shape, which unifies seemingly disparate surface material. The
analyses presented here are, again, not intended to supplant those of previous authors,
87
Injection function, Generalized Musical Intervals and Transformations (New
Haven: Yale University Press, 1987, reprint New Haven: Yale University Press,
2007), Chapter 6. 88
Lewin 2007, 178.
79
but may be used in conjunction with them to provide a richer interpretation of the
works.
In another view, the goal of the analytical method presented here is to
determine the underlying pitch-class voices, which make up the surface of post-tonal
pieces. This is not unlike the analysis of tonal music; in Schenkerian analysis, the
voices demonstrate ways in which the overarching tonic is prolonged using various
diminution techniques. As in tonal music, the interpreted voices in the following
analyses are understood as unfolding multiple pitch-class voices in tandem.
Compound melodies in tonal music are typically interpreted as projecting multiple
stepwise lines. In the following analyses, the voices of polyphonic melodies are
determined by their relationship to the main voice-leading motive.
Schoenberg Op. 11 n. 1
The [002] voice-leading interpretation presented in Figures 2-8a and b
highlight a recursive connection between melody and accompaniment in
Schoenberg’s piece. The voice leadings between the pairs of accompanimental
trichords, c-d (<442>) and g-h (<113>), reinforce the intervallic contrast presented by
the opening trichords of the antecedent and consequent phrases, ms 1-3 (<t99>) and
9-11 (<388)>) respectively. In ms. 1-2, the antecedent (trichord a) descends by pc int
3, followed by int 1; in ms. 9-10, the consequent (trichord e) descends by int 4,
followed by int 2. These interval pairs then form the basis for the voice leading
between the melodies’ accompanimental trichords: c moves by vlset <442> to d, and
g moves by vlset <113> to h. Thus, the intervals within the melodic trichords are
80
replicated as voice leadings between accompanimental trichords; the technique of
intervals becoming T-levels is a standard recursive one in tonal music as well as in
post-tonal music.89
Many properties of vlclass [002] are highlighted in the passage, emphasizing
its prominence as a featured voice-leading set-class. As a closer examination of
Figure 2-8 will show, voice leading by any member of [002] preserves the whole-tone
content of a pcset. In particular, even members of [002] preserve whole-tone
membership among the pcs of a set, while odd members of [002] modulate all pcs
from one collection to the other. For example, pcset c contains two members of the
odd whole-tone collection, {5e}, and one even {6}; because vlset <442> is
exclusively even, pcset d has the same combination: two odd pcs, {91}, and one even
{t}. Likewise, pcset g is entirely even <468>, and moves by vlset <113> to odd pcset
h {57e}.
Given the whole-tone preserving property of vlclass [002], the paired pcsets in
Figure 2-8 must contain the same ratio of whole-tone members to one another.
Trichords e and f, which make up the second melodic statement in ms. 9-11, do not
share the same whole-tone ratios, and therefore cannot be spanned by a member of
vlclass [002]; e is a member of [026] and f is the chromatic trichord [012]. The
melody as a whole is made up of a whole-tone pentachord with one additional pc,
{689t02}, and is accompanied by the whole-tone trichords g {468} and h {57e}. The
consequent’s predominantly whole-tone sonorities, and change from [002]-based
89
This recursion could be shown in Klumpenhouwer-network analysis.
81
voice leading contrast with the antecedent’s mix of whole-tone collections; this
contrast closes the opening section of the piece.90
Despite the whole-tone content retention aspects of vlset [002], each of the
seven members of vlclass [002] shown in Figure 2-8 create a unique voice-leading
space where adjacent pcsets are mapped onto one another by the generating vlset.
For example, Figures 4-1a and b show the spaces produced by vlset <442> with
pcsets {000} and {001} at the origin, respectively. The three-dimensional spaces
form six by six hypercubes, cubes whose opposing faces are adjacent to one another.
This diversity is built on top of a unity, however; as shown in chapter 3, all spaces
produced by members of vlclass [002] reduce by rotation, transposition, and inversion
to one of two ordered set-class spaces, shown in Figures 4-1c and d. These spaces
partition the trichordal tn-classes by their whole-tone contents, Figure 4-1c includes
ordered set-classes made up of members of one whole-tone scale, while the ordered
set-classes in 14-1d are made up of both whole-tone scales. Both spaces are three-
dimensional; the two sides of Figure 4-1c fold onto one another, and Figure 4-1d
wraps around itself to form an irregular hexagonal prism, as shown.
Each of the voice leadings shown in Figure 2-8 may be traced as a single line
segment along one of the vlclass [002] ordered set-class spaces shown in Figures 4-1c
and d; the ordered tn-class and ordered set-class interpretations of these chords are
shown in Figure 4-1e. It is important to note that these spaces are not reduced
90
George Perle argues that the melody’s shift to a predominantly whole-tone
collection is analogous to the variation of a melody in a tonal composition, where
additional statements are presented within new harmonic contexts in Twelve-tone
Tonality 2nd
ed. (Berkeley: University of California Press, 1996) 162-163.
82
through permutational equivalence, therefore ordered tn-classes [014] and [041] are
not represented by the same point. When reduced by rotation and ordered inversion,
however, [041] is equivalent to [034]. As a result, the spaces are very flexible
analytical tools, as pcsets may be reordered and represented by different points on the
space. For example, ordered set-classes [014] and [015] are not adjacent on the
space, but [034] and [015] are. The first voice leading shown in Figure 2-8 interprets
chord a as ordered tn-class [041], which is an inversion of ordered set-class [034].
This same chord is reinterpreted as an ordered tn-class [014] in order to show its
voice-leading mapping onto chord c. Such multiple interpretations are part of the
richness of vlset analysis. 91
Webern Op. 5 n. 2
In his discussion of the second of Anton Webern’s Five Pieces for String
Quartet, David Lewin shows a network of relationships interpreted from the various
forms of the opening gesture, G-B-G-C#, pcset {G,B,C#}, tn[046], sc [026],
presented in the first three measures of the piece (as is typical for Lewin, he does not
label the tn or sc). As Lewin shows, these trichords, whether melodic, harmonic, or
some combination of the two, all feature pcs from the odd, or C#, whole-tone scale.
The transformations upon the network, therefore, preserve whole-tone membership:
91
In my opinion, it is the analyst’s responsibility to determine if the relationships
formed by multiple interpretations are meaningful and worthy of inclusion in an
analysis. All of the transformational mappings shown in Figure 4 contribute to my
understanding of the opening section of Schoenberg’s piece.
83
T4, T6, T8, Tt, I0, I4, and I6, and are taken from the T0-t, I0-t dihedral subgroup.92
Figures 4-2a and b show the statements of [026] Lewin culls from the score, as well
as the network of transformations among them. Lewin identifies another important
note group, adding a C to G-B-G-C# (tn [0456] and set-class [0126]), formed by
filling in the whole-step of each [026]; the transformational network can also be
considered in the context of this set. Lewin’s article is intended to demonstrate the
construction of a network and its use in a portion of the piece, not to present an
analysis of the piece as a whole; therefore, he does not discuss the later portions of
the piece.93
The following analysis of Webern’s Op. 5 n. 2 will show that the piece is
governed by the transformations of vlclass [043]. These transformations create
networks of relationships among both pcsets and vlsets in Webern’s piece. Just as
Lewin focuses on the viola’s opening [026] and its intervals, the primacy of vlclass
[043] is determined by the ordered transformational voice leading between the viola’s
first and second trichords—vlset <910>, which is T9(<043>).94
The two trichords are
made up of pcs <7e1> and <401> in score order. Vlset <910> preserves this
92
Lewin uses the letter names I, J, and K as labels for “inversion about” D (I4), F#
(I0), and A (I6), respectively. “Transformational Techniques in Atonal and other
Music Theories” in Perspectives of New Music 21, no 1/2 (1982): 312-324. Another
study of this short piece comes from Bruce Archibald, who interprets a series of
symmetries spanning the piece, in various states of “completeness” which can be
thought of analogously as “unbalanced” and “balanced” – terms Lewin uses in his
study of inversional balance in Schoenberg’s music and also a term from
Schoenberg’s formal theories; see “Some Thoughts on Symmetry in Early Webern:
Op. 5 N. 2” in Perspectives of New Music, 10, n. 2 (1972): 159-163. 93
Lewin 1982, 312-325. The Figures appear on pgs 318 and 320, respectively. 94
Although the viola melody begins with a tetrachordal multiset (G-B-G-C#), the
second G is interpreted as the arpeggiation of a single pc voice in my analysis.
84
ordering, with mappings 7 – 4, e – 0, and 1 – 1. This voice leading is then varied
through transposition and inversion, and used to span pcsets in melodic and
accompanimental gestures throughout the piece. While Lewin’s [026]/whole-tone
network provides a good model for the opening phrase, and is also relevant to the
third phrase, it is not applicable to the remainder of the piece. By focusing on
transformational voice leading, rather than harmonic similarity, the network of
relationships formed by members of vlclass [043] can be applied to the entire work.
The texture of Webern’s piece is largely melody and accompaniment, and it is
divided into four distinct phrases which are marked by changes in the
accompaniment: mm. 1-4, 4-7, 7-10, and 10-13. The first, second, and last phrases
feature chordal accompaniment, while the third consists of a slow Eb-F trill in the
second violin as a counterpoint to the melody. The second phrase is demarcated from
the first and third by semitonal sixteenth-note motives in mm. 4-5 and 6-7.95
Figure 4-3 shows the statements of vlclass [043] in the melody in the
remainder of the piece. As shown, the melody is transferred to the 2nd
violin in mm.
5-6, the 1st violin in mm. 7-12, and returns to the 2
nd violin in mm. 12-13. The pitch-
class voices are shown below the staff, with lines demonstrating the pc mappings.
95
Although Lewin does not demonstrate a transformational network for the last three
phrases of the piece, the third phrase, mm. 7-10, fits comfortably with Lewin’s
[026]—whole-tone interpretation of mm. 1-3. The second violin alternates Eb and F,
while the first violin supplies the remaining members of the odd whole-tone scale. If
G#, D, and E are seen as non-harmonic tones, the first violin presents three
overlapping [026]s: {9e3}, {391}, and {917}. It is possible to interpret many more
[026]s in conjunction with the second violin.
85
Pcs are listed vertically in registral order for ease of reading, with the corresponding
vlset shown below.96
The pitch-class voices shown in Figure 4-3 correspond to the polyphonic lines
expressed by the melody. For example, each pc of the opening trichord, {7e1},
represents a voice, which moves by a different interval to the next pc; G moves by T9
to E, B moves T1 to C, and C# moves T0 C#, although descending an octave in pitch.
These pc voice motions correspond to the mappings induced by vlset <910>. There
are many possible vlset interpretations of any two pcsets, and there are many possible
pcset interpretations of any given musical passage. My analysis focuses on the
ordered voice leading produced by the viola’s opening gesture, and shows how that
voice leading is transformed throughout the piece.
Each of the subsequent voice leadings listed in Figure 4-3 is a member of
vlclass [043] under transposition or inversion. Therefore, the pc voices project
variations of the opening voice leading which unifies the melody. The second vlset
for example, <845>, is an I5 inversion of the opening voice leading. In other words,
my analysis differs from those of previous analysts by focusing on the harmonic
consistency expressed by pc voices, rather than pcsets.97
Figure 4-4 shows the
ordered harmonic relationships given by the vlsets highlighted in Figure 4-3.
96
This method of pitch-class voice-leading representation is indebted to Joseph
Straus, who employs it in multiple articles, including 2003. 97
As stated above, this is not intended to supplant pcset analysis, but rather to provide
an additional perspective. It is valuable to point out, for instance, that {589} is a
subset of the opening accompanimental tetrachord, and that the two pcsets in question,
{014} and {589} are inversions of one another. This is expressible as vlset <999>i,
86
Two of the vlsets, <265>i and <401>i in mm. 5 and 8, are inversional
mappings. That is, the vlsets map the individual pcsets onto one another by
inversional sums. <265>i maps ordered pcset <t40> onto <425>, which is formed by
the last three pcs of m. 6. <401>i maps <89e> onto <832>.98
If transpositional and
inversional vlsets are thought of as two transformational modes, we may define an
operation, changemode, which produces a mode change from one vlset to another,
along with transposition or inversion. As shown on Figure 4-4, CI0 maps
transpositional vlset <t67> onto inversional vlset <265>i; the index numbers are
transformed by I0, and the mode is changed by C. Likewise, CT6 maps vlset <401>i
onto <t67>.99
In order to produce the voice-leading consistency shown in Figure 4-4, the
surface of the piece must be interpreted as projecting three pc voices, that is,
trichords. Some gestures, such as mm. 5-6 are easily interpreted as trichords. The
first two trichords in score order, <t40> and <625>, are inversions of the opening
trichords, {7e1} and {401}, by Ie and I6 respectively. As a result, the vlset spanning
the trichords in mm. 5-6, <t67>, is an I7 inversion of the opening <910>; I7 results
from the difference between inversional sums: 6 - e = 7.
but the purpose of this analysis is to demonstrate the variations of vlset <910> that
underlie the pitch material of Op. 5 n. 2. 98
As the pitch-class voices demonstrate, the common-tone G# is mapped onto itself.
This echoes the common C# in the opening voice-leading. 99
C forms a group with transposition and inversion of order 48 on the set of 48
transpositional and inversional members of vlclass [043]. CI operations are
involutions, and CT operations are not, excepting CT0 and CT6. T0 is the identity.
87
Elsewhere, the melody is more difficult to parse into trichords. As Figure 4-3
shows, in both mm. 3 and 9, vlclass [043] maps a trichord from the melody onto a
melodic dyad plus a pitch-class “borrowed” from the 2nd
violin—A and F,
respectively. In addition, many successive trichords are overlapping. The first
example occurs in m. 3, where G and C# are each members of two trichords.
The trichordal interpretation of mm. 10-11 is shown by beams in Figure 4-3.
The melody begins and ends with chromatic [012] trichords, <543> AND <e0t>
which are T7-related; these bookend an imbricated -4/-1 interval cycle <G#ED#B>
(<G#ED#> AND <EDB>), which produces an [0158] tetrachord. As the beams
show, the repeated E and D# in m.11 are interpreted as members of pcset <345>, and
the G# forms pcset <8e0> with B and C, producing vlset <487>.100
B and C are
common tones with the next trichord, although the C is repeated on the downbeat of
m. 12.
Vlclass [043] is also expressed in the accompaniment of the piece. Figure 4-5
reduces the score to two staves, and demonstrates some of the pc voices formed by
vlclass [043] in the accompaniment, as well as between the melody and
accompaniment. The opening is accompanied by two chords sustained by the cello
and second violin—{2589} and {379}. As shown in the voice-leading mapping
below the staff of Figure 4-5, pc A is interpreted as belonging to the second chord,
and not the first, the two opening trichords are <258> and <379>; the A in the
opening is interpreted as an anticipation, a member of the following chord. The vlset
100
In terms of polyphonic melody, the G# is an anticipation of the next pcset, {8e0}
88
spanning the two trichords is <7et>, a Tt transformation of the <910> voice leading it
accompanies. This voice leading follows the registral mapping on the surface of the
piece. As Figure 4-5 shows, the pc voices do not cross in pitch space. Two
interpretations of the symmetrical tetrachords in mm. 4-5 (<F#GBbB> and
<FF#BC>) are shown below the staff. Both demonstrate the voice-leading mapping
between ordered tn-classes [015] to [016].
Figure 4-5 also demonstrates vlclass [043] mappings between the melody and
accompaniment. The second and third accompanimental trichords (<GEbA> and
<GBC#>), are the same tn-class, [046], as the opening melody. These pcsets interact
with the melody in mm. 2 and 3. As shown, the vlset which maps pcset <012> onto
<7e1>, <7et>, is identical to the first accompanimental voice-leading mapping.
Two of the voice leadings produced between the melody and accompaniment
are identical to the opening vlset, <910>. In m. 6, <910> maps accompanimental
pcset {519} onto melodic pcset {256}. The same vlset maps {590} onto {e34} in m.
11. These repetitions of the opening vlset help to punctuate the piece’s phrases, as
they correspond to clear variations of the opening melody.
Figures 4-6 displays an alternative transformational graph of Webern’s Op. 5
n. 2. The nodes of this graph are the vlsets presented by the piece, and the arrows
demonstrate some of the relationships among them. Like Lewin’s network, the graph
follows the outline of the score. This graph of relationships demonstrates the
transformation of the basic voice leading throughout the piece, and the way in which
this transformation unifies the piece. While Lewin’s network focuses on a small
89
portion of the piece, and the whole-tone D6 subgroup of the T/I group, the arrows of
my graph reflect the use of the full group formed by transposition, inversion, and
modechange.101
Webern Op. 5 n. 3
Webern’s Op. 5 n. 3 features one predominant voice leading—vlclass [038].
This voice leading is presented in various forms throughout the piece. As in the
analysis of Op. 5 n. 2 above, the primary voice leading is first presented in prominent
melodic material at the opening of the piece; vlclass [038] is produced by the ordered
pc voices in the violin 1 melody (A) in m. 4. As Figures 4-7a and b show, A may be
broken up into two trichords, <254> and <06t>—members of tn-classes [023] and
[046] respectively. As in the analysis above, the pc voices expressed by the motive
are interpreted in score order; D maps onto C, F - F#, and E - Bb. Vlset <t16> recurs
whenever A is repeated, provided both trichords are transposed by the same interval.
For example, the viola enters one eighth note after the first violin, transposed by T7;
this is also shown in Figures 4-7a and b.
The opening melody, A, is repeated in mm. 10-11 in the cello, followed by the
liquidation of A in all parts through m. 14, shown in Figure 4-8. In mm. 10-11 and
12-13, the cello and second violin repeat the T7 pairing from m. 4. As Figure 4-8
shows, this produces two statements of vlset <t16>. The viola and second violin
phrases that follow each present a new variation of A. In the viola, the first trichord
101
A smaller portion of the graph, with alternative node-connecting arrows, may be
used to demonstrate smaller subgroups; this may not model the piece as effectively,
however. Alternative voice-leading interpretations of the piece would also produce
alternative graphs, of course.
90
is expanded by repeating the opening minor third, creating a 3, -1 pitch interval cycle;
this cycle produces two possible tn-class [023]s, the voice leading from the second,
{t01}, is shown on the Figure. In addition, the variation omits the final interval. The
first violin’s statement also omits the final interval, but does not add an additional
interval 3, shortening A to five pitches. Both the viola and first violin voice leadings
have common-tones which map onto themselves. This is reflected by the T0 in vlset
<470>, T6 of the opening <t16>.
The opening melody, and trichordal [023]/[046] juxtaposition it presents,
gives way to another melody (B)—first presented in mm. 9-10—which presents a
new trichordal pair: {t12} and {59t}, members of tn-classes [034] and [045]. Both A
and B are varied through the second half of the piece, and the movement closes with a
fortississimo statement of B in octaves in all instruments. Although B is made up of
different tn-classes than A, it projects the same vlset. As Figure 4-9 shows, pcset
{2t1} maps onto {t95} by vlset <8e4>; like vlset <t16>, <8e4> is a member of
vlclass [038]. Unlike A, this voice-leading mapping does not take place in score
order, as the second and third pcs of the first pcset are reversed in the second pcset.
This rotation on the surface is a variation of the opening voice leading.
One may choose to interpret melodies A and B in a variety of ways. They
may be antecedent and consequent, conflicting themes, or B may be viewed as a
variation of A, among many other interpretations. However the melodies are viewed,
the main interpretation in this analysis is that the voice leading presented in A is
subjected to variation in its realization in B. A and B are the two main melodic ideas
91
presented by the piece. Throughout the work, they are mined for motivic material.
Vlclass [038] forms the common thread between these various gestures.
Figure 4-10a shows the ordered tn-class space formed by vlclass [047]. As
discussed in chapter 3, the ordered tn-class spaces are reduced through rotational
equivalence, therefore the space represents the various rotations of vlclass [047],
including [038].102
The voice-leading mappings projected by A and B are highlighted
on the space; A’s <t16> voice leading maps ordered tn-class [032] onto [046], while
B’s <8e4> maps [034] onto [045]. The voice leadings are understood as the motion
from one ordered tn-class to another along the line segment that links them. As
discussed in chapter 3, the tn-classes on the space are equivalent under transposition
and rotation, but not permutation, as vlsets such as <t16> do not operate consistently
upon the two distinct permutations of many pcsets. For example, as the space shows,
vlclass [038] maps [023] onto ordered tn-classes [016], [054], [042], [046], [007], and
[063], and maps [032] onto [006], [046], [054], [025], [034], and [072].
B contains an intermediate trichord, <t0e>, between <2t1> and <t59>. This
interpolation may be seen as a way of extending and elaborating the voice leading
between the two chords, <8e4>, which governs the phrase. Vlclass [038] cannot
provide the mapping between this intermediate trichord and the other two members of
B. Ordered tn-class [021] is at least four steps away from both [034] and [045] on the
[038] tn-class space shown in Figure 4-10a. One possible route from [034] to [021]
102
My analysis of Webern’s piece highlights the specific ordered voice-leading
presented by melody A, therefore I have chosen to describe the piece’s voice-leadings
using [038].
92
begins with the vertical motion of three consecutive inverse voice leadings, <094>,
followed by a rotation <380>, forming a total voice leading of <3e0>, of vlclass
[014]. The specific transposition and rotation is given by <2t1> + <t19> = <0et>.
The voice leading from <0et> to <t59> is the combination of two <038>s and two
<940>s, for a total of <624>, of vlclass [024]; <t0e> + <79e> = <59t>; see Figure 4-
10b.
The intermediate voice leadings in B, <t19> and <79e>, are interpreted by the
[038] tn-class space in the way in which voice leadings on the traditional orbifold are
interpreted in terms of the semitone. Vlclass [038] forms the metric by which the
voice-leading distance is judged. Because [038] forms a complete tn-class space, it is
possible to represent any trichordal voice leading culled from the piece in terms of
[038] distance.
In addition to interpolating an additional trichord, B varies A’s voice leading
through the introduction of invariance, as all three trichords contain Bb. For the first
and last trichords, this is a combination of the transposition of the vlset and the tn-
class changes. Vlset <8e4> maps the first trichord, in score order, onto the last, and
the trichord begins with pc interval 8. In addition to the intermediary trichord, this
invariance helps to contrast B with A. The distinguishing features of B—the added
trichord and use of invariance with vlclass [038]—are immediately used in the
variation of A which begins in the next measure, discussed briefly above. As Figure
4-8 shows, in m. 11, the viola’s overlapping [023]s, {8et} and {t10}, share Bb, the
same pc shared by all three of melody b’s trichords. Vlclass [038] maps both pcset
93
<8et> and <t10> onto <280>, by vlsets <692> and <470>, respectively. Note that the
transpositional difference between the two vlsets is Tt. This is the same difference
between A’s vlset <t16>, and B’s <8e4>. All three pcsets have some kind of
invariant relationship with the other two. {t10} and {082} overlap with pc 0, and
{8et} and {082} both contain pc 8. Therefore, both characteristics of melody B are
incorporated into this variation of melody A.
The accompaniment to melody B also introduces invariance in m. 9-10. As
Figure 4-9 shows, the viola presents trichords {e03} and {643}, pc 3 is shared. Vlset
<470> spans the trichords, a T6 transposition of the original vlset in m. 4, while the
vlset spans a new pair of tn-classes, [014] and [013].
The accompanimental material in the second violin in m. 9-11 demonstrates
that vlclass [038] may also be used to navigate between melodies B and A, as it
echoes the features of melody B, and combines them with melody A. As shown on
Figure 4-9, the second violin begins with pcset {034}, followed by {234}, T2 and T4
of the first two trichords of B. The trichords share two pcs, 3 and 4. The gesture
continues to a straightforward statement of melody A in m. 11, a Tt transposition of
the first violin in m. 4, shown in Figure 4-8. The first tn-class of melody b, pcset
{034}, is mapped onto the first pcset of melody A, {023}, by vlset <e83>, an inverted
form of [038].
Melody B’s tn-class juxtaposition, [034] and [045] is foreshadowed by the
opening three measures as well as the material following melody A. The piece begins
with the upper three parts sounding [034]s, shown in a reduction in Figure 4-11a.
94
Excluding the [014] in m. 3, the three string parts all move by the same intervals—pc
int 8, followed by int 5. These intervals form three [045]s horizontally, and are
repeated in m. 10 in retrograde: B’s [045] is retrograde-inverted.
A more extensive variation of A occurs in the first violin in mm. 12-14. A is
presented in a new contour—the gesture is entirely descending, with the intervals in
their smallest forms, and the pcsets are re-ordered (see Figure 4-8). Vlset <149>
maps the first pcset, <032> onto the second, <17e>. The melody continues with an
ascending [026], pcset {359}, and ends with a [023], pcset {467}. Melody A’s
original [023]/[046] juxtaposition is thus presented in retrograde, with the second
trichord inverted. The resulting voice leading cannot be a transpositional member of
vlclass [038], and it is inversional vlset <74e>i which maps <395> onto <476>.
The accompaniment in mm. 13-15 continues to demonstrate a derivation of
material from melodies A and B. The second violin and cello both provide
statements of A, followed by 8/6 pc-interval cycles in the viola and cello. Each cycle
is four pcs long, creating [0268] tetrachords, which may be seen as overlapping [046]
and [026] trichords, matching the material in the first violin. These are followed by
ascending pitch interval 8s, {08} and {95}, in the second violin and viola, and a
descending [015]. The {5890} tetrachord may be viewed as overlapping [034] and
[014] trichords. Although there are no explicit tn-class [034]s, the two [0268]s in the
previous measures are separated by semitone, and may be combined to form both
[014]s and [034]s. This serves to revisit the juxtaposition presented in the opening, as
well as in melody B.
95
The variation of A in mm. 12-14 is extended in the phrase beginning with the
pickup to m. 18. The melody begins with the tn-classes of melody A, pcsets {578}
and {046}, linked by vlset <149>. This is followed by a series of inversions with pc
invariance between [046]s and [026]s. The melody ends in m. 21 with pcset {9e0}.
Vlset <529>i maps pcset {8t2} at the end of m. 20 onto {9e0}.
As in the earlier sections of the piece, the melody in mm. 18-21 is
accompanied by gestures based on the voice leadings presented by melodies A and B.
The accompaniment begins in m. 15, three bars earlier, where the viola maps pcset
{89e} onto {489} by vlset <850>i, while the second violin and cello arpeggiate
(separately) [014]s and [034] ; a reduction is shown in Figure 4-11b. When the first
violin enters, the second violin and viola both present arpeggios of melody a’s tn-
class juxtaposition [023]/[046] with invariance. Both use the voice leading <470>.
As mentioned earlier, the piece closes with a statement of melody B in all four parts.
The preceding analysis of Webern Op. 5 n. 3 demonstrates that the piece is
unified by the use of vlclass [038] in both transpositional and inversional forms.
Presented by the opening and closing trichords in the piece’s main melodic
statements, A and B in m. 4 and mm 9-10, respectively, these voice leadings can also
be found throughout the piece in accompanimental gestures. Much like the variation
of pitch material, vlset <t16> forms a motive which is transposed, inverted,
reordered, fragmented, and condensed throughout the piece. The consistent use of
vlclass [038] can be visualized using the ordered tn-class voice-leading space it
creates.
96
Berg Op. 5 n. 1
Alban Berg’s Four Pieces for Clarinet and Piano Op. 5, have been described
by earlier analysts as a miniature four-movement sonata.103
In keeping with this
interpretation, Douglas Jarman interprets the opening clarinet gesture, shown in
Figure 4-12, as the source of motivic material for all four pieces. Jarman first divides
the melody into an [012569] hexachord and [037] trichord, then extracts five
trichords from the [012569] hexachord: [012], [014], [015], [016], and [037]. He then
demonstrates how material in the remainder of the movement, as well as other
movements, may be derived from transformations and combinations of these
principal set-classes.104
Dave Headlam also shows a way in which the opening
clarinet gesture may serve as generative material for much of the remainder of Op. 5
n. 1. In lieu of Jarman’s trichords, Headlam points out that the gesture employs
primarily interval-classes 1, 4, and 5. Nearly all of the adjacent pcs are separated by
ic 4 or 5, and each of the first three notes are linked by pitch-interval 1 to some later
pitch, Ab5 – A5, Eb5 – E5, and G4 – F#4.105
Headlam identifies six additional
generative features presented by the gesture: 1) pcset {8349} and its set-class [0156];
2) pcset {7689} and its sc [0123]; 3) the wedge motion between the first and third
trichords, Ab5 – A5, Eb5 – D5, and G4 – F#4; 4) the ordered interval series it forms,
<-5, (-5+-8=-1), 2, 3, 4, 5, -7, -8, -9>, the final -9 interval occurs between the clarinet
F#4 and Piano A3; 5) an emphasis on D; and 6) sc [012569], the SCHoenBErG
103
Douglas Jarman, NG. 104
The Music of Alban Berg (Berkeley: University of California, 1979) 23-27. 105
The Music of Alban Berg (New Haven: Yale University Press, 1996) 183-185.
97
hexachord. Headlam then discusses how these gestures are featured in the remainder
of the piece.106
My analysis also focuses on the opening clarinet gesture, and demonstrates
that the voice leading presented therein forms the primary motivic material for the
piece. As Figure 4-13a shows, the first and third trichords, <837> and <926>, are
linked by contour and register, and vlset <1ee> maps the first trichord onto the third
in score order. Vlclass [200], of which <1ee> is a member, forms the basic cell
which unifies the remaining material presented on the surface of the piece. Figure 4-
13b revisits the vlclass [002] ordered set-class space. As in the Webern analysis
above, [002] and [200] are equivalent under rotation, and their spaces are identical.
The arrows on the space represent two of the voice leadings presented by the opening
clarinet gesture: the <1ee> mapping between the first and third trichords, ordered set-
classes [045] and [047], and the <022> mapping between the second and third
trichords, <904> and <926>, ordered set-classes [037] and [047]. The remaining
voice leadings may be traced upon the space as well.
The first and third trichords of the piece are also linked to the second trichord
through vlclass [200]. The second trichord, <904>, maps onto the third, <926>, in
score order by vlset <022>. The first trichord cannot be mapped onto the second
using a transpositional vlset member of vlclass [200], but requires an inversional
vlset, <577>i, which inverts the first note of each trichord at I5, {8} – {9}, and the
common ic4 at I7, {37} – {40}.
106
ibid, 183-186.
98
The opening trichords are echoed harmonically in the left hand of the piano in
m. 2, as shown in Figure 4-14. The <1ee> voice leading originally projected by the
clarinet’s [045] and [047] trichords is now featured vertically between their
inversions, [015] and [037]. Vlset <133> maps <67e> onto <7t2>. As the Figure
shows, the I2 mapping between vlsets <1ee> and <133> is equal to the difference
between the inversional sums of the respective trichords; {378} inverts by I2 onto
{67e}, and {269} by I4 onto {7t2}.
The registral and dynamic climax of Op. 5 n. 1 in mm. 7-8 presents a new set
of three trichords in the clarinet, pcsets {912}, {023}, and {e34}, or tn-classes [045],
[023], and [045]. The second and third trichords imitate the descending gestures of
the first and third trichords at the opening. The trichords are mapped onto one
another in succession by inversionally-related vlsets; <311> maps <912> onto <023>,
and <e11> maps <023> onto <e34>. See Figure 4-13.
Using vlsets, prominent motivic features that are not clearly related to one
another can be shown as expressing the same underlying voice leading. For example,
the right hand piano gesture in m. 2 can be divided into two trichordal pcsets, {9t8}
and {334}. As Figure 4-15 shows, vlset <577> maps <t98> onto <343>. This voice
leading requires both a re-ordering of the first pcset as well as the interpretation of a
doubled Eb. The doubling requirement is ameliorated by the echoing gesture by the
clarinet in m. 3. As Figure 4-13 shows, vlset <355> maps <t89> onto <112>.
The remainder of the clarinet melody expresses vlclass [200] in a variety of
ways. The voice leading is subject to motivic variation techniques such as
99
transposition, inversion, and rotation. Some of these voice leadings are obvious,
while others require more subjective interpretation of the surface of the piece.
As Figure 4-13 shows, in m. 4 pcset <121> is mapped onto <7t9> by vlset
<688>, and in turn <t79> onto <435> by <688>. A rotated version of the voice
leading is unfolded in mm. 5-6, as the clarinet features two ic3 dyads followed by ic
1. This extends the <99e> voice leading between <519> and <2t8> to the space of a
full measure. Vlset <8tt> maps pcset <t76> onto <654> in the second half of m. 6;
F# is a common tone. The ascending gesture in m. 7 is made up of two inversionally
related [0148], inverted in register about Ab4, followed by an ascent of five
semitones. The imbricated trichords are closely related to the opening clarinet
gesture, although they are presented in a new contour; the first trichord is an
augmented triad which shares two pcs {37} with the opening [015], and it is followed
by two triads—Gb major and D minor. The D minor triad shares two pcs, {29}, with
the D major triad in the opening gesture. The juxtaposed triads in m. 7 present a
retrograde member of the vlclass; the Gb triad, pcset {803} is mapped onto the D
minor triad in score order by vlset <ee1>. The climactic gesture begun in the pickup
to m. 8 is discussed above. The third trichord, pcset <e34>, is mapped onto pcset
<t45> by vlset <e11>. Finally, the clarinet ends by unfolding a rotated version of the
opening vlset, <1e1>, with a series of ordered dyads <45> – <98> – <67>
representing pcsets <496> and <587>.
100
The piano accompaniment following the fermata in m. 9 is based on the
opening clarinet gesture, as Headlam shows in his analysis.107
The opening
hexachord is transposed down four semitones in pitch, and the two trichords are
layered vertically. In the opening of the piece, this hexachord is followed by a D
major triad, which shares a common A with the minor triad of the hexachord. In m.
9, pcs G and E, which follow the F minor triad in the left hand, create a C major triad
with C as a common tone. This is a Tt pitch-class transposition of the opening
gesture’s D major triad. Therefore, the three trichords of the opening are mapped
onto the piano accompaniment in m. 9 by T8, T8, and Tt, a retrograded member of
vlclass [200].
The piano accompaniment in Berg’s Op. 5 n. 1 is contrapuntal in texture, and
there are many ways in which vlclass [200] may be embedded. Some of the more
salient examples are given by wedge motions, characteristic features of Berg’s atonal
music. These wedge motions are especially relevant given that they replicate the
wedge motion of the clarinet’s three pc voices at the outset of the piece. As shown in
Figure 4-17 in m. 3 pcset <t01> is mapped onto <912> by vlset <e11>, <912> is then
mapped onto <803> by <ee1>. <e11> maps pcset <382> in m. 4 across the barline
onto <293> on the downbeat of m. 5. There are many further examples in the piece.
In addition to its use in wedge motions, vlclass [200] unifies the contrapuntal
surface of the piano accompaniment in more abstract ways. As Figure 4-18a shows,
the last beat of m. 5 includes tn-classes [045] and [047], echoing the first and third
107
Headlam 1996, 132-133.
101
trichords of the opening clarinet melody. These trichords are followed by a tn-class
[016] in the left hand, and [045] at the beginning of m. 6, followed by another wedge-
motion <1ee>. This sequence of trichords is produced by the mappings of vlclass
[200], as shown on the figure. Another example occurs in m. 7. As Figure 4-18b
shows, pcset <e04> is mapped onto <e26> by vlset <022>, <e26> onto <7t0> by
<886>, and <654> onto <898> by <244>. In the left hand, vlset <577> maps pcset
<834> onto <1te>.
As shown by the previous analysis, the [200] voice leading presented by the
clarinet at the outset of the piece is manipulated and varied in both melody and
accompaniment throughout the piece. Although the piece is highly contrapuntal in
texture, the underlying pitch-class counterpoint is derived from vlclass [200].
Berg Op. 5 n. 2
As mentioned above, Jarman and Headlam discuss multi-movement
connections in Berg’s Op. 5 pieces. As both point out, the opening clarinet
hexachord of n. 1, pcset <837904> sc [012569], is included in the clarinet line in mm.
1-4 of n. 2. The remaining pcs, D and F#, are given to the piano at the opening.108
Headlam also points out that the clarinet begins with pcset <310874>, member of
[013478], which is the Z-pair of [012569], as well as another [012569] hexachord,
pcset {108749}; this hexachord combines three ic5s—the clarinet’s E – A at the end
of m. 3 and the right hand Ab – Db and G – C.109
Another important harmonic
connection is the partitioning of pcset {01378}, a subset of [013478], into tn-classes
108
Jarman 1979, 24. Headlam 1996, 94-95. 109
Headlam 1996, 95-96.
102
[037] and [045] in the right hand of the piano in mm. 2-4. This replicates the
trichords of the opening clarinet gesture of the first piece, and preserves maximal
common tones Ab, Eb, G, and C.
Alban Berg’s Four Pieces for Clarinet and Piano Op. 5 n. 2 is static in texture.
The left hand features dyads nearly exclusively, while the right hand is entirely made
up of trichords. Each hand is nearly entirely homophonic, forming a clear harmonic
accompaniment to the clarinet melody. The clarinet and two hands of the piano
create 3 voice-leading strata, each of which projects its own pc voices. These strata
are unified by the nearly exclusive use of whole-tone vlsets.
The left hand features only three dyadic set-classes, [04], [02], and [06] in
order of appearance. These dyads are all subsets of the whole-tone scale, therefore
the voice leading among them is drawn from the whole-tone scale. The voice-leading
sets featured in the left hand are members of voice-leading classes [02], [04], and
[06]. Figure 4-19 arranges all of the pcsets in the left hand into two horizontal pitch-
class voices. As shown, all of the voice leadings may be interpreted as some member
of vlclass [02], [04], or [06]. Furthermore, the four [04] vlsets, <7e> <̧e3>, and
<37>, form a network where each vlset is related to the next by T4. This mirrors the
augmented triad which closes the piece, as well as the [04] dyads featured throughout.
The piece begins with a repeated {26} dyad in the left hand. As shown in
Figure 4-19, this moves by T3 to {59} in m. 4, or vlset <33>. If the second dyad is
reordered, {95}, the mapping instead produces vlset <7e>, a member of vlclass [04].
Because dyads are inversionally symmetrical, the same inversional voice-leading
103
classes map {26} onto {59}, <73>i and <ee>i, respectively. In the second half of m.
4, the left hand oscillates between {59} and {26}. <59> is mapped onto <26> by
vlsets <99> and <73>i, and <59> onto <62> by <15> and <ee>i.
In m. 5, a harmonic change occurs, as the [04] dyads give way to [02] dyads;
these are spelled as minor sevenths in pitch space. As Figure 4-19 shows, pcset <59>
is mapped onto <68> by vlset <1e>, a member of vlclass [02]. The minor sevenths
then ascend by half-step. If the voice leading in this passage is interpreted as a series
of voice crossings, each successive seventh is produced by vlset <e3>. Vlset <e3> is
another member of vlclass [04].
The final left hand dyad, {06}, is introduced following the ascending minor
sevenths in m. 5. The left hand then alternates between three pcsets, {06}, {15}, and
{9e}, which summarize the whole-tone set-classes featured in the piece, [02], [04],
and [06]. As Figure 4-19 shows, <9e> is mapped onto <06> by vlset <37>.
The right hand voice-leading stratum is made up entirely of trichords. In mm.
2-4, the right hand alternates between pcsets {037} and {801}. Although these
trichords are not whole-tone subsets, they may be linked by similar voice leading.
Vlset <155> maps pcset <037> onto <180>. This voice leading demonstrates the
common [04] dyad—{37} and {80}—between the two pcsets. Vlclass [044] is very
similar to the [04] voice leadings in the left hand. The I0 inversion of the opening
vlset, <e77>, maps pcset <180> back onto <037>.
As Figure 4-19 shows, additional trichords are introduced in m. 5. Two of
these, {913} and {t04}, are whole-tone subsets. All other trichords in the right hand
104
are made up of members of both whole-tone scales. Vlclass [044] preserves the
whole-tone contents of a set, therefore it is impossible for any vlset members of [044]
to map {037} onto {913}, or {t04} onto {812}. Each pair of pcsets has an interval in
common, however, so it is possible to show their mappings using another multiset
vlclass, [033]. Vlset <366> maps <037> onto <391>, and <1tt> maps <04t> onto
<128>. The two whole-tone trichords, however, are spanned by a member of vlclass
[044]; <391> is mapped onto <t04> by vlset <733>. M. 5 returns to pcset {037} on
beat three. Vlset <ee1>, a member of vlclass [002], maps pcset <812> onto <703>.
An inversional vlset, <155>i, maps <370> onto <tt5>, and <t66> maps <tt5> onto
<84e>. This succession is repeated on beat two of m. 6, and m. 6 ends with pcsets
{48e} and {037}. These trichords may be mapped onto one another by members of
vlclass [002]; vlset <ee1> maps <48e> onto <370>, and vlset <e11> maps <037>
onto <e48>.
As the previous discussion has shown, the left and right hands of the piano
part can be interpreted as featuring predominantly whole-tone voice-leading. The
clarinet melody matches the piano part, as it utilizes whole-tone vlclasses [006],
[0066], and [044]. As discussed above, the clarinet line is interpreted as a
contrapuntal melody, which projects multiple pc voices.
Figure 4-20 shows the way in which the clarinet melody expresses the above
vlclasses. The first clarinet phrase, from the pickup to m. 2 to the downbeat of m. 3,
may be divided into two trichords, pcsets <013> and <487>. Vlset <771> maps the
first trichord onto the second; the first two pcs of the second trichord, Ab and G, are
105
reversed in order. A T1 transposition of the first vlset, <882>, maps <847> onto
<409> in m. 4; the trichords share E at the end of m. 3. The clarinet melody in mm.
5-6 is shaped by three pitch interval-cycles: -4/-1, -2/-3, and -1/-1. Each cycle forms
a successively more compact tetrachod, and the first cycles overlap by one note, F#.
The first tetrachord, <3t6e>, is mapped onto the next tetrachord, <4e16>, by vlset
<1177>. T1 of this vlset, <2288>, maps <641e> onto the next tetrachord, <8697>,
which extends to the downbeat of m. 7. Another pcset interpretation of mm. 6-7,
which follows the phrase markings, shows that pcset <987> is mapped onto <698> by
vlset <911>. While the repetition of the descending trichord <987>, and its
replacement <986> suggests <00e> voice leading on the surface of the piece, the
vlclass [044] interpretation matches the whole-tone voice-leadings and harmonies of
the accompaniment. Another vlclass [006] vlset, <228>, maps <698> onto <8e4>,
which, in score order <4e8>, is mapped onto the final trichord, <652>, by vlset
<266>.
As the previous discussion has shown, Berg’s Op. 5 n. 2 is easily divided into
three voice-leading strata. While two of these layers are homophonic and one is
melodic, all three are unified through their use of whole-tone voice-leading classes.
This forms a recursive structure with the whole-tone pcsets used on the surface of the
piece, especially the augmented triad. The whole-tone voice-leading consistency
among pc voices and pcsets provides another multi-movement connection in Berg’s
Op. 5, as the first piece, discussed above, also features a whole-tone voice-leading
class, [200].
106
Chapter 5: Summary, Conclusion, and Future Work
As discussed above, many studies of voice leading in post-tonal music focus
on harmonic similarity and semitonal offset. While this works well for some musical
contexts, it is not satisfactory for the intervallic diversity displayed by many post-
tonal pieces. Voice-leading sets provide a way in which to separate the study of voice
leading from harmony, and demonstrate that pc voices may unify disparate musical
surfaces.
The generalized definition of voice-leading parsimony proposed by this study
is best exemplified by the voice-leading spaces generated in chapter 3. The unit
distance on each space is defined by its generative vlset. For example, motion by one
unit of distance, therefore the “smoothest” voice leading, upon the vlset <23> space is
motion by the vlset itself. Less-smooth voice leadings are defined by combinations of
this generative vlset. Individual pieces may be modeled by one or more vl spaces;
representation on multiple spaces may be proposed to change voice-leading
interpretations and thereby change the metric for relative smoothness in different
formal sections. These spaces are useful tools for the type of analyses shown in
chapter 4, as adjacent pcsets or set-classes on the space reflect some version of the
underlying voice leading. The voice-leading spaces are also useful compositional
tools, as they provide a composer with a clear method for voice-leading manipulation
and variation.
Future Work
107
The method for generating voice-leading spaces in chapter 3 should be
expanded to higher cardinalities. This is difficult, due to the number of dimensions
required, as well as the exponential expansion of the number of members of the
ordered pcset space. This expansion can be reduced through equivalence classes, as
shown by the ordered tn-class and set-class spaces above, but the representational
power of the voice-leading spaces may be diminished by further layers of abstraction,
such as multiplication-class equivalence.
The voice-leading spaces shown in chapter 3 may be used for the
representation of voice leadings interpreted from a variety of musical works. These
voice leadings are understood as the motion between pcsets or set-classes on the
space; this motion corresponds to the pc voices expressed by the voice-leading
mapping. Therefore, animation is required to adequately represent these voice
leadings. The next step towards improving both the spaces’ visual representation and
explanatory power is the development of user-friendly software which will allow
users to animate voice leadings within the wide variety of spaces introduced here.110
Such animation could prove particularly useful in the representation of larger
cardinalities and their corresponding higher-dimensional structures; additionally, the
program could be designed to display only a small portion of a larger space at a given
time.
110
I have produced a number of visualization tools using Matlab, and plan to expand
them to encompass the full range of voice-leading sets. For an example of
transformational animation, see Stephanie Lind and John Roeder’s “Transformational
Distance and Form in Berg’s Schlafend traegt man mich,” in Music Theory Online 15,
1 (2009), as well as other contributions to the volume.
108
Segmentation Techniques
The problem of segmentation in the analysis of post-tonal music is
exacerbated by vlsets, as they exponentially increase the number of possible
relationships between musical events. The analyses shown in the previous chapter
address this issue by identifying some prominent voice-leading motive or motives,
and demonstrating the transformations of that motive throughout the piece. As is the
case for pcset theory and K-nets, vlsets are merely analytical tools; it is the analyst’s
responsibility to sort through the large quantity of data a piece presents in order to
construct an analysis from the most meaningful information.
One way in which to limit the great number of possible voice-leading
interpretations of a piece is to use consistent interpretive techniques. John Roeder’s
emphasis on pitch voice leadings provides a good example. Pitch space is ordered by
register, and therefore adjacent chord tones are well-defined using this ordering.
These chord-tone adjacencies are relative distances, however, as the “lowest” pitch
may be any frequency, so long as it is lower than the remaining pitches. Pitch-class
space is modular, eliminating the low-to-high ordering pitch space provides.
Harmonies in pitch-class space are ordered with respect to the chromatic scale,
however; just like pitch space, pitch-class space is ordered by relative frequency, that
is, the distance from one chord member to another. Just as the “lowest” pitch is
determined by its relationship to the remaining chord members, the members of a
109
pitch-class set are ordered with respect to one another. Normal order thus forms a
natural “register” for pitch-class space.111
Describing voice leadings among normal orders, rather than ordered pcsets,
greatly reduces the volume of information a piece presents. For example, if restricted
to normal orders, all minor triads are mapped onto major triads by vlclass [010]; all
tn-class [0147]s are mapped onto tn-class [0246]s by vlclass [0101], etc. This
reduction of pc space may be essential for the voice-leading analysis of larger
cardinalities, as one unordered pcset may have hundreds of possible mappings onto
another. There are only two possible mappings between two pcsets in normal order;
either the pc voices are transpositions or inversions. For example, <014> maps onto
<379> by either <365> or <38e>i.
In an analytical example, Figure 5-1 reinterprets the voice leading of the
accompaniment to Berg’s Op. 5 n. 2, with its right-hand trichords and left-hand
dyads, as a series of pentachords. The pc voices are produced by mapping
corresponding normal-order members onto one another. The two possible vlsets for
each mapping are shown below the score. The varied vlsets from this mapping show
some patterning, and can be considered a “first pass” through the piece.
Spaces of Voice leadings
Each of the voice-leading spaces explored in this study organize pcsets or
their ordered equivalence classes in a space defined by a specific vlset or vlclass.
111
Pitch-class space may also be ordered by the circle of fifths, where pc numerals
stand for the number of fifths from pc C. This would simply map the set of pcsets
onto itself through multiplication.
110
Because vlsets are isomorphic to the set of ordered pcsets, the objects on these spaces
could also be interpreted as organizing voice leadings, and demonstrating
relationships among them. This produces a metric for transformations between voice
leadings, and greatly expands the spaces’ analytical usefulness.
The generalization of the spaces from pcsets to vlsets is similar to the
generalization of voice leading proposed in this study. The analyses shown in chapter
4 privilege harmonic similarity among voice leadings, just as most contemporary
studies of voice leading privilege harmonic similarity among pcsets. The spaces may
therefore be used to model pieces which demonstrate many distinct voice leadings.
Figure 5-2 revisits mm. 1-4 of the clarinet melody from Alban Berg’s Op. 5 n.
1. The segmentation into trichords is the same as that shown in chapter 4, but the
pcsets are ordered by normal order; all are interpreted as transpositional vlsets. In the
figure, the pcsets are listed in normal order (ascending), showing the three pc voices
in pc register. Figures 5-3a and b show the unordered vlsets as locations on the
trichordal orbifold.112
The lines on the figure represent distances between vlsets. The
full orbifold is shown in Figure 5-3a for perspective. Figure 5-3c shows the vlsets on
the [013] ordered set-class space. The sets on this space are equivalent through
rotation, ordered transposition, and ordered inversion. As shown, the four ordered
vlclasses presented by the clarinet, [023], [001], [035], and [002], are located in one
112
This space is reduced by permutation, so the vlsets are unordered.
111
region of the space, demonstrating that the vlsets in the passage are very similar,
when [013] is used as a metric.113
The arrows on the space shown in Figure 5-3c imply that some transformation
other than transposition or inversion may map one vlset onto another. The set of
vlsets themselves form a model for the construction of such transformations.
Following Lewin’s creation of recursive K-net transformations, a hyper-vlset is an
ordered series of transpositions or inversions which maps one vlset onto another.
Hyper-vlsets are listed with double angle brackets to differentiate them from vlsets,
such as <<013>>. Figure 5-4 shows a graph of the vlsets in question. As shown,
each hyper-vlset is equivalent to <<013>> under rotation, ordered transposition, and
inversion, except for the transformations between vlclasses [001] and [002]; this
transformation is measured in two <<013>>s.
Serial Music
The study of voice leading as an ordered set of transformations finds its
natural extension in serial music. The ordered intervals of a series of any length may
be expressed using vlset notation. Ordered pc transformations may also be used to
demonstrate how rows are mapped onto one another. For example, the sixth
movement of Alban Berg’s Lyric Suite uses two rows: Row 1, or 1PF, is
<56t41928730e> and Row 2, or 2PF, is <54120e6t9873>. These rows, and their It
inverted forms—which causes all four rows begin on F—are layered successively in
mm. 1-6, shown in Figure 5-5.
113
This is the trichordal space shown in Tymoczko 2006.
112
Row 1 is mapped onto Row 2 by <0t3te2422574>.114
This mapping is
expressed vertically between the rows when the viola enters in m. 2. In m. 6, when
the inverted forms of the rows are given the same treatment in the violins, the same
vertical dyads are sounded. In mm. 4-5, the first violin plays the inverted form of
Row 2—2IF—above 1PF in the cello. If the mapping of 1PF onto 2IF is interpreted
as an ordered series of inversions, rather than transpositions, the ordered set is
<t070e8688536>i, an It inversion of the transpositional mapping. The remaining pair
is in m. 6 between the viola and second violin, 2PF and 1IF, respectively, which
produce an ordered inversional series, <t81890200352>i, a Tt transposition of the
original series, and an I8 inversion of the preceding pair. Inversion by index t is
important in other ways in the passage. Of course, row pairs {1PF 1IF} and {2PF
2IF} map onto themselves by It; in addition, in 1PF, order position-pairs 4-5, 6-7, and
8-9 are sum-t dyads, therefore they exchange places under It, and the first and last pcs
of the row, 5 and e, map onto themselves under It.115
The ordered mappings between the two rows of the movement form a third
row, which is subjected to transformation and manipulation just as the two pc rows
through which it is formed are. For example, the end of the movement returns to the
opening’s dyadic texture and reverses the layering used at the beginning; instruments
are removed one by one over the last five bars, until only the viola is left. The score
highlights invariant collections among horizontal and vertical row forms in this
114
Following Berg’s notes on the Lyric Suite given to the Kolisch Quartet and his
letter to Schoenberg in the derivation of material, George Perle derives the
hexachords of Row 2 from Row 1 through an ordered partition. Perle 2001, 11. 115
Various row relationships from this movement are given in Headlam 1996.
113
section. In m. 40, the second violin and viola play rows 1PC# and 2IEb, respectively;
the dual transformations T8 and I8 from 1PF and 2IF transform the ordered mapping
from Row 1 onto Row 2 to <461652022e90>i, an I4 transformation of the original
transpositional row. As Figure 5-6 shows, the cello and 2nd
violin present the original
form of the transformational row, <0t3te2422574>, although they are offset by an
eighth note. The viola and violin 1, offset by two eighth notes, begin a tritone apart,
and therefore present T6 of the row, <649458t88e1t>. This row is It of the row
between the second violin and viola.
Due to the pitch-class/order-position exchange, many more mapping
relationships among rows are possible.116
For example, the order position mapping
from 1PF onto 2PF is expressed by <055tt3922266>; each interval in the series
demonstrates the op interval from one row to another. This mapping only produces a
form of Row 2 which begins on the same pitch. If this harmonic relationship between
the rows is altered, the op mapping must be used in conjunction with some other
transformation. For example, in m. 40, Row 1IA maps onto 2IA under the op
mapping, and then may be transposed by T6 to 2IEb.
Diatonic and Cyclic Voice leading
116
For example, see Walter O’Connell’s “Tone-spaces” in Die Reihe, 8 (1968): 35-67,
Andrew Mead “Some Implications of the Order-number/Pitch-class Isomorphism
Inherent in the Twelve-tone System” in Perspectives of New Music27, n. 1 (1989):
180-233, Larry Solomon “New Symmetric Transformations” in Perspectives of New
Music 2, n. 2 (1973): 257-264, and Michael Stanfield “Some Exchange Operations in
Twelve-tone Theory: Part One” in Perspectives of New Music 23, n. 1 (1984): 258-
277.
114
The method for interpreting voice leadings proposed here may also be applied
to tonal music, whether using the full chromatic, or diatonic collection. Voice leading
in tonal music is typically the purview of Schenkerian analysis, as it provides a means
for showing the prolongation of the tonic triad at all levels of a piece. However,
voice-leading sets may be used to provide a unique contrapuntal perspective.
A four-voice chorale includes a limited number of diatonic sets, as mod-seven
space is considerably smaller than the full chromatic. In diatonic sets, root-doubled
triads are members of [0024], and seventh chords [0135]. In harmonic analysis, triads
and seventh chords may be presented in a number of different guises, with options for
doublings and incomplete chords. While these sonorities may be interpreted as
expressing the same basic harmony, they have distinct voice leading characteristics.
For example, the tripled root cadence results in multiset [0002], which is not capable
of the same types of voice leading as [0024].
A typical step towards harmonic analysis identifies non-harmonic tones, such
as appogiaturas, suspensions, passing tones, etc. All other notes fit into some
consonant sonority—either a triad or seventh chord. Voice-leading sets provide a
way in which to include non-tertian sonorities in the contrapuntal analysis of diatonic
voice leading. For example, Figure 5-7 shows that a 4-3 suspension projects two
diatonic set-classes: [0034] and [0024]. This motion can be expressed by vlset
<0060>, which shows that three pc voices are static, and one moves by diatonic
interval 6. Just as in the earlier post-tonal demonstrations, this vlset may be subjected
to variation through transposition and inversion. If the suspension figure recurs, and
115
the vlset is transposed by T1 to <1101>, the suspended voice remains motionless,
while the three remaining chord members ascend by diatonic step; see Figure 5-8.
Figures 5-7 and 5-8 have very different harmonic implications, but present the same
underlying contrapuntal motion through the relationships between the voices. Just as
in their application to post-tonal music, diatonic vlsets are largely independent of the
harmonic objects they operate upon.
The process used to create voice-leading spaces shown in chapter 3 may be
used to create diatonic voice-leading spaces. One advantage of doing this is the
reduction in the size of the space involved; there are only 2,401 ordered diatonic
tetrachords, compared to 20,736 ordered chromatic tetrachords. These produce much
more manageable spaces.
Just as voice leading among diatonic sets may be described by vlsets, voice
leadings among cyclic collections such as the octonic, hexatonic, whole-tone scale,
Olivier Messiaen’s Modes of Limited Transposition, etc, may be represented by vlsets
and voice-leading spaces. The sets and voice leadings are simply defined modulo the
collection; for example, the hexatonic collection is made up of mod-6 sets and voice
leadings. Vlset <112> maps hexatonic set {023} onto {135}, which correspond to
pcsets {045} and {159}, respectively.
Contour Spaces
Voice-leading spaces may be created for any modulus. If the sets in these
spaces are interpreted as scale-degree sets, rather than pc sets, they may be populated
by any collection of pcs which are mapped onto the scale-degrees. For example, if a
116
mod-7 vlset <013> is applied to a C major triad, diatonic set {024}, it maps the triad
onto {030}, a C-E multiset. If the diatonic sets are mapped onto some other
collection, say a chromatic septachord, the same mapping will produce different
musical results. Generalized contour spaces may be created, through which any
pitch-class collections may be mapped, just as pitch-classes are mapped onto an
ordered series in serial music.
Transformation Vectors
The analyses presented in chapter four demonstrate some of the ways in which
pitch-class voices may be defined in post-tonal music using vlsets—vectors of
transformational mappings. Vectors of transformations need not be limited to pitch-
class mappings, however. Any musical parameter, such as rhythm, meter,
articulation, texture, etc, may be described as an ordered series of mappings.117
Let the positive integers be mapped onto durations such that the integers
correspond to sixteenth-note durations, that is, 0 = no duration, 1 = one sixteenth-
note, 2 = one eighth-note, etc. Just as vlsets provide a method for mapping pcsets
onto one another regardless of harmonic similarity, vectors of transpositions and
inversions may be used to map various rhythmic patterns onto one another. As in the
analyses shown above, these transformations may be used to depict the variation of
rhythmic motives throughout a piece. The idea of duration voices is more difficult to
117
In GMIT, Lewin describes a number of GIS constructions whose objects are non-
pitch musical parameters, such as rhythm and timbre. Roeder 1994 demonstrates
some additional ways in which vectors of transformations may be employed.
117
conceptualize, but nevertheless, the simultaneous transformations of individual
rhythms forms a type of duration counterpoint.
For example, returning to Anton Webern’s Op. 5 n. 2, the rhythm of the
opening viola melody is expressed by durationset {4624}, or quarter, dotted-quarter,
eighth, quarter. This durationset is varied in the melody in m. 5; beginning with Bb
and continuing to F# on the downbeat of m. 6, the 2nd
violin plays durationset
{2313}. This is very nearly half of the opening durationset. The first three values of
this set are exactly half of the opening three durations, while the final value is
extended by one sixteenth-note.
If the opening durationset is extended to include the eighth rest in m. 1, the
ordered set is {46240}, where 0 denotes no duration. Although the rest consists of
two sixteenth notes, there is no melody duration during the rest. Simply for clarity of
demonstration, let us assume mod-twelve.118
This pentachord is mapped onto the
{22244} gesture which immediately follows by <t8004>.
The rhythm of the melody in the final measure of the piece consists of three
eighths and a quarter, or {2224}; this interpretation ignores the eighth rest in the
middle of the measure. The opening gesture is mapped onto the closing gesture by
<t800>. This mapping represents the expanding and contracting of the rhythmic
values as they are transformed from the opening of the piece to the end.
Another tetrachordal rhythmic collection occurs in the melody in m. 6, in the
2nd
violin, {3133}. As an un-ordered set, it is an inversion of the durationset
118
The longest rhythm appears to be the tied chords in the accompaniment at the end,
which consists of 18 sixteenth-notes, so mod-nineteen may be a better choice.
118
highlighted in the final measure of the piece <5555>i maps one set onto the other. It
is difficult to imagine how a duration may be inverted, but it is important to
remember that the mapping takes place between the numbers, which are
representatives of the durations. If the durations are arranged, like pitch-classes, in a
mod-12 circle, <5555>i forms the axis of inversion which the durations exchange
places around.
Any musical feature may be treated transformationally the way durations are
in the above example, provided it is parameterized, that is, mapped onto an ordered
series. This creates a feature vector; a space which maps musical objects onto
locations given by coordinates. The voice-leading spaces discussed above are feature
spaces.
As the above discussion shows, vlsets offer great promise for the analysis of
post-tonal and other musics. The analytical techniques outlined here and in chapter 4
reflect some possible methods for identifying pc voices in post-tonal pieces. These
voices may be represented as motion in some of the voice-leading spaces developed
above. Using vlsets, voice-leading parsimony may be redefined contextually for
individual musical contexts. The voice-leading spaces then show this relative
parsimony through the proximity of their objects.
119
Figure 1-1: Arnold Schoenberg’s Drei Klavierstuecke Op. 11 n. 2, mm. 11-13
a b c d e
{1467} {e235} {4580} {569tt1} {578t01}
Figure 1-2a: fuzzy transposition as in Straus 2003
6 3
7 e
4 5
1 2
*Tt(1)
a b
Figure 1-2b: pitch voice leadings as in Roeder 1994
6 -3 3
7 +4 e
4 +1 5
1 +1 2
<1,1,4,-3>
a b
120
Figure 1-2c: vlsets from chord a to b and d to e
9 0
5 5
6 3 6 3 1 t
7 e 7 e t 8
4 5 4 5 6 1
1 2 1 2 t 7
<tt9t> <0690>i <01e020>
a b a b d e
121
Figure 1-3: values for the moving voice when retaining two common tones,
from Richard Cohn.
Set-class “Root” “Third” “Fifth”
[012] 3 0 9 {012}-{312}; {012}-{012}; {012}-{01e}
[013] 4 1 5 {013}-{413}; {013}-{023}; {013}-{01t}
[014] 5 2 5 {014}-{514}; {014}-{034}; {014}-{019}
[015] 6 3 3 {015}-{615}; {015}-{045}; {015}-{018}
[016] 5 2,4,6* 1 {016}-{716}; {016}-{056}; {016}-{0e6};
{016}-{076}; {016}-{017}
[024] 6 0 6 {024}-{624}; {024}-{024}; {024}-{02t}
[025] 5 1 4 {025}-{725}; {025}-{035}; {025}-{029}
[026] 4 2,4,6* 2 {026}-{826}; {026}-{046}; {026}-{0t6};
{026}-{086}; {026}-{028}
[027] 3 3 0 {027}-{927}; {027}-{057}; {027}-{027}
[036] 3 6 3 {036}-{936}; {036}-{096}; {036}-{039}
[037] 2 1 1 {037}-{t37}; {037}-{047}; {037}-{038}
[048] 0 0 0 {048}-{048}; {048}-{048}; {048}-{048}
*Because the tritone divides the octave evenly, four members of [016] and [026]
share the same tritone, resulting in more common-tone preserving transformations
when moving the other voice. These additional transformations are between
transpositionally-related sets. Symmetrical sets, such as [024], have only two
possible operations, as the interval between “root” and “fifth” can only form a
member of the set-class with one other pitch-class; [036] has three possible
transformations because of the symmetry of the tritone.
122
Figure 1-4a: the trichordal orbifold.
123
Figure 1-4b: one-quarter of the trichordal orbifold
124
Figure 2-1: vlsets in each cardinality
1: 12 * 2 = 24
2: 12 * 12 * 2 = 288
3: 12 * 12 * 12 * 2 = 3,456
4: 12 * 12 * 12 * 12 * 2 = 41,472
5: 12 * 12 * 12 * 12 * 12 * 2 = 497,664
6: 12 * 12 * 12 * 12 * 12 * 12 * 2 = 5,971,968
7: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2 = 71,663,616
8: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2 = 859,963,392
9: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2
= 10,319,560,704
10: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2
= 123,834,728,448
11: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2
= 1,486,016,741,376
12: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2
= 1.78322009 * 1013
13: 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 12 * 2
= 2.13986411 * 1014
∞: (12^∞) * 2 = ∞
125
Figure 2-2: vlsets spanning reordered pcsets
<014> - <444> - <458> <014> - <460>i - <458>
<014> - <471> - <485> <014> - <499>i - <485>
<014> - <570> - <584> <014> - <598>i - <584>
<014> - <534> - <548> <014> - <550>i - <548>
<014> - <831> - <845> <014> - <859>i - <845>
<014> - <840> - <854> <014> - <868>i - <854>
126
Figure 2-3: parallel pitch-class voice leading
Figure 2-4: oblique pitch-class voice leading
Figure 2-5: contrary pitch-class voice leading
x = pcset <24> <39>i(x) <2t>i(x) <1e>i(x) <00>i(x) <e1>i(x)
<222> <777>i
<05> <10>i
<39> <1e> <et96>
127
Figure 2-6: similar pitch-class voice leading
<21> <78>i
<1122> <011>i
Figure 2-7: vlset transposition, or ordered vlclass [01]
<02> - <01> - <03> <02> - <67> - <69>
<02> - <12> - <14> <02> - <78> - <7t>
<02> - <23> - <25> <02> - <89> - <8e>
<02> - <34> - <36> <02> - <9t> - <90>
<02> - <45> - <47> <02> - <te> - <t1>
<02> - <56> - <58> <02> - <e0> - <e2>
128
Figure 2-8a: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 1-5
Figure 2-8b: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 6-13
e 9 e 1 e e 9 1
8 5 5 9 8 5 5 9
7 4 6 t 7 6 4 t
<tt8> <442> <tt0> <446> a b c d a c b d
8 5 0 8 6 8 8 0
4 e e 2 2 4 t e
6 7 3 9 0 6 9 3
<113> <99e> <442> <335> g h i j e g f i
129
Figure 2-9: ordered pcset tn-classes paired by ordered vlclasses
a). [01] b). [10]
[00] – [01] [06] – [50] [00] – [10] [06] – [05]
[01] – [02] [50] – [40] [01] – [00] [50] – [06]
[02] – [03] [40] – [30] [02] – [01] [40] – [50]
[03] – [04] [30] – [20] [03] – [02] [30] – [40]
[04] – [05] [20] – [10] [04] – [03] [20] – [30]
[05] – [06] [10] – [00] [05] – [04] [10] – [20]
c). [02] d). [20]
[00] – [02] [06] – [40] [00] – [20] [06] – [04]
[01] – [03] [50] – [30] [01] – [10] [50] – [05]
[02] – [04] [40] – [20] [02] – [00] [40] – [06]
[03] – [05] [30] – [10] [03] – [01] [30] – [50]
[04] – [06] [20] – [00] [04] – [02] [20] – [40]
[05] – [50] [10] – [01] [05] – [03] [10] – [30]
e). [03] f). [30]
[00] – [03] [06] – [30] [00] – [30] [06] – [03]
[01] – [04] [50] – [20] [01] – [20] [50] – [04]
[02] – [05] [40] – [10] [02] – [10] [40] – [05]
[03] – [06] [30] – [00] [03] – [00] [30] – [06]
[04] – [50] [20] – [01] [04] – [01] [20] – [50]
[05] – [40] [10] – [02] [05] – [02] [10] – [20]
130
g). [04] h). [40]
[00] – [04] [06] – [20] [00] – [40] [06] – [02]
[01] – [05] [50] – [10] [01] – [30] [50] – [03]
[02] – [06] [40] – [00] [02] – [20] [40] – [04]
[03] – [50] [30] – [01] [03] – [10] [30] – [05]
[04] – [40] [20] – [02] [04] – [00] [20] – [06]
[05] – [30] [10] – [03] [05] – [01] [10] – [50]
i). [05] j). [50]
[00] – [05] [06] – [10] [00] – [50] [06] – [01]
[01] – [06] [50] – [00] [01] – [40] [50] – [02]
[02] – [50] [40] – [01] [02] – [30] [40] – [03]
[03] – [40] [30] – [02] [03] – [20] [30] – [04]
[04] – [30] [20] – [03] [04] – [10] [20] – [05]
[05] – [20] [10] – [04] [05] – [00] [10] – [06]
k). [06] l). [00]
[00] – [06] [06] – [00] [00] – [00] [06] – [06]
[01] – [50] [50] – [01] [01] – [01] [50] – [50]
[02] – [40] [40] – [02] [02] – [02] [40] – [40]
[03] – [30] [30] – [03] [03] – [03] [30] – [30]
[04] – [20] [20] – [04] [04] – [04] [20] – [20]
[05] – [10] [10] – [05] [05] – [05] [10] – [10]
131
Figure 2-10: ordered tn-class cycles
a). [01]
([00] – [01] – [02] – [03] – [04] – [05] – [06] – [50] – [40] – [30] – [20] – [10])
b). [10]
([00] – [10] – [20] – [30] – [40] – [50] – [06] – [05] – [04] – [03] – [02] – [01])
c). [02] d). [20]
([00] – [02] – [04] – [06] – [40] – [20]) ([00] – [20] – [40] – [06] – [04] – [02])
([01] – [03] – [05] – [50] – [30] – [10]) ([01] – [10] – [30] – [50] – [05] – [03])
e). [03] f). [30]
([00] – [03] – [06] – [30]) ([00] – [30] – [06] – [03])
([01] – [04] – [50] – [20]) ([01] – [20] – [50] – [04])
([02] – [05] – [40] – [10]) ([02] – [10] – [40] – [05])
g). [04] h). [40]
([00] – [04] – [40]) ([00] – [40] – [04])
([01] – [05] – [30]) ([01] – [30] – [05])
([02] – [06] – [20]) ([02] – [20] – [06])
([03] – [50] – [10]) ([03] – [10] – [50])
132
i). [05]
([00] – [05] – [20] – [03] – [40] – [01] – [06] – [10] – [04] – [30] – [02] – [50])
j). [50]
([00] – [50] – [02] – [30] – [04] – [10] – [06] – [01] – [40] – [03] – [20] – [05])
Figure 2-11: an ordered tn-class [04] tn-class cycle
<34> – <15> – <49> – <37> – <74> – <t2> – <56>;
tn-classes: [01] – [05] – [30] – [01]
Figure 2-12: ordered pcset and tn-class cycles formed by vlset <14>
<12> – <14> – <26> – <14> – <3t> – <14> – <42> – <14> – <56> – <14> – <6t> –
<14> – <72> – <14> – <86> – <14> – <9t> – <14> – <t2> – <14> – <e6> – <14> –
<0t> – <14> – <12>
tn-classes:
[01] – [04] – [50] – [20] – [01] – [04] – [50] – [20] – [01] – [04] – [50] – [20] – [01]
133
Figure 2-13: transpositions of vlset <01>i and <0e>i
a). <01>i b). <0e>i
<04> - <01>i - <09> <04> - <0e>i - <07>
<04> - <12>i - <1t> <04> - <10>i - <18>
<04> - <23>i - <2e> <04> - <21>i - <29>
<04> - <34>i - <30> <04> - <32>i - <3t>
<04> - <45>i - <41> <04> - <43>i - <4e>
<04> - <56>i - <52> <04> - <54>i - <50>
<04> - <67>i - <63> <04> - <65>i - <61>
<04> - <78>i - <74> <04> - <76>i - <72>
<04> - <89>i - <85> <04> - <87>i - <83>
<04> - <9t>i - <96> <04> - <98>i - <94>
<04> - <te>i - <t7> <04> - <t9>i - <t5>
<04> - <e0>i - <e8> <04> - <et>i - <e6>
134
Figure 2-14: ordered tn-classes paired by ordered vlclasses [01]i and [10]i
a). [01]i b). [10]i
[00] – [01] [00] – [10]
[02] – [10] [01] – [20]
[03] – [20] [02] – [30]
[04] – [30] [03] – [40]
[05] – [40] [04] – [50]
[06] – [50] [05] – [06]
[00]
[06]
[03] [30]
[01]
[02]
[04]
[05]
[10]
[20]
[40]
[50]
[00]
[06]
[03] [30]
[01]
[02]
[04]
[05]
[10]
[20]
[40]
[50]
[01]i [10]i
135
Figure 2-15: ordered pcset-classes paired by ordered vlclasses [04]i and [40]i
a). [04]i b). [40]i
[00] – [04] [00] – [40]
[01] – [03] [10] – [30]
[02] – [02] [20] – [20]
[05] – [10] [50] – [01]
[06] – [20] [06] – [02]
[50] – [30] [05] – [03]
[40] – [40] [04] – [04]
[00]
[06]
[03] [30]
[01]
[02]
[04]
[05]
[10]
[20]
[40]
[50]
[00]
[06]
[03] [30]
[01]
[02]
[04]
[05]
[10]
[20]
[40]
[50]
[04]i [40]i
136
Figure 2-16: tn-class pairs
a). [02]i b). [20]i c). [03]i d). [30]i
[00] – [02] [00] – [20] [00] – [03] [00] – [30]
[01] – [01] [01] – [30] [01] – [02] [01] – [40]
[03] – [10] [02] – [40] [04] – [10] [02] – [50]
[04] – [20] [03] – [50] [05] – [20] [03] – [06]
[05] – [30] [04] – [06] [06] – [30] [04] – [05]
[06] – [40] [05] – [05] [50] – [40] [20] – [10]
[50] – [50] [10] – [10]
a). [05]i b). [50]i c). [06]i d). [00]i
[00] – [05] [00] – [50] [00] – [06] [00] – [00]
[01] – [04] [01] – [06] [01] – [05] [01] – [10]
[02] – [03] [02] – [05] [02] – [04] [02] – [20]
[06] – [10] [03] – [04] [03] – [03] [03] – [30]
[50] – [20] [40] – [10] [50] – [10] [04] – [40]
[40] – [30] [30] – [20] [40] – [20] [05] – [50]
[30] – [30] [06] – [06]
137
Figure 2-17: ordered tn-classes paired by ordered vlclass [025]
[000] – [025] [010] – [035] [020] – [045] [030] – [055] [040] – [065] [050] – [075]
[001] – [026] [011] – [036] [021] – [046] [031] – [056] [041] – [066] [051] – [076]
[002] – [027] [012] – [037] [022] – [047] [032] – [057] [042] – [067] [052] – [077]
[003] – [028] [013] – [038] [023] – [048] [033] – [058] [043] – [068] [053] – [078]
[004] – [029] [014] – [039] [024] – [049] [034] – [059] [044] – [069] [054] – [079]
[005] – [02t] [015] – [03t] [025] – [04t] [035] – [05t] [045] – [06t] [055] – [07t]
[006] – [02e] [016] – [03e] [026] – [04e] [036] – [05e] [046] – [06e] [056] – [07e]
[007] – [020] [017] – [030] [027] – [040] [037] – [050] [047] – [060] [057] – [070]
[008] – [021] [018] – [031] [028] – [041] [038] – [051] [048] – [061] [058] – [071]
[009] – [022] [019] – [032] [029] – [042] [039] – [052] [049] – [062] [059] – [072]
[00t] – [023] [01t] – [033] [02t] – [043] [03t] – [053] [04t] – [063] [05t] – [073]
[00e] – [024] [01e] – [034] [02e] – [044] [03e] – [054] [04e] – [064] [05e] – [074]
[060] – [085] [070] – [095] [080] – [0t5] [090] – [0e5] [0t0] – [005] [0e0] – [015]
[061] – [086] [071] – [096] [081] – [0t6] [091] – [0e6] [0t1] – [006] [0e1] – [016]
[062] – [087] [072] – [097] [082] – [0t7] [092] – [0e7] [0t2] – [007] [0e2] – [017]
[063] – [088] [073] – [098] [083] – [0t8] [093] – [0e8] [0t3] – [008] [0e3] – [018]
[064] – [089] [074] – [099] [084] – [0t9] [094] – [0e9] [0t4] – [009] [0e4] – [019]
[065] – [08t] [075] – [09t] [085] – [0tt] [095] – [0et] [0t5] – [00t] [0e5] – [01t]
[066] – [08e] [076] – [09e] [086] – [0te] [096] – [0ee] [0t6] – [00e] [0e6] – [01e]
[067] – [080] [077] – [090] [087] – [0t0] [097] – [0e0] [0t7] – [000] [0e7] – [010]
[068] – [081] [078] – [091] [088] – [0t1] [098] – [0e1] [0t8] – [001] [0e8] – [011]
[069] – [082] [079] – [092] [089] – [0t2] [099] – [0e2] [0t9] – [002] [0e9] – [012]
[06t] – [083] [07t] – [093] [08t] – [0t3] [09t] – [0e3] [0tt] – [003] [0et] – [013]
[06e] – [084] [07e] – [094] [08e] – [0t4] [09e] – [0e4] [0te] – [004] [0ee] – [014]
138
Table 2-1: subgroups of the group of dyadic vlsets
1 {<00>} 6 {<42>} 12 {<23>} 24 {<06>, <12>}
6 {<46>} 12 {<31>} 24 {<06>, <13>}
2 {<06>} 6 {<62>} 12 {<32>} 24 {<06>, <14>}
2 {<60>} 6 {<64>} 12 {<34>} 24 {<06>, <15>}
2 {<66>} 6 {<82>} 12 {<41>} 24 {<21>, <60>}
12 {<43>} 24 {<31>, <60>}
3 {<04>} 8 {<03>, <60>} 12 {<61>} 24 {<41>, <60>}
3 {<40>} 8 {<06>, <30>} 12 {<81>}
3 {<44>} 8 {<06>, <33>} 12 {<91>} 36 {<01>, <40>}
3 {<48>} 12 {<t1>} 36 {<02>, <20>}
9 {<04>, <40>} 12 {<02>, <60>} 36 {<04>, <10>}
4 {<03>} 12 {<06>, <20>} 36 {<04>, <11>}
4 {<30>} 12 {<01>} 12 {<06>, <22>} 36 {<04>, <13>}
4 {<33>} 12 {<10>} 12 {<06>, <24>} 36 {<04>, <21>}
4 {<36>} 12 {<11>} 36 {<12>, <20>}
4 {<39>} 12 {<12>} 16 {<03>, <30>}
4 {<63>} 12 {<13>} 48 {<01>, <30>}
4 {<06>, <60>} 12 {<14>} 18 {<02>, <40>} 48 {<03>, <10>}
12 {<15>} 18 {<04>, <20>} 48 {<03>, <11>}
6 {<02>} 12 {<16>} 18 {<04>, <22>} 48 {<03>, <12>}
6 {<20>} 12 {<17>}
6 {<22>} 12 {<18>} 24 {<01>, <60>} 72 {<01>, <20>}
6 {<24>} 12 {<19>} 24 {<02>, <30>} 72 {<02>, <10>}
6 {<26>} 12 {<1t>} 24 {<03>, <20>} 72 {<02>, <11>}
6 {<28>} 12 {<1e>} 24 {<06>, <10>}
6 {<2t>} 12 {<21>} 24 {<06>, <11>} 144 {<01>, <10>}
139
Table 2-2: group table for subgroup generated by <17>
140
Figure 2-18: subgroup <17> with three sets of pcsets as objects
If originating with pc {00}:
{00}---{17}---{22}---{39}---{44}---{5e}---{66}---{71}---{88}---{93}---{tt}---{e5}
If originating with pc {02}:
{02}---{19}---{24}---{3e}---{46}---{51}---{68}---{73}---{8t}---{95}---{t0}---{e7}
If originating with pc {01}:
{01}---{18}---{23}---{3t}---{45}---{50}---{67}---{72}---{89}---{94}---{te}---{e6}
141
Table 2-3: subgroup <02> with inversion
142
Table 2-4: examples of four-member subgroups of [C12 C12] C2
<00> <06> <00>i <06>i
<00> <00> <06> <00>i <06>i
<06> <06> <00> <06>i <00>i
<00>i <00>i <06>i <00> <06>
<06>i <06>i <00>i <06> <00>
<00> <06> <a,b>i <a,b+6>i
<00> <00> <06> <a,b>i <a,b+6>i
<06> <06> <00> <a,b+6>i <a,b>i
<a,b>i <a,b>i <a,b+6>i <00> <06>
<a,b+6>i <a,b+6>i <a,b>i <06> <00>
<00> <66> <29>i <83>i
<00> <00> <66> <29>i <83>i
<66> <66> <00> <83>i <29>i
<29>i <29>i <83>i <00> <66>
<83>i <83>i <29>i <66> <00>
<00> <06> <12>i <18>i
<00> <00> <06> <12>i <18>i
<06> <06> <00> <18>i <12>i
<12>i <12>i <18>i <00> <06>
<18>i <18>i <12>i <06> <00>
143
Table 2-5: examples of two-member subgroups of [C12 C12] C2
<00> <20>i
<00> <00> <20>i
<20>i <20>i <00>
Table 2-6: C12 C12 subgroups combined with inversional vlsets
2 {<00>}, <a,b>i 144 24 {<23>}, <a+2n, b+3n>i 12
24 {<31>}, <a+3n, b+n>i 12
4 {<06>}, <a, b+6n>i 72 24 {<32>}, <a+3n, b+2n>i 12
4 {<60>}, <a+6n, b>i 72 24 {<34>}, <a+3n, b+4n>i 12
4 {<66>}, <a+6n, b+6n>i 72 24 {<41>}, <a+4n, b+n>i 12
24 {<43>}, <a+4n, b+3n>i 12
6 {<04>}, <a, b+4n>i 48 24 {<61>}, <a+6n, b+n>i 12
6 {<40>}, <a+4n, b>i 48 24 {<81>}, <a+8n, b+n>i 12
6 {<44>}, <a+4n, b+4n>i 48 24 {<91>}, <a+9n, b+n>i 12
6 {<48>}, <a+4n, b+8n>i 48 24 {<t1>}, <a+tn, b+n>i 12
24 {<02>, <60>} + <a,b>i 12
8 {<03>}, <a, b+3n>i 36 24 {<06>, <20>} + <a,b>i 12
8 {<30>}, <a+3n, b>i 36 24 {<06>, <22>} + <a,b>i 12
8 {<33>}, <a+3n, b+3n>i 36 24 {<06>, <24>} + <a,b>i 12
<00> <15>i
<00> <00> <15>i
<15>i <15>i <00>
144
8 {<36>}, <a+3n, b+6n>i 36
8 {<39>}, <a+3n, b+9n>i 36 32 {<03>, <30>} + <a,b>i 9
8 {<63>}, <a+6n, b+3n>i 36
8 {<06>, <60>} + <a,b>i 36 36 {<02>, <40>} + <a,b>i 8
36 {<04>, <20>} + <a,b>i 8
12 {<02>}, <a, b+2n>i 24 36 {<04>, <22>} + <a,b>i 8
12 {<20>}, <a+2n, b>i 24
12 {<22>}, <a+2n, b+2n>i 24 48 {<01>, <60>} + <a,b>i 6
12 {<24>}, <a+2n, b+4n>i 24 48 {<02>, <30>} + <a,b>i 6
12 {<26>}, <a+2n, b+6n>i 24 48 {<03>, <20>} + <a,b>i 6
12 {<28>}, <a+2n, b+8n>i 24 48 {<06>, <10>} + <a,b>i 6
12 {<2t>}, <a+2n, b+tn>i 24 48 {<06>, <11>} + <a,b>i 6
12 {<42>}, <a+4n, b+2n>i 24 48 {<06>, <12>} + <a,b>i 6
12 {<46>}, <a+4n, b+6n>i 24 48 {<06>, <13>} + <a,b>i 6
12 {<62>}, <a+6n, b+2n>i 24 48 {<06>, <14>} + <a,b>i 6
12 {<64>}, <a+6n, b+4n>i 24 48 {<06>, <15>} + <a,b>i 6
12 {<82>}, <a+8n, b+2n>i 24 48 {<21>, <60>} + <a,b>i 6
48 {<31>, <60>} + <a,b>i 6
16 {<03>, <60>} + <a,b>i 18 48 {<41>, <60>} + <a,b>i 6
16 {<06>, <30>} + <a,b>i 18
16 {<06>, <33>} + <a,b>i 18 72 {<01>, <40>} + <a,b>i 4
145
72 {<02>, <20>} + <a,b>i 4
18 {<04>, <40>} + <a,b>i 16 72 {<04>, <10>} + <a,b>i 4
72 {<04>, <11>} + <a,b>i 4
24 {<01>}, <a, b+n>i 12 72 {<04>, <13>} + <a,b>i 4
24 {<10>}, <a+n, b>i 12 72 {<04>, <21>} + <a,b>i 4
24 {<11>}, <a+n, b+n>i 12 72 {<12>, <20>} + <a,b>i 4
24 {<12>}, <a+n, b+2n>i 12
24 {<13>}, <a+n, b+3n>i 12 96 {<01>, <30>} + <a,b>i 3
24 {<14>}, <a+n, b+4n>i 12 96 {<03>, <10>} + <a,b>i 3
24 {<15>}, <a+n, b+5n>i 12 96 {<03>, <11>} + <a,b>i 3
24 {<16>}, <a+n, b+6n>i 12 96 {<03>, <12>} + <a,b>i 3
24 {<17>}, <a+n, b+7n>i 12
24 {<18>}, <a+n, b+8n>i 12 144 {<01>, <20>} + <a,b>i 2
24 {<19>}, <a+n, b+9n>i 12 144 {<02>, <10>} + <a,b>i 2
24 {<1t>}, <a+n, b+tn>i 12 144 {<02>, <11>} + <a,b>i 2
24 {<1e>}, <a+n, b+en>i 12
24 {<21>}, <a+2n, b+n>i 12 288 {<01>, <10>} + <a,b>i 1
146
Table 2-7: group table for {<04>, <40>} + <61>i
<00> <04> <08> <48> <40> <44> <84> <88> <80>
<00> <00> <04> <08> <48> <40> <44> <84> <88> <80>
<04> <04> <08> <00> <40> <44> <48> <88> <80> <84>
<08> <08> <00> <04> <44> <48> <40> <80> <84> <88>
<48> <48> <40> <44> <84> <88> <80> <00> <04> <08>
<40> <40> <44> <48> <88> <80> <84> <04> <08> <00>
<44> <44> <48> <40> <80> <84> <88> <08> <00> <04>
<84> <84> <88> <80> <00> <04> <08> <48> <40> <44>
<88> <88> <80> <84> <04> <08> <00> <40> <44> <48>
<80> <80> <84> <88> <08> <00> <04> <44> <48> <40>
<61>i <61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i
<65>i <65>i <69>i <61>i <t1>i <t5>i <t9>i <29>i <21>i <25>i
<69>i <69>i <61>i <65>i <t5>i <t9>i <t1>i <21>i <25>i <29>i
<t9>i <t9>i <t1>i <t5>i <25>i <29>i <21>i <61>i <65>i <69>i
<t1>i <t1>i <t5>i <t9>i <29>i <21>i <25>i <65>i <69>i <61>i
<t5>i <t5>i <t9>i <t1>i <21>i <25>i <29>i <69>i <61>i <65>i
<25>i <25>i <29>i <21>i <61>i <65>i <69>i <t9>i <t1>i <t5>i
<29>i <29>i <21>i <25>i <65>i <69>i <61>i <t1>i <t5>i <t9>i
<21>i <21>i <25>i <29>i <69>i <61>i <65>i <t5>i <t9>i <t1>i
147
Table 2-7 continued
<61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i
<61>i <65>i <69>i <t9>i <t1>i <t5>i <25>i <29>i <21>i
<69>i <61>i <65>i <t5>i <t9>i <t1>i <21>i <25>i <29>i
<65>i <69>i <61>i <t1>i <t5>i <t9>i <29>i <21>i <25>i
<25>i <29>i <21>i <61>i <65>i <69>i <t9>i <t1>i <t5>i
<21>i <25>i <29>i <69>i <61>i <65>i <t5>i <t9>i <t1>i
<29>i <21>i <25>i <65>i <69>i <61>i <t1>i <t5>i <t9>i
<t9>i <t1>i <t5>i <25>i <29>i <21>i <61>i <65>i <69>i
<t5>i <t9>i <t1>i <21>i <25>i <29>i <69>i <61>i <65>i
<t1>i <t5>i <t9>i <29>i <40> <25>i <65>i <69>i <61>i
<00> <04> <08> <48> <48> <44> <84> <88> <80>
<08> <00> <04> <44> <48> <40> <80> <84> <88>
<04> <08> <00> <40> <44> <48> <88> <80> <84>
<84> <88> <80> <00> <04> <08> <48> <40> <44>
<80> <84> <88> <08> <00> <04> <44> <48> <40>
<88> <80> <84> <04> <08> <00> <40> <44> <48>
<48> <40> <44> <84> <88> <80> <00> <04> <08>
<44> <48> <40> <80> <84> <88> <08> <00> <04>
<40> <44> <48> <88> <80> <84> <04> <08> <00>
148
Figure 2-19: cyclic groups and rotational symmetry, C3: T0, T4, T8
Figure 2-20: dihedral groups, D3, T0, T4, T8, In, In+4, In+8
A
B C
T8 T4
A
B C
IB IC
IA
149
Figure 2-21: T/I group representation
0 e 1
t 2
9 3
8 4
7 5 6
150
Figure 2-22: the dodecagonal torus
151
Figure 2-23: interlocking dodecagons
a). pcset {000}
b). + vlset <216> c). + vlset <358>i
0
e 1
t 2
9 3
8 4
7 5
6
0 e 1
t 2
9 3
8 4
7 5 6
0 e 1
t 2
9 3
8 4
7 5 6
2
1 3
0 4
e 5
t 6
9 7
8
1 0 2
e 3
t 4
9 5
8 6 7
6 5 7
4 8
3 9
2 t
1 e 0
T2
T1
T6
3
4 2
5 1
6 0
7 e
8 t
9
5 6 4
7 3
8 2
9 1
t 0 e
8 9 7
t 6
e 5
0 4
1 3 2
I8
I5
I3
152
Figure 2-24: vlset <t27> on interlocking dodecagons;
{238} - <t27> - {053}
2
1 3
0 4
e 5
t 6
9 7
8
3 2 4
1 5
0 6
e 7
t 8 9
8 7 9
6 t
5 e
4 0
3 1 2
0
e 1
t 2
9 3
8 4
7 5
6
5 4 6
3 7
2 8
1 9
0 t e
3 2 4
1 5
0 6
e 7
t 8 9
153
Figure 3-1: Straus’ trichordal set-class space
154
Figure 3-2a: the dyadic orbifold
Figure 3-2b: dyadic permutational equivalence
155
Figure 3-2c: dyadic Moebius strip
156
Figure 3-3: torus knot of singleton dyads
157
Figure 3-4a: Moebius strip dyadic orbifold
158
Figure 3-4b: dyadic set-class line
[00] – [01] – [02] – [03] – [04] – [05] – [06]
Figure 3-4c: dyadic tn-class circle
Figure 3-5a: <05> space
[00]
[09] [03]
[06]
[0e] [01]
[07] [05]
[0t] [02]
[08] [04]
159
Figure 3-5b: <05> Moebius strip
160
Figure 3-5c: adjacent voice leadings along the <05> Moebius strip
0 0 5 5 0 e e e
9 4 4 e e 7 2 9
<07> <05> <07> <07> <07> <07> <07>
Figure 3-5d: adjacent and compound moves along the <05> Moebius strip
0 5 4 e 6 6 1 8
9 9 e 9 5 0 7 2
<05> <67> <05> <87> <07> <77> <77>
(7 moves) (5 moves) (2 moves) (2 moves)
161
Figure 3-6a: <05> tn-class circle
Figure 3-6b: <05> set-class line
[00] – [05] – [02] – [03] – [04] – [01] – [06]
[00]
[09] [03]
[06]
[07] [05]
[0e] [01]
[02] [0t]
[04] [08]
162
Figure 3-7a: <2e>, {00} toroidal voice-leading space
163
Figure 3-7b: <2e>, {00} toroidal voice-leading space
164
Table 3-1: dyadic vlset spaces
Vlset Sum Number of
Pcsets
Tn-class
cycles
Vlset Sum Number of
Pcsets
Tn-class
cycles
<01> 1 144 1 <0e> e 144 1
<2e> 1 48 3 <1t> e 48 3
<3t> 1 144 1 <29> e 144 1
<49> 1 144 1 <38> e 144 1
<58> 1 48 3 <47> e 48 3
<67> 1 144 1 <56> e 144 1
<05> 5 144 1 <07> 7 144 1
<14> 5 48 3 <8e> 7 48 3
<23> 5 144 1 <9t> 7 144 1
<6e> 5 144 1 <16> 7 144 1
<7t> 5 48 3 <25> 7 48 3
<89> 5 144 1 <34> 7 144 1
<02> 2 36 2 <0t> t 36 2
<11> 2 12 12 <ee> t 12 12
<3e> 2 36 4 <19> t 36 4
<4t> 2 12 6 <28> t 12 6
<59> 2 36 4 <37> t 36 4
<68> 2 36 2 <46> t 36 2
<77> 2 12 12 <55> t 12 12
<03> 3 16 3 <09> 9 16 3
<12> 3 48 1 <te> 9 48 1
<4e> 3 48 1 <18> 9 48 1
<5t> 3 48 1 <27> 9 48 1
<69> 3 16 3 <36> 9 16 3
<78> 3 48 1 <45> 9 48 1
<04> 4 9 4 <08> 8 9 4
<13> 4 36 2 <9e> 8 36 2
<22> 4 6 12 <tt> 8 6 12
<5e> 4 12 6 <17> 8 12 6
<6t> 4 18 4 <26> 8 18 4
<79> 4 36 2 <35> 8 36 2
<88> 4 3 12 <44> 8 3 12
<00> 0 1 12 <06> 6 4 6
<1e> 0 12 2 <15> 6 12 4
<2t> 0 6 4 <24> 6 12 2
<39> 0 4 6 <33> 6 4 12
<48> 0 3 4 <7e> 6 12 4
<57> 0 12 2 <8t> 6 12 2
<66> 0 2 12 <99> 6 4 12
165
Figure 3-8: <02>, {00} toroidal voice-leading space
Figure 3-9: <02>, {11} toroidal voice-leading space
166
Figure 3-10: <02>, {01} toroidal voice-leading space
Figure 3-11: <02>, {10} toroidal voice-leading space
167
Figure 3-12: <02>, {11} Moebius strip
168
Figure 3-13: vlset <02> spaces
a). transposition-class cycles
b). set-class lines
[00] – [02] – [04] – [06] [01] – [03] – [05]
Figure 3-14: <68>, {00} toroidal voice-leading space
[00]
[06]
[0t] [02]
[08] [04]
[01]
[07]
[0e] [03]
[09] [05]
169
Figure 3-15: <67>, {00} toroidal voice-leading space
170
Figure 3-16: <67> Moebius strip
171
Figure 3-17: <12>, {47} toroidal voice-leading space
Figure 3-18a: <42>, {11} toroidal voice-leading space
172
Figure 3-18b: <42>, {11} toroidal voice-leading space
11 77 17 71
9e 53
59 95
35 e9
e3 3e
173
Figure 3-19: <12>, {47} Moebius strip
174
Figure 3-20: <42>, {11} permutation space {e3} {9e} {77} {17} {59} {35} {11} Figure 3-21: <06>, {11} permutation space {11}—{17}—{77} Figure 3-22: <06>, {01} voice-leading space {61} {67} {01} {07} Figure 3-23: <57>, {02} permutation space {77}—{02}—{59}—{t4}—{3e}—{86}—{11} Figure 3-24: <26>, {11} toroidal voice-leading space
175
Figure 3-25: <001>, {000} hypercube
176
Figure 3-26: the trichordal orbifold
177
Table 3-2: trichordal voice-leading spaces
Vlset Inversional Pair Sum Number of
Pcsets
Tn-class
spaces
<001>, <445>, <889> <00e>, <443>, <887> 1/e 1728 1
<02e>, <346>, <78t> <01t>, <245>, <689> 1/e 1728 1
<03t>, <247>, <68e> <029>, <146>, <58t> 1/e 1728 1
<049>, <148>, <058> <038>, <047>, <48e> 1/e 1728 1
<067>, <4te>, <238> <056>, <49t>, <128> 1/e 1728 1
<11e>, <355>, <799> <1ee>, <335>, <779> 1/e 432 2
<12t>, <256>, <69t> <2te>, <236>, <67t> 1/e 1728 1
<139>, <157>, <59e> <39e>, <137>, <57e> 1/e 432 2
<166>, <5tt>, <229> <66e>, <3tt>, <227> 1/e 1728 1
<337>, <77e>, <3ee> <599>, <119>, <155> 1/e 108 5
<005>, <449>, <188> <007>, <44e>, <388> 5/7 1728 1
<014>, <458>, <089> <08e>, <034>, <478> 5/7 1728 1
<023>, <467>, <8te> <09t>, <124>, <568> 5/7 1728 1
<06e>, <34t>, <278> <016>, <45t>, <289> 5/7 1728 1
<07t>, <24e>, <368> <025>, <469>, <18t> 5/7 1728 1
<113>, <557>, <99e> <9ee>, <133>, <577> 5/7 432 2
<122>, <566>, <9tt> <tte>, <223>, <667> 5/7 1728 1
<15e>, <359>, <179> <17e>, <35e>, <379> 5/7 432 2
<16t>, <25t>, <269> <26e>, <36t>, <27t> 5/7 1728 1
<33e>, <377>, <7ee> <199>, <115>, <559> 5/7 108 5
<002>, <446>, <88t> <00t>, <244>, <688> 2/t 216 2
<011>, <455>, <899> <0ee>, <334>, <778> 2/t 864 1
<03e>, <347>, <78e> <019>, <145>, <589> 2/t 864 1
<04t>, <248>, <068> <028>, <046>, <48t> 2/t 216 2
<059>, <149>, <158> <037>, <47e>, <38e> 2/t 864 1
<077>, <4ee>, <338> <055>, <499>, <118> 2/t 864 1
<12e>, <356>, <79t> <1te>, <235>, <679> 2/t 864 1
<13t>, <257>, <69e> <29e>, <136>, <57t> 2/t 864 1
<167>, <5te>, <239> <56e>, <39t>, <127> 2/t 864 1
<22t>, <266>, <6tt> <2tt>, <226>, <66t> 2/t 54 5
<004>, <448>, <088> <008>, <044>, <488> 4/8 27 5
<013>, <457>, <89e> <09e>, <134>, <578> 4/8 432 1
<022>, <466>, <8tt> <0tt>, <224>, <668> 4/8 108 2
<05e>, <349>, <178> <017>, <45e>, <389> 4/8 432 1
<06t>, <24t>, <268> <026>, <46t>, <28t> 4/8 108 2
<079>, <14e>, <358> <035>, <479>, <18e> 4/8 432 1
<112>, <556>, <99t> <tee>, <233>, <677> 4/8 432 1
<15t>, <259>, <169> <27e>, <36e>, <37t> 4/8 432 1
<23e>, <367>, <7te> <19t>, <125>, <569> 4/8 432 1
<277>, <6ee>, <33t> <55t>, <299>, <116> 4/8 432 1
<000>, <444>, <888> 0 1, 3
<01e>, <345>, <789> 0 144 3
178
<02t>, <246>, <68t> 0 36 6
<039>, <147>, <58e> 0 16, 48 4
<048> 0 9
<057>, <49e>, <138> 0 144 3
<066>, <4tt>, <228> 0 8, 12 10
<11t>, <255>, <699> <2ee>, <336>, <77t> 0 16, 48 4
<129>, <156>, <59t> <237>, <67e>, <3te> 0 144 3
<003>, <447>, <88e> <009>, <144>, <588> 3/9 64, 192 4
<012>, <456>, <89t> <0te>, <234>, <678> 3/9 576 3
<04e>, <348>, <078> <018>, <045>, <489> 3/9 576 3
<05t>, <249>, <168> <027>, <46e>, <38t> 3/9 576 3
<069>, <14t>, <258> <036>, <47t>, <28e> 3/9 64, 192 4
<111>, <555>, <999> <eee>, <333>, <777> 3/9 4, 12
<13e>, <357>, <79e> <19e>, <135>, <579> 3/9 144 6
<159> <37e> 3/9 36
<177>, <5ee>, <339> <55e>, <399>, <117> 3/9 16, 48 10
<22e>, <366>, <7tt> <1tt>, <225>, <669> 3/9 64, 192 4
<23t>, <267>, <6te> <29t>, <126>, <56t> 3/9 576 3
<006>, <44t>, <288> 6 8, 24 10
<015>, <459>, <189> <07e>, <34e>, <378> 6 288 3
<024>, <468>, <08t> 6 72 6
<033>, <477>, <8ee> <099>, <114>, <558> 6 32, 96 4
<123>, <567>, <9te> 6 288 3
<16e>, <35t>, <279> 6 288 3
<17t>, <25e>, <369> 6 32, 96 4
<222>, <666>, <ttt> 6 2, 6
<26t> 6 18
179
Figure 3-27a: vlclass [047] ordered tn-class space
Figure 3-27b: Motion among adjacent ordered tn-classes in [047] ordered tn-class
space
[000] [047] [012] [056] [043] [044] [007] [042]
7 4 1 1 8 4 5 e
7 1 0 0 9 0 0 9
7 9 e 7 5 4 5 7
<269> <730> <7e2> <158> <3e8> <158> <269>
180
Figure 3-28: vlclass [047] ordered set-class space
181
Figure 3-29: examples of complete ordered set-class spaces
a) vlclass [056]
b) vlclass [025]
182
c) vlclass [014]
d) vlclass [013]
183
e) vlclass [001]
184
Figure 3-30: vlclass [002] ordered set-class spaces
a)
[000]
[002] [002]
[004] [024] [004]
[006] [026] [046] [006]
[048]
b)
[023] [025] [027]
[045] [047] [037]
[016] [036] [056]
([045]) [014] [034] [015] ([037])
([023]) [012] [013] [035] ([027])
[001] [003] [005]
([023] [025] [027])
[001]
[003]
[005]
[012] [014]
[027] [037]
[056]
[015]
[034]
185
Figure 4-1a: <442>, {000} voice-leading space
186
Figure 4-1b: <442>, {001} voice-leading space
187
Figure 4-1c: vlclass [002] whole-tone ordered set-class space
[000]
[002] ([002])
[004] [024] ([004])
[006] [026] [046] ([006])
[048]
188
Figure 4-1d: vlclass [002] mixed whole-tone ordered set-class space
[023] [025] [027]
[045] [047] [037]
[016] [036] [056]
([045]) [014] [034] [015] ([037])
([023]) [012] [013] [035] ([027])
[001] [003] [005]
([023] [025] [027])
[001]
[003]
[005]
[012] [014]
[027] [037]
[056]
[015]
[034]
189
Figure 4-1e: ordered set-class interpretations from Figure 2-8
a <7 e 8> <t t 8> b <5 9 4> a <7 8 e> <t t 0> c <5 6 e>
[041] – [034] [015] [014] [016]
c <5 6 e> <4 4 2> d <9 t 1> b <5 9 4> <4 4 6> d <9 1 t>
[016] [014] [015] [041] – [034]
g <4 6 8> <1 1 3> h <5 7 e> e <0 2 6> <4 4 2> g <4 6 8>
[024] [026] [026] [024]
i <e 0 3> <9 9 e> j <8 9 2> f <8 9 t> <3 3 5> i <e 0 3>
[014] [016] [012] [014]
190
Figures 4-2a and b: [026] in Webern’s Op. 5 n. 2 and a network of transformations linking them
191
Figure 4-3: melodic statements of vlclass [043] in Webern’s Op. 5 n. 2
Vla
Vln II
Vln II
Vln I
Vln II
Vln I Vln II
191
192
Figure 4-4: Network of melodic vlsets
I5 I6 I9
<910> <845> <t21> <e78>
Te
CI0 I6 CT6 I1
<t65> <265>i <401>i <t67> <376>
T1
I7 T1
<487> <3e0> <401>
193
Figure 4-5: vlclass [043] in the accompaniment to Webern’s Op. 5 n. 2
194
Figure 4-5 continued
195
Figure 4-6: vlclass [043] transformation graph
196
Figure 4-7a: Webern Op. 5 n. 3, mm. 1-4
Figure 4-7b: vlset <t16> mappings
5 0 0 7
2 t 9 5
4 6 e 1
<t16> <t16>
T7
{90e} {715}
A
{254} {06t}
197
Figure 4-8: variations of A
8 3 e 4
5 1 8 0
7 9 t 6
<t16> <t16>
e 1 2
t t 8
8 0 0
<470>
3 t 4 e
0 8 1 9
2 4 3 5
<t16> <t16>
8 9 3 1 5 7
5 3 2 e 9 6
7 7 0 7 3 4
<470> <149> <74e>i
T7 T7
198
Figure 4-9: B, mm. 9-10
B
2 t
1 5
t 9
<8e4>
0 6
e 4
3 3
<470>
{034} {234}
199
Figure 4-10a: [038] voice-leading space, equivalent to [047] under rotation
200
Figure 4-10b: melody B’s complete voice-leading {2t1} - {0et} - {t59}
201
Figure 4-11a: reduction of opening [034]s in Violins and Viola, mm. 1-3
Figure 4-11b: reduction of Violin II, Viola, and Cello, mm. 15-17
{014} {901}
[014] [034]
e 9
9 8
8 4
<850>i
{7te}
[034]
202
Figure 4-12: Berg Op. 5 n. 1, opening clarinet gesture, in C
{789034} {269}
[012569] [037]
203
Figure 4-13a: Berg’s Op. 5 n. 1, Clarinet melody in C
203
204
Figure 4-13b: vlclass [002] ordered set-class spaces
[000]
[002] [002]
[004] [024] [004]
[006] [026] [046] [006]
[048]
[023] [025] [027]
[045] [047] [037]
[016] [036] [056]
([045]) [014] [034] [015] ([037])
([023]) [012] [013] [035] ([027])
[001] [003] [005]
([023] [025] [027])
205
Figure 4-14: vlclass [022] in m. 2 LH
{837} {67e} e 2
I2 7 t
{926} {7t2} 6 7
I4 <133>
<1ee> I2
206
Figure 4-15: vlclass [022] in m. 2 RH
Figure 4-16: opening clarinet gesture in the accompaniment, m. 9
{e34} {590}
[012569]
Clar.
{047} {837} {4e3}
[037] {904} {590}
{926} {740}
<t88>
t 4
9 3
8 3
<577>
207
Figure 4-17: accompaniment, mm. 3-5
0 2 3 2 3
t 1 0 8 9
1 9 8 3 2
<e11> <1ee> <200> <e11>
208
Figure 4-18a: vlclass [002] in mm. 5-6
Figure 4-18b: vlclass [002] in m. 7
t 0 t 3 4
9 7 3 7 6
5 4 9 8 7
<977> <533> <0tt> <1ee>
e 6 0 6 9
4 2 t 5 8
0 e 7 4 8
<022> <688> <244>
{348} <577> {1te}
209
Figure 4-19: Berg’s Op. 5 n. 2
209
210
Figure 4-20: Berg’s Op. 5 n. 2 melody
211
Figure 5-1: pentachordal vlsets in Berg’s Op. 5 n. 2
0 1 0 0 1 0
7 8 7 7 8 7
3 0 3 3 0 3
6 6 6 9 6 9
2 2 2 5 2 5
<0ee01> <0110e> <33435> <98798> <34534>
<03505>i <03505>i <37t37>i <36938>i<36938>i
212
Figure 5-1 continued
0 3 t 1 7 t 4 7 t 4 7 4 6
7 9 4 8 3 5 e 3 5 e 3 e 2
3 1 0 2 0 t 8 0 t 8 0 8 t
9 6 7 8 9 6 5 9 6 5 6 5 t
5 8 9 t e 0 1 e 0 1 0 1 t
<etee9> <41132> <t9et9> <34312> <eet11> <45556> <65235>
<34323> <e11e1> <eet11> <t9et9> <87776>
<48159>i <037e2>i <039t3>i <e27e8>i <9e691>i <45e58>i <23617>i
<5t369>i <35915>i <9e691>i <039t3>i <45e58>i
213
8 4 9 t 2 t 5
7 0 6 9 1 9 4
3 9 2 8 1 7 3
<658> <565> <631> <544> <688> <877>
Figure 5-2 vlsets among normal orders in Berg’s Op. 5 n. 1 melody
214
Figure 5-3a vlsets on the trichordal orbifold
215
Figure 5-3b: vlsets on the trichordal orbifold
216
Figure 5-3c: vlsets on the [013] ordered set-class space
217
Figure 5-4 hyper-vlset graph
<<e0t>> <<198>> <<t23>> <<10t>>i <<134>>i
<658> <565> <631> <544> <688> <877>
(<<301>>+<<t97>>) (<<235>>+<<8et>>)
218
Figure 5-5: Alban Berg’s Lyric Suite, VI, mm. 1-6
Viola: Row 2PF 5 4 1 2 0 e 6 t 9 8 7 3
Cello: Row 1PF 5 6 t 4 1 9 2 8 7 3 0 e
T-row <0 t 3 t e 2 4 2 2 5 7 4>
Vln I: Row 2IF 5 6 9 8 t e 4 0 1 2 3 7 Vln II: Row 1IF 5 4 0 6 9 1 8 2 3 7 t e
Cello: Row 1PF 5 6 t 4 1 9 2 8 7 3 0 e Vla: Row 2PF 5 4 1 2 0 e 6 t 9 8 7 3
I-row <t 0 7 0 e 8 6 8 8 5 3 6>i I-row <t 8 1 8 9 0 2 0 0 3 5 2>i
219
Figure 5-6: Lyric Suite, mm. 39-46
Vln II: Row 1PC# 1 2 6 0 9 5 t 4 3 e 8 7
Viola: Row 2IEb 3 4 7 6 8 9 2 t e 0 1 5
I-row <4 6 1 6 5 2 0 2 2 e 9 0>i
Vln I: Row 1IA 9 8 4 t 1 5 0 6 7 e 2 3
Viola: Row 2IEb 3 4 7 6 8 9 2 t e 0 1 5
T-row <6 4 9 4 5 8 t 8 8 e 1 t>
220
Figure 5-7: 4-3 suspension as diatonic vlset
4 4
3 2
0 0
0 0
Diatonic vlset <0060>
Figure 5-8: transposition of 4-3 diatonic vlset
4 5
3 3
0 1
0 1
T1(<0060>) = <1101>
221
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