morris - voice-leading spaces

Society for Music Theory

Voice-Leading SpacesAuthor(s): Robert D. MorrisSource: Music Theory Spectrum, Vol. 20, No. 2 (Autumn, 1998), pp. 175-208Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/746047Accessed: 22/01/2010 05:20

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Voice-Leading Spaces

Robert D. Morris

Voice-leading in tonal music-the progression of pitches to form horizontal "voices" or "lines"-is normative. Even in single-voice writing there are "rules" for the way a melody should progress. In the composition of a cantus firmus in modal counterpoint, for example, a leap is limited to certain intervals and must be followed either by a step in the opposite direction or by another leap, provided the two successive leaps outline one of a few permissible three-note sonorities. In multi-voice contexts, the leading of a voice is determined even further. As I compose, for instance, I ask: Will the next note I write down form a consonance with the other voices? If not, is the dissonance correctly prepared and resolved? What scale degrees and harmonies are involved? (And the answers to such questions will of course depend on whether the note is in the bass, soprano, or an inner voice.) But these voice-leading rules are not arbitrary, for their own sake; they enable the listener to parse the ongoing musical fabric into meaningful units. They help me to determine "by ear" whether the next note is in the same voice, or jumps to another in an arpeggiation, or is ornamental or not, and so forth.

Many composers and analysts have sought some extension or generalization of tonal voice-leading for non-tonal music. Analysts such as Felix Salzer, Roy Travis, and Edward Laufer have attempted to apply linear concepts such as Schenkerian

prolongation to music that appears to have little to do with tonality or even pitch concentricity.l Joseph N. Straus and others have however called such work into question.2 Other theorists have obviated voice-leading as a criterion for dis- tinguishing linear aspects of pitch structure. For example, in my own theory of compositional design, ensembles of (un- interpreted) pc segments, often called lynes, are realized in pitch, time, and other musical dimensions, using some means of musical articulation to maintain an association between the components of a given lyne.3 For instance, a lyne might be associated with a register, an instrument, a dynamic level, a mode of articulation, or any combination of these, thereby separating it out from other simultaneously unfolding lynes. Another approach to linear pitch continuity is found in the various transformational models of musical structure devised

'See for instance, Felix Salzer, Structural Hearing (New York: Dover, 1982); Roy Travis, "Directed Motion in Schoenberg and Webern," Perspec- tives of New Music 4/2 (1966): 85-89; and Edward Laufer, "Schoenberg's Klavierstiick Opus 33a: a Linear Approach," paper delivered at the joint meeting of the Music Theory Society of New York State and the Arnold Schoenberg Institute, Barnard College, Columbia University, 4 October 1991.

2See Joseph N. Straus "The Problem of Prolongation in Post-Tonal Mu- sic," Journal of Music Theory 31/1 (1987): 1-21.

3Robert Morris, Composition with Pitch-Classes: a Theory of Composi- tional Design (New Haven: Yale University Press, 1987).

176 Music Theory Spectrum

by David Lewin, Henry Klumpenhouwer, and others.4 A simple but effective approach to voice-leading as transformation is to examine the relations of the intervals between the pcs in chords (from low to high) and their influence on the intervals between different chords. This generalization of the relation of figured bass to voice-leading has allowed Alan Chapman and John Roeder to discover many interesting voice-leading constraints, although without providing any normative definition of how voices should move.5 More general and abstract transformational theories consider paths of transformation between and among pcs. The network of such paths has suggested an analogy to combinations of lines in common-practice music.6 Transformational theories have also been applied to chord grammars in tonal music in which the transformations can be harmonic functions such as tonic and dominant.7 While much of this work tends to ignore

4See David Lewin, "Klumpenhouwer Networks and Some Isographies that Involve Them," Music Theory Spectrum 12/1 (1990): 83-120; and Henry Klumpenhouwer, "A Generalized Model of Voice-Leading for Atonal Music" (Ph.D. diss., Harvard University, 1991).

5See Alan Chapman, "Some Intervallic Aspects of Pitch-Class Set Re- lations," Journal of Music Theory 25/2 (1981): 275-90; John Roeder, "Har- monic Implications of Schoenberg's Observations of Atonal Voice-Leading," Journal of Music Theory 33/1 (1989): 27-62; and idem, "Voice-Leading as Transformation," in Musical Transformation and Musical Intuition: Essays in Honor of David Lewin, ed. Raphael Atlas and Michael Cherlin (Roxbury, Mass.: Ovenbird Press, 1994), 41-58.

6See Joseph N. Straus, "Voice-Leading in Atonal Music," in Music Theory in Concept and Practice, ed. James Baker, David Beach, and Jonathan W. Bernard (Rochester: University of Rochester Press, 1997), 237-74.

7A number of theorists have applied (mathematical) groups of transformations to gain a better understanding of key relations in late nineteenth- century music. See David Lewin, "A Formal Theory of Generalized Tonal Functions," Journal of Music Theory 26/1 (1982): 23-60; idem, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987); Brian Hyer, "Reimag(in)ing Riemann," Journal of Music Theory 39/1 (1995): 101-38; David Kopp, "A Comprehensive Theory of Chromatic Me- diant Relations in Mid-Nineteenth Century Music," (Ph.D. diss, Brandeis

issues of voice-leading, tonal or otherwise, Richard Cohn has placed what he calls smooth or parsimonious voice-leading at the center of his approach.8 Some new work by David Lewin expands on Cohn's and touches on many other voice-leading topics in twentieth-century music.9 A good summary of some of these approaches is found in a recent article by Straus.'1

In any case, in much of the literature cited above the analysis of voice-leading is simply a matter of non-prescrip- tively describing certain and perhaps typical motions or transformations of pitches or pitch classes, one to another. Of course, many types of post-tonal music simply do not treat pitches or sounds as anything like voices; consequently, there may be no need to model them with some concept of voice- leading. On the other hand, the constraints and invariances of serial music often involve forms of voice-leading or sub- stitute other processes to achieve similar ends. In this music, the order of pcs and pc intervals in a series helps furnish a norm for progression. For instance, if one uses a row that does not contain a particular interval class adjacently, if that interval class occurs in the music one may usually assume that the two notes forming it are from different row forms. Or, if the first interval of a row is unique, then the appearance of that interval class in the music often indicates that a row form is just beginning or ending."

University, 1995): Richard Cohn, "Maximally Smooth Cycles, Hexatonic Sys- tems, and the Analysis of Late-Romantic Triadic Progressions," Music Anal- ysis 15/1 (1996): 9-40; and idem, "Neo-Riemannian Operations, Parsimo- nious Trichords, and their Tonnetz Representations," Journal of Music Theor) 41/1 (1997): 1-66.

"Cohn, "Neo-Riemannian Operations." 9David Lewin, "Cohn Functions," Journal of Music Theor) 40/2 (1996):

181-216, and idem, "Some Ideas on Voice-Leading Between Pcsets" (un- published paper, 1997).

"'Straus, "Voice-Leading in Atonal Music." 'For example, consider the row of Webern's Symphony. Its INT, its series

of adjacent ordered pc intervals, is <3BB4B618119>. It has no member of

Voice-Leading Spaces 177

But in non-serial, non-tonal contexts, neither serial ordering nor tonal voice-leading helps provide criteria for how one note or interval leads to another. Hence the non- prescriptive and diverse approach to voice-leading in post- tonal music. In fact, the situation is so eclectic that there seems to be little agreement about what a "voice" is or how it "leads," or about what kinds of long-range structures or listening strategies follow from the various definitions.

In this article I provide some general definitions and contexts for the discussion of voice-leading. First I set limitations on what I call the "total voice-leading" between two pcsets, and then I spell out an exhaustive taxonomy of two-voice contrapuntal motions. The second part of the paper begins with an examination and generalization of Cohn's explicit coupling of transformation and proximate voice-leading of major and minor triads as represented on Hugo Riemann's Tonnetz. The tonnetz is a traditional example of what I have previously called a compositional space;l2 such spaces are out-of-time networks of pcs that can underlie compositional or improvisational action. After studying various transformations of the tonnetz, voice-leading is implemented by the use of another type of compositional space, two-partition graphs. The result forms the third part of the paper: a collection of voice-leading spaces, a new category of compositional space. In the last part, I describe methods that allow one to construct voice-leading spaces of any degree of complexity or closure..

ON THE VOICE-LEADING BETWEEN TWO PCSETS

Perhaps the most fundamental distinction among either simultaneous or successive pitch intervals is that of step ver-

ic 5, the only members of ic 3 at its beginning and end, and a unique ic 6 at its middle.

12See Robert Morris, "Compositional Spaces and Other Territories," Per- spectives of New Music 33/1-2 (1995): 328-58.

sus leap. Steps are the basis for continuity within melodies or contrapuntal lines, whereas leaps often partition the me- lodic flow into simultaneously evolving strata, with the non- temporally adjacent notes in each strata connected by steps. Lines with multiple strata are called compound melodies; Example la provides an illustration. Example lb shows an analogous instance from post-tonal music. While the instrumental parts move variously from chord to chord, the chords are connected by semitones. Thus we may hear the succession of the chords form three parallel lines in spite of the instrumental lines, which are, of course, also audible. In post-tonal music the distinction between step and leap is a relative one, with smaller intervals leading linear continuity and larger intervals segregating different registral strata.l3 Nevertheless, the smallest intervals possible will tend to function as steps in almost all contexts: in twelve-tone temperament the semitone usually functions as a step, and in tonal music, one or two semitones define a step.14 Therefore, I use the term

13Notes further apart than a distance D will be considered connected via leap, those less distant than D will be considered connected by step. D can be a combination of pitch and time such that D = pitch-interval plus time- interval (city-block metric) or D = V\/(pitch-interval)2 + (time-interval)2 (Pythagorean metric), or others. See James Tenney and Larry Polansky, "Temporal Gestalt Perception in Music," Journal of Music Theory 24/2 (1980): 205-41. While notes can be segregated into pitch and pc sets according to this criterion-notes connected by step are within a set, notes connected by leap are in different sets-such a criterion is too rigid in general since D will change in different musical contexts, and other criteria for the relatedness of notes will interact with it.

14While the augmented second may be defined as a step in the harmonic minor scale, it almost always functions as a leap in other tonal contexts. Outside of Western music, steps and leaps may be variously defined; for instance, the South Indian Scale "Jhalavarali" (C, Dl, El', F#, G, Al, B, C as notated in and tempered to Western pitch classes), contains a step of (approximately) four semitones between its third and fourth scale degrees, while the "leap" from the first degree to the third is of (about) two semitones. In diatonic scale theory, scales that have leaps that are smaller than steps are said to have contradictions. See Jay Rahn, "Coordination of Interval Sizes in Seven-Tone Collections," Journal of Music Theory 35/1-2 (1992): 33-60.


Example la. A compound melody

p.iA .. . J I J

i-- _ - I 0f t Ltr r , f rI F Fj*l ' l IIX e 1

'I I I

Example lb. Instrumental versus registral voice-leading in Schoenberg, Pierrot lunaire, "Eine blasse Waischerin," m. 4

instrumental voice-leading

flute

clar.

violin

registral voice-leading

fcn U U t ~ J

I # i- -I-t

_~~~~

v I - ' #w

proximate voice-leading to denote a pitch or pitch-class motion by ics 0, 1, or 2.

But what exactly is voice-leading between pcsets in post- tonal music? How does one pcset progress to the next? I address these questions by allowing any possible connection, but with a definition. Given two pcsets A and B, the total voice-leading from A to B includes any and all moves from any pcs of A to any pcs of B--that is, all the ways one can associate the pcs of A with those of B in as many voices as necessary or desired. Each voice will be a path from a pc or subset of A to B. I shall not presently consider whether these voices are heard as paths or connections between the pcs of A and B, but I will demand that they be suggested or denoted by musical notation. Example 2 suggests what total voice-

leading entails. Example 2a presents one possible voice- leading where pcset A = {2AB} and B = {017}. It has three voices, defined by staff and step direction. By traditional standards of voice-leading, this example is pathological for any of the following reasons. (1) The top voice is a wedge so that two pcs occur in the voice at once. This voice is not be considered to imply the two-voice texture in the upper clef of voice-leading (b). (2) The B~ in voice 2 (of voice-leading (a)) has no goal note. (3) Pc B of pcset A is doubled. Whether it is heard at all will depend on what instruments play this example. (4) The pc 1 in pcset B is not found in the example. While (b) repairs (a) so that it conforms to more usual standards of voice-leading, we may want to take (a) as it is, for any number of analytic or compositional reasons.

However, it is often useful to place limitations on the total voice-leading of two pcsets. For present purposes, I now define three increasingly restrictive limits, designated Ri, R2, and R3, on total voice-leading plus another optional restriction.

RI: Each voice is a series of single pcs or rests; voices cannot contain wedges or forks.15 The single voice of Ex- ample 2c, if it is to comply with R1, has to be written or conceived as the three-voice texture in 2d. There are no other restraints under R1: pcsets need not contribute all of their pcs to a voice-leading; pcs may be doubled (i.e., used in more than one voice). The voice-leadings in Examples 2e, 2f, and 2g satisfy R1 and are members of the total voice-leading of A = {128} and B = {0378}. In 2e, pc 3 of B is omitted. 2f doubles pc 2 of A, but leaves out a pc from both pcset A and B. In both 2e and 2f the intersecting pc 8 of pcsets A and B is realized as a common pitch. In the voice-leading of Ex- ample 2g, the same pc is omitted from the second verticality

'5R1 places limits on how one interprets a lyne in a serial array. For instance, playing two pcs of a row at once now implies two voices.


Example 2. Restrictions on total voice-leading

A = {2AB}; B = {017}

(a) (b) 1- I I 1-1.

A={47};B= {5};C= {238}

(d) Rl 21. 2.

(c) 1. (no restrictions)

-Y I I I- I ,JI I

7 I r I - 4.

,- 3. r _r

The following voice-leadings comply with R1: A ={128}; B = {0378}

(e) 1 (f)1 I 1 (g). A 1. I 1f). I . . I l

3r- i"

2. r 2. t

2. r IL I II.

' a l t w _ 1_ ,T- 'f

3.] I 3.'

The following voice-leadings (plus (e) and (g) above) comply with R2:

A = {128}; A = {128}; A= {047}; B = {0378} B = {0378} B = {257B}

A = {03B}; B = {478}

Examples (e), (h), (i2), (j), and (k) comply with R3.

3.

1, 2 . 1. 2. 1 $ 23. i[

A ~~~~~~I


so that pcset B is represented by the pcs that differentiate it from pcset A.

R2: Each pc of A is connected via a voice to one and only one pc of B; or each pc of B is connected via a voice from one and only pc of A. Thus either A or B, but not both, may omit pcs in the voice-leading. Note that, of the illustrations just cited in reference to R1, Examples 2e and 2g also comply with R2. (Only 2f is purely R1 since it omits pcs from both A and B.) Other examples of voice-leadings that satisfy R2 are also given in Example 2. Example 2h has each pc of A connected to one and only pc of B. Since B is larger than A, one pc of B is omitted, namely 0. Examples 2il and 2i2 show two ways to handle the missing pc 0: it is realized in a fourth voice that either rests during A or doubles a note in A. The voice-leading in Example 2j shows how the voice-leading of sonorities with different cardinalities is managed in tonal music; the one with fewer pcs employs doubling. The last voice- leading (Example 2k) is one-to-one and onto, but does not use anything near proximate voice-leading.16

R3: A voice-leading that satisfies R2 is further restricted to voices that do not contain rests. Thus the result is in first- species (homorhythmic) counterpoint. If A and B are of the same cardinality, then the voice-leading is one-to-one and onto; only in this situation are all pcs of pcsets A and B obliged to occur in the voice-leading. If A and B are of different cardinalities, then the pcset with the smaller cardinality is forced to employ doubling to satify R3.

Finally, when all the pcs of A and B occur in the voice- leading we say it is definitive.

16A one-to-one and onto mapping between two sets X and Y entails that each member xn of X is mapped to a unique member Ym of Y and that #X = #Y (see note 46).

A TAXONOMY OF VOICE-LEADING TYPES

Certain distinctions in tonal voice-leading can be applied to post-tonal music. Here I have in mind the categories of voice motions-such as parallel or contrary-and the disposition of contrapuntal voices-such as voice-crossing, the treatment of unisons, and the like. While these concepts are clear enough in their implementation in tonal voice-leading, outside of any particular stylistic prescription they intersect in various and subtle ways. To illustrate their interdepen- dence, I now formally define voice motions, dispositions, and intersections.

Consider two voices X and Y of two notes each. The successive notes of X are xl and x2, and those of Y are yl and y2. xl and yl occur together, as do x2 and y2. Only one further stipulation is given: xl is either identical to or lower than yl.

Four categories of voice motion are defined. In parallel motion, the directed interval from xl to x2 is not zero and is the same as that from yl to y2 (both voices ascend or descend by the same amount). In similar motion, the direction (up or down) of the directed interval from xl to x2 is the same as that of yl to y2, but the intervals are not identical (both voices ascend or descend but not by the same amount). In contrary motion, the direction of the directed interval from xl to x2 is the opposite of that of yl to y2 (one voice ascends while the other descends). And in oblique motion, the interval from xl to x2 is zero while the interval from yl to y2 is not zero, or vice versa (one voice repeats or holds a note, while the other moves either up or down). The four voice motions are mutually independent.

I now define four other relations between the voices X and Y, denoted collectively by the term contrapuntal conditions. Depending on the disposition of their notes, the counterpoint of X and Y may or may not be subject to a particular contrapuntal condition. Unlike the voice motions, the conditions


are not mutually exclusive; but, as we will see, neither are they completely dependent. The four conditions are named by small case letters: c, i, s, and u. The first condition, c (for voice-crossing17), applies if x2 is greater than yl, or y2 is less than xl (the second note of X is higher than the first of Y, or the second note of Y is lower than the first of X; remember that in all cases the first note of X is lower than or equal to the first note of Y). The second condition is called i (for invert18). It is met when xl< yl and x2 >y2 (the voices X and Y contrapuntally invert).

While these first two conditions concern the pitch dispositions of notes between voices X and Y, the other two conditions concern pitch intersections between the two voices. Condition three is s (for shared notes) and indicates that either xl = y2 or yl = x2, but not both. (The first note of X is the same as the last note of Y, or the first note of Y is the same as the last note of X, but not both.) Finally, condition u (for unison) specifies that either xl = yl or x2 = y2, or both. (The first notes of X and Y are the same, and/or the last notes of X and Y are the same.)

As mentioned above, the four conditions are not independent. For instance, u and i are incompatible, since if xl = yl, then xl is not less than yl as specified for condition i; or if x2 = y2, then y2 is not greater than x2 as specified for condition i; and if neither xl = yl nor x2 = y2, then condition u does not apply. In fact, only one more than half of the sixteen sets of all combinations of contrapuntal conditions are logically consistent. These are the nine condition sets. We write the conditions sets in braces: {c} means c alone, {s} s alone, {i} i alone, {u} u alone, {ic} i and c together, {cu} c and u together, {su} s and u together, {is} i and s together, and { } (no conditions, the null set).

17This contrapuntal condition is also known as "overlap." '8The term "voice-crossing" is sometimes used for this condition.

When we examine the interaction of the four voice motions and the four contrapuntal conditions, we see that not all combinations of one of the four motions and one of the nine condition sets are possible. Example 3 provides a chart of all consistent combinations of motions and conditions. Voice motions are arranged in columns in the chart, while condition sets appear in rows. A position of the chart is filled if the combination of a motion and a condition set is consistent. The notes that fill the consistent combinations are coded on the staff to show the notes of voice X as whole notes and the notes of Y as quarter-note heads. Most of the filled positions of the chart show more than one counterpoint of voices X and Y, illustrating distinct types of voice-leading for each combination of voice motion and condition set. These voice-leading motion types are labeled with capital letters, usually followed by super- and/or subscripts. Upper-case P, S, and C stand for parallel, similar, and contrary motions. O stands for "over" and U for "under" in cases of oblique motion. D stands for doubling and E for voice exchange. Subscripts indicate the pertinent condition set. Superscripts further describe the character of a voice-leading type's voice motion: +, -, =,<, and > respectively symbolize ascend-

ing, descending, remaining constant, expanding, and contracting.

For instance, type P+ indicates a parallel ascending voice- leading without any contrapuntal conditions applied. 0- indicates an oblique motion, where the top voice Y descends over the static lower voice X; no conditions apply to this type. Type S i denotes an ascending similar motion subject to the conditions i and c; thus the voices invert and the second note of voice X is higher than the first note of Y-in fact, x2 is the highest note of the counterpoint. Cc occurs when voices move in contrary motion such that the interval between between the second notes of the voices is greater than the interval between the first notes and conditions i and c apply. Finally, Cic- indicates a contraction of the two successive


Example 3. Chart of voice-leading motion types

Parallel Similar

()

(c)

{s)

(i)

(u)

(ic)

{cu)

{su}

(is}

Contrary Oblique

o ? a |, ' U o .. 8 < " ? ? .

? P+ P S+ S~ C> C< 0 0+ U U+

PC PC SC SC

,~ A . -. I I P+ PS s S

I- -.I 0,- ? I

I - -oo+

(doubling) icic ic 1 C ic

1~- +

St D_c - D" .!.. S^-C -.(x e U S

(doublingUI ic ICralles by IC. IC i'

??!? {~~~ I I .l~oe I" 0o

(exchange) OUS OSU U US Uu

1 -- o |?, -? 0

>s Cs E Is is


vertical intervals while both the lowest notes of the coun-

terpoint (xl and y2) and the highest notes of the counterpoint (yl and x2) descend. The reader may decipher the other labels in a similar manner.

Example 3 illustrates a number of interesting facts about voice-leading at the present level of generalization. (1) All four voice motions are possible only when none of the contrapuntal conditions apply. As depicted on the first row of the chart, there are always two voice-leading types of parallel, similar, and contrary motion (either ascending/descending or

expanding/contracting) and four types of oblique motions. (2) Condition sets {c}, {s}, or {cu} limit voice motion to parallel or similar. (3) Condition i is only possible as one species of contrary motion. C> represents the only application of i, inversion, that does not also satisfy condition c, voice crossing. (4) Oblique motion is only possible if condition sets {ic} or {u} or {su} apply, or if no conditions apply. (5) It is interesting that no set of two-element condition sets partition the four motions. Yet some condition sets permit mutually exclusive motions; for instance, {cu} is associated with only parallel and similar motion, while {is} is associated with only contrary motion. (6) The familiar contrapuntal phenomenon of "parallels by contrary motion" is exemplified by both Cic and Cc+ . These types mirror the less familiar "parallels by similar motion" shown on the same row of the chart.

We may group the voice-leading types into equivalence classes under the serial operations of inversion and retrograde, where inversion (I) exchanges the two voices, xl <

yl and x2 + y2; retrograde (R) exchanges the first and last note in each voice, xl +- x2 and yl +- y2; and retrograde inversion (RI) exchanges the first and last note of opposite voices, xl + y2 and yl - x2. The equivalence classes have one, two, or four members, depending on the invariances of the types. Types D are E invariant under all transformations: identity, I, R, and RI. RI invariance is exhibited by types employing parallel motion and null or singleton conditions

sets, such as P+. Types P+ and P- form a two-element class since the latter is the R or I of the former. Remarkably, in all but one instance the types in a class have the same voice motion and condition set. The exception involves the inversionally-invariant C', which is equivalent to C< under R or RI.

Our voice-leading types are useful in varying degrees of precision in analysis or composition. One can simply peel away the superscripts and subscripts to reveal voice motion alone. Ignoring only the superscripts reveals the class of a voice-leading type. The whole label indicates the exact voice- leading. Example 4 analyzes a first-species three-part counterpoint, where each vertical sonority is a member of SC 3-5 [016]. The analysis takes the voices in pairs, producing three streams of labels. In keeping with tradition, one could hi- erarchize the streams. The relation of the two highest voices might be given less analytical weight than the relation of the middle voice to the lowest, with greatest attention paid to the voice-leading between the lowest and highest voices.

While the chart of voice-leading types represents a high level of generality, it is not the highest. We can take one more step by removing the distinction between parallel and similar motions. We thereby recognize only the relative direction of voice movement and ignore the exact size of intervals between the notes either within or between voices. Of course, we still distinguish a unison from a non-unison interval.

Under such conditions, the distinction between voice- leading types differentiated only by similar and parallel motion disappears, except in the types that apply the {cu} condition set; here the combination of unison with dyads differentiates the Suc type19 from D + and D-, which represent voice identity. In any case, by making no distinction between exact sizes of non-zero intervals we leave pitch-space and

19Note that the order of the u and c in the subscripts differentiates S+c which has its unison first, from S+, which has the unison last.


Example 4. A sample voice-leading motion analysis

1.

2.

3.

1. and 2

1. and 3

2. and 3

-oI - J a ^is I; I

S+ O- O0

S+ C> C> i

E

C<- IC

P+ U+ Ci- S- iC l

enter contour-space. Thus the chart in Example 3 now provides a repertoire of motions for the counterpoint of two or more simultaneously presented contours. One could go on to investigate contour combinatoriality, partial ordering relations, and the like. Moreover, since the application of contour theory is limited neither to pitches of any kind nor even to events in time, the voice-leading motions can provide a framework for many other structures and experiences far removed from any sonic similarity to counterpoint or voice- leading per se. As inviting as a pursuit of generalized voice- leading might be, however, we now return to the more mun- dane land of pitch and pitch class.

THE TONNETZ AS A VOICE-LEADING SPACE

The inquiry into modeling twentieth-century voice-leading as actions within or on a compositional space is much more

modest than the previous generalization of voice-leading to within contour. Our only constraints on voice-leading between pitch and/or pitch-class verticalities will be the specification of limits on the intervals of the voices that bridge those verticalities. We will either specify the repertoire for the intervals (for instance, stipulate that voices can only move from one verticality to the next by only three semitones down, and one or two semitones up), or place boundaries on the sizes of intervals (for instance, require that the intervals in voices between two verticalities must not be less than ic 2, that is, less than two semitones, either up or down). As mentioned above, most of the previous work on voice-leading as transformation places no limits on interval size, being con- cerned rather with the effect of transformations such as transposition and inversion on the connection of successive pcsets. However, Richard Cohn has explicitly coupled transformation and proximate voice-leading in a particularly suggestive manner.20

Cohn observes that any major or minor triad A can be inverted around any two of its notes to form a new inversionally related triad B, but with a special property: the pitch class that differentiates A from B changes only by ic 1 or 2, a half or whole tone. Since two notes remain the same and one changes minimally, Cohn calls this "smooth" or "parsimonious" voice-leading. Cohn then shows that very few SCs have this property for more than one inversion operator and that the ones that do are exactly the pcsets that participate in the structure of tonal music. In the case of a major or minor triad (a member of SC 3-11), each of three different con- stituent ics is preserved by its own distinct inversion operator. Each operator produces parsimonious voice-leading. The three inversion operations are named L, P and R, originally

20See Cohn, "Neo-Riemannian Operations."

!.

V . v-


defined with slightly different names by Lewin.21 Example 5 formally defines and illustrates the L, P, and R operations.22 L preserves the ic 3, P preserves the ic 5, and R preserves the ic 4. To these are added what I call their "obverse" transforms, L', P', and R'.23 In an obverse operation, one note is held invariant while the other two change. L' retains one note while the complementary ic 3 in the triad changes and is therefore related to L. P' and R' are similarly related to P and R.

Cohn further shows how the transformational group generated by L, P, and R can be represented on Riemann's Tonnetz, a two-dimensional pc array wrapping around vertically and horizontally.24 The tonnetz writes the pcs vertically

21The L, P, and R transformations were originally posited and defined by Lewin in Generalized Musical Intervals and Transformations. Note that these operations are examples of Lewin's "context sensitive" transformations. By defining the transformation as "inversion that preserves ic 4," we avoid the difficulty of variable n-values for invariance in context-neutral TnI operations. For instance, given two members of SC 3-11, X = {047} and Y = {148}, T4I preserves the pcs of X's ic 4 (T4IX = {049}), while ToI preserves the pcs of Y's ic 4 (ToIY = {B84}). For a discussion and analytic application of context- sensitive inversion, see David Lewin, Musical Form and Transformation: 4 Analytic Essays (New Haven: Yale University Press, 1993), passim.

22Only the L and P transformations are maximally parsimonious or smooth, as they involve changes of only one pc by one semitone. In "Max- imally Smooth Cycles," Cohn implies that the mathematical group generated by all sequences of L and P acts on a triad to produce only three other pcs, which, together with the triad, form a member of SC 6-20 [014589], the type E all-combinatorial hexachord. The group generated by L, P, and R, as discussed in Cohn's "Neo-Riemannian Operations" is much more complex and ramified.

23Cohn, in "Neo-Riemannian Operations," discusses the obverses as composite operations of L, P, and R. For instance, L' is equivalent to RLP, as can be deduced from Example 5.

24In "Neo-Riemannian Operations" Cohn borrows from Balzano the no- tion of using the tonnetz to display triads as triangles, but the display of the group operations as flips is his own. Cohn also points out that the tonnetz did not originate even with Riemann, but can be traced back to Euler and Oettingen.

in cycles of T4 (bottom to top) and horizontally in cycles of T3 (left to right). Example 6 provides three copies of the tonnetz, illustrating how the six transformations are represented. A triad is shown by a triangle, the vertices of which are its pcs on the tonnetz. Each transformation flips the triangle a different way, keeping two vertices--a side -in common, or only one vertex in common. A sequence of transformations can be depicted as a series of flips on the tonnetz.25

There is an invariance that permits preservation of the triad's set class under flipping: transpositionally related triads are represented by all triangles of the same shape and size, under the geometric transformation called translation. Tri- angles of the same size but related by a geometric RI-flip are related by inversion.26

The tonnetz can be interpreted as an example of a compositional space.27 One follows, or traces, so to speak, various action-paths on the space to provide a series of pcs that have particular properties due to the structure of the space. On the tonnetz, the flipping of triangles correlates to the progression of triads in proximate voice-leading, as indicated in Example 7a.

However, as nice a representation of voice-leading as the flips might be, there is a complication. This is brought out in Example 7b, where the progression of triads in Example 7a is broken into three parts, each of which follows the indi- vidual pitch-class mappings enacted by the transformations.

25Pcsets of other cardinalities can also be shown on the tonnetz; for instance, rectangles, trapezoids, squares, or other four-sided figures can represent pcsets with four distinct pcs. Although flips representing context-free inversion operators can be defined for these sets and ones of higher cardinality, their number and complexity increases to the degree that the visual clarity offered by the tonnetz with trichords decays into a tangle of swaps and moves.

26In addition, triangles related by an R-flip are related by TnMI, and triangles related by an I-flip are related by TnM.

27See Morris, "Compositional Spaces."


Example 5. P, L, R and their obverses, P', L', R'

transformation triad mapping defining inversion example operation

L {abc} -- {a'cb} a' = I a {047}->{B74}

14 -TBI

P {abc} - {cb'a} b' = Ic b {047} -{ 730}

I =T7I

R {abc} {bac' c' = Ig c {047}-{409}

Io T4I

L' {abc} -{ab'c'} b' = Ia b {047}- {085}

c'=Ia c I0 -ToI P' {abc} -{a'bc'} a'= b a {047}-{841}

c' = I c 14=T8I

R' {abc} - {a'b'c} a' =I a {047}) {2A7}

b' = Ic b b = T2I

a' =I a means a is inverted around the pcs x and y yielding a'

note: L'=RLP P'=RPL R'=LRP


Example 6. The tonnetz representation of the LPR transformations

L,L P, P R,R

I I I I I 5 8 B 2 5

- 1 4-_7 A 1-

-9 0 3 6 9

-5---8 B 2 5-

1 1 1 l l

thick line = {047}

thin line = {B74} = L{047}

dotted line = {085} = L'{047}

I I I I I 5 8 B 2 5

? .

- 1--- 4 7 A 1-

-9 --3 6 9-

-5 8 B 2 5-

I I I I l

thick line = {047}

thin line = {730} = P{047}

dotted line = {841} = P {047}

I I I I ! 5 8 B 2 5

-1 46 7-'-A 1-

-9--0 3 6 9-

-5 8 B 2 5-

I I l I I

thick line = {047}

thin line = {409} = R{047}

dotted line = {2A7} = R {047}

Note how the common pcs between triads are not presented by the parts of Example 7b; in other words, the transformational voice-leading frequently "contradicts" the proximate voice-leading. This might remind us of the Schoenberg passage in Example lb, where the instrumental voice-leading complicated the proximate voice-leading. In short, we can deduce from Example 7b that the triangle flips on the tonnetz are really flips-with-twists. For example, reconsidering the left side of Example 7a, the flip of the thick triangle into the thin triangle is not really the flip of 4 to 4, 7 to 7, and 0 to B, but actually the flip and twist of 4 to 7, 7 to 4, and 0 to B.

The representation of voice-leading, proximate or otherwise, on the tonnetz can be generalized in at least two ways.

First, one might choose other, non-adjacent points, to compose the triangles that are flipped. But can we be sure that the flipped triangle will be the L, P, or R of its unflipped generator? The answer is yes, and this property can be proven from our previous observation that the tonnetz is constructed of cycles of T4 from bottom to top and T3 from left to right. The second, perhaps more interesting approach starts with the question, Are there other tonnetz spaces with the same or similar properties? The answer here is also yes; indeed, some of them are derivable from our present tonnetz. To imagine such a derivation, we first recognize the property shared by each member of the family of all tonnetz spaces: each of its distinct columns, diagonals, and rows contains a


Example 7. Triad transformation as triangle flips on the tonnetz [4,3,1,7]

a 47B

B 2

2

7a v 4 7^7

0o

L R P R' 047 > 47B -- 27B -> 27A > 047 Jk II W%-f%0-.I"

7b

3 "( o 8 b8 ?

A , 0 0o 'i

I t I

~

complete and ordered Tn cycle.28 We can use transformations usually applied to (Stravinskian) rotational arrays to transform one tonnetz space into another.29 Two transformations X and Y are described below; they can be used together or alone. A set of tonnetz spaces related by X and/or Y will be called an XY family.

28This guarantees that on a tonnetz constructed from cycles of interval s bottom to top and cycles of interval t left to right, and given a point labeled k on the tonnetz, a point n positions up from k and m positions to the right of k is labeled ns + mt + k.

29The X and Y transforms described here have been used to study Stravin- skian and other types of rotational arrays. See Robert D. Morris, "Gener-

alizing Rotational Arrays," Journal of Music Theory 32/1 (1988): 75-132.

Now that we have many tonnetz spaces to consider, we give each one a name based on its combination of Tn cycles. So, a tonnetz space whose upward verticals are of Tx cycles, whose left-to-right horizontals are of Ty cycles, whose southeast to northwest diagonals are of Tz cycles, and whose southwest to northeast diagonals are of Tw cycles, is given the tonnetz space descriptor [x,y,z,w]. The descriptor for our basic tonnetz is [4,3,1,7]. Actually, only the first two entries in the descriptor need be specified since the last two are derived from the first such that z = x-y and w = x +y; thus, a descriptor may be written: [x,y,x-y,x+y].

Example 8 shows how the X and Y transformations work. Given a tonnetz, X leaves the top row alone, rotates the next-to-top row one position to the left wrapping around, rotates the second row from the top two positions to the left, and so on, as shown in Example 8a.30 This has the effect of twisting the tonnetz space, so that its northwest-to-southeast diagonals become its verticals. Furthermore, if the original tonnetz was composed of interval cycles s vertically and t horizontally, the X-transformed tonnetz will be constructible from s-t vertically and t horizontally. The Y transformation is similar to X, except that the southwest to northeast diagonals of a tonnetz space become the horizontals of the transformed space. Thus the first column remains the same, the second column is rotated one position and wrapped around, the third column is rotated two positions and wrapped around, and so on, as demonstrated in Example 8b.31

Two points need to be emphasized here. First, both the X and Y transformations change the way the original tonnetz space wraps around. The X-transformed tonnetz space aligns

30Formally, let E(, ) represent a position on a tonnetz space of m rows and n columns with i = row and j = column (and rows and columns are numbered from 0). Then E( j) = X(E(,j_,,), j-i taken mod n.

31Using the same definitions in the previous note, E(,.,) = X(E(,_j,,). i-j taken mod m.


Example 8. The X and Y transformations on tonnetz spaces

[x,y,x-y,x+y]

a b c d

X: e f g h

i j k I

[x-y,y,x-2y,x]

(1) (i) (j) (k)

a b c d

f g h

k 1 i

e

J

(d) (a) (b) (c)

(NW/SE diagonals become verticals.)

[x,x+y,-y,2x+y]

(h) a j g d

> (1) e b k h

(d) i f c 1

(SW/NE diagonals become horizontals.)

its columns so that the cycle in the first column is continued in the fourth. This is shown in Example 8a by the pc d in parentheses listed under the first column, indicating that the first column a, f, k continues with d, e, j. The other columns are associated similarly: the cycle in the fourth column continues with the cycle in the third, the third is continued by the second, and the second by the first. An analogous transformation happens after the Y transformation is applied, as illustrated in Example 8b. Here the cycle in the first row is

continued by the bottom row, then by the middle row, and then by a return to the top row.32

32This cyclic mapping of columns one to another might make it seem that the Riemann tonnetz is more "natural" than its transformations under X and Y. This is however simply an artifact of notation. Any tonnetz, written as an array of pcs that wrap around, is really a finite portion of infinite planar space, consisting of an infinite set of repeating and repeated columns and rows. This space is called the covering space of the tonnetz. For instance, below on the left is a 12 by 12 portion of the covering space of the Riemannian tonnetz

8a

[x,y,x-y,x+y] 8b

a

Y e i

b c d h

1

f g j k

(i)

(a) (e)


The second point is that the tonnetz space descriptor changes in predictable ways under the X and Y transforms. This is also shown in Example 8. Our descriptor here, [x,y, x-y,x + y], is transformed by X into [x-y,y,x-2y,x] and by Y into [x,x+y,-y,2x+y].

Either X or Y may each be repeated until it returns to identity. The two transformations may also be interlaced and multiply applied. For instance, the string of symbols X2y3X describes a transformation on a tonnetz space by first doing X to the space, then Y three times, then X twice. Even though X and Y do not commute, we can calculate how many tonnetz spaces there are in the XY family of our original tonnetz space [4,3,1,7]. Since X4 = X and Y3 = Y, there are twelve tonnetz spaces in this family.33

Once we have transformed a tonnetz space into another we can flip adjacent triangles just as before, except that the pcsets, SCs, and voice-leading intervals involved will be dif-

with the (wraparound) tonnetz written within it underlined. To the right is the X-transform of the Riemannian covering space with the X-transformed tonnetz also underlined; note that the columns in the covering space do not

map to each other. 58B2 58B258B2 147 A147A147A 903690369036 58B2 58B258B2 147A147A147A 903690369036 58B2 58B258B2 147A147A14 7A 903690369036 58B258B258B2 147A147A1 4 7A 903690369036

5 8 B25 8B2 58B2 4 7A147A1 47A1 36 9 036903690 258B258B258B 1 47A14 7A1 47A 036903690369 B2 58 B2 5 8B2 58 A1 47A1 47A1 47 903690369036 8B258B258B25 7A147A147A14 690369036903

33Since X4 = X, the periodicity of X is four; thus there are four distinct Xm transformations. Similarly, there are three distinct yn transformations. So there are twelve (4 x 3) distinct XmYn transformations and twelve members of the XY family.

ferent. For instance, a new tonnetz is presented in Example 9. This space is an X-transform of the one in our previous examples. Since the old descriptor was [4,3,1,7], under X the new descriptor is [4-3,3,4-6,4] = [1,3,A,4]. Now each triangle represents a member of SC 3-3 [014]. The P, P', R, and R' transformations are also shown, but only P and P' guarantee proximate voice-leading by major second to another member of the same class. In this and most other tonnetz spaces, the six flip transformations do not always produce proximate voice-leading.

One advantage of transforming a tonnetz space into another is that the X and Y transformations are isomorphic to a non-standard one-to-one and onto operation on pcs. Such pc operations have been studied by Andrew Mead and my- self.34 Looking at the tonnetz space in Example 9 and comparing it with its progenitor in Example 6, we see that the X transformation has had the effect of mapping pcs as follows: pcs 2, 5, 8, and B remain the same; 4->7, 7->A, A-l1, 1--4; and 0<--6 and 3<->9. The resultant pc operator would be notated as (47A1)(06)(39) in group theory, and its special properties define the nature of the pc transformation induced by X.35

I conclude this brief introduction to tonnetz space transformations by examining two other spaces. The first of these in Example 10a is a tonnetz space, but from a different XY family than the one that includes tonnetz spaces [4,3,1,7] and [1,3,A,4]. The descriptor for this space is [6,1,5,7], involving

34Andrew Mead, "Some Implications of the Pitch-Class/Order-Number

Isomorphism Inherent in the Twelve-Tone System; Part Two: The Mallalieu

Complex, Its Extensions and Related Rows," Perspectives of New Music 12/1 (1989): 180-233; Morris, Composition with Pitch-Classes; idem, "Set Groups, Complementation, and Mappings Among Pitch-Class Sets," Journal of Music

Theory 26/1 (1982): 101-44; and idem, "Pitch-Class Complementation and its Generalizations," Journal of Music Theory 34/2 (1990): 175-245.

35This operator can be shown to preserve the SC affiliations of pcsets related by T0, T3, T6, Tg, ToMI, T3MI, T6MI, and T9MI.


Example 9. LPR on tonnetz space [1,3,A,4]

P,P

I I l 5 ,8 B

-3 6-9

I 2 5

1 4-

0 3

2 5 8 B 2-

thick line = (67A}

thin line = {A96} = P{67A}

dotted line = { 874} = P { 67A}

cycles of T6. It is a two-row, six-column array that wraps around so that its first row maps to its second in the horizontal dimension; this is shown by the pcs in parentheses as before. Aside from these differences, all the properties of tonnetz spaces described above apply. The triangles are all of the same SC, in this case 3-5 [016]. As shown in the example, the L and R transforms do their work. However, only P'

produces proximate voice-leading. The reader may wish to perform the X or Y transforms on [6,1,5,7] to produce other members of this XY family.

The specimen in Example lOb is not a tonnetz space, but has many of the same features. I call it a Perle space because

I I I I 5 8 B 2

-4 7.iA---

- 3-6 9 0

-2 5

I I

5

4 -

3-

8 B 2-

I I I

thick line = (67A)

thin line = {763} = R{67A}

dotted line = {21A} = R {67A}

it is a way of interpreting and generalizing George Perle's cyclic sets.36 Spaces of this type use not only cycles of T, operations, but also TnI operations. They may also include more than one instance of each pc. Note also that this space does not wrap around vertically and that its two rows contain complementary Tn cycles, T5 on the top and T7 on the bottom. A simple adaptation of the tonnetz descriptor can be used to distinguish Perle spaces from one another. Now the four places in the descriptor give the operations for the cycles

36See George Perle. Twelve-Tone Tonality (Berkeley: University of Cal- ifornia Press. 1977).

R, R


Example lOa. LPR on tonnetz space [6,1,5,7]

(7)

(B) 6- -12 3 4 5 (6)

(5) 6 7-8 9 A B (0)

(0)- (1)

thick line = 127)

thin line = {278} = P{ 127}

dotted line = {017} = PI{ 127}

(9)

(B) 0 1-2- -3 4 5 (6) /I/

(5) 6-7 8 9 A B (0)

(2)-(3)

thick line = 127}

thin line = {167} = R{ 127)

dotted line = {239} = R'{ 127)

Example lOb. Triangles on Perle space [T,/7,T,I,T7I,T5I]

0 5-A 3 8-1 6-B 4 9-2 7

I/ I/ I/ I/ 0 7 2 9 4 B 6 1 8 3 A5

T5 T5 T5 T5 5->A 8 1 6 B 9 >2

ToI T ToI T ToI T , ToI t ^

7 T5I 4 T5I 6 T5I 3 T51

SC 3-7[025] SC 3-11[037] SC 2-5[05] SC 3-5[016]

Each triangle is a Klumpenhouwer network with strong isography, yet no triangle is related to another by Tn or TnI.

Perle cyclic set:00 5 7 A 2 3 9 8 4 1 B 6 6 B 1 4 8 9 3 2 A 7 5...


found respectively: left-to-right, bottom to top, diagonally southeast to northwest, and diagonally southwest to northeast. So this space has the descriptor [T57,ToI,T7I,T5I].

Drawing triangles on a Perle space does not preserve set class, however. Instead, all triangles of the same shape and size have strong transformational isography. Four instances of this fact are shown on Example 10b. Each triangle is a

Klumpenhouwer network with exactly the same transformations among its pcs as the other triangles.37 Thus, this com-

positional space provides a means of progression between

configurations of notes that are members of different SCs, but that are internally related by the same transformations. One can either flip or move triangles on this space to generate musical passages with stable degrees of transformational relatedness among pcsets not related by the Tn or T,I operations. Since this kind of space models Perle's cyclic sets, we see that Perle's sets are not of the same compositional cat-

egory as rows or pc series, which function rather as "things" in a compositional space or are generated by tracing pcs in a space such as a tonnetz space. That Perle is aware of this crucial distinction is demonstrated by his compositional use of his own generalization of cyclic sets in his system of twelve- tone tonality.38

37See Lewin, "Klumpenhouwer Networks." Indeed, there are only two different but related Klumpenhouwer networks derivable from minimally small triangles on the Perle space. Triangles with hypotenuses in a southwest/ northeast direction produce the networks illustrated in Example lOb, networks containing T5, ToI, T5I. Triangles with hypotenuses in a northwest/ southeast direction contain the transformations T7, T7I, and ToI. These operations are the inner automorphisms under ToI of the operations in the triangles of the example: i.e., T7 = ToIT5ToI. ToI = ToIToIToI, T7I =

To0TIToI. The reason for this invariance is due to this Perle space's geometric RI symmetry.

38My elaboration of Perle's cycle sets as Perle spaces is only a step in the direction taken by Paul Lansky, "Affine Music" (Ph.D. diss., Princeton Uni- versity, 1973), which is a comprehensive expansion of Perle's twelve-tone tonality using concepts and techniques from matrix algebra and affine ge- ometry.

A SYSTEM OF VOICE-LEADING SPACES

While a generalized connection of tonnetz spaces to proximate voice-leading is not secured, we have seen that it is profitable to study atonal voice-leading in the broader context of the compositional space. Moreover, the tonnetz spaces have a direct connection with a well-understood compositional space, the literal two-partition graph, which I intro- duced elsewhere in the context of non-aggregate combina-

toriality.39 Example 11 shows the Riemann Wreath, a network of nodes each containing a dyad or single pc such that pairs of nodes connected by lines always form a member of 3-11 [037]. A walk on this graph following the lines produces a series of imbricated major or minor triads. This is demonstrated in Example 12 with dotted lines mapping a

portion of the wreath to a sequence of pcs on the bottom staff. Then the pc sequence is overlapped on the top staff to form a series of triads exhibiting Cohn's parsimonious voice- leading, most of which use the LPR transforms.

The connection between flipping triangles on the tonnetz space and tracing the lines on a literal two-partition graph is not fortuitous. In a two-partition graph like the wreath, members of a generating trichordal set class are partitioned into two parts, a pc and a dyad. Parts of members of the generating SC that intersect are represented by a single node on the graph, and the lines connecting pairs of nodes represent the partitioned members of the SC. On the tonnetz, two triangles related by the L, P, R flips share two pcs. The pcsets represented by the two triangles on the tonnetz are shown on the two-partition graph by two lines each connected to a dyad node; the other ends of the two lines connect to a node containing a singleton pc. With the L', P', and R' flips of triangles on the tonnetz, the triangles intersect at only one

39See Robert Morris, "Combinatoriality Without the Aggregate," Per- spectives of New Music 21/1 (1983): 432-86, and Morris, "Compositional Spaces."


Example 11. A compositional space: a literal two-partition graph

The content of any two connected nodes is a member of set-class 3-11 [037].


Example 12. Traversing a path on the Riemann Wreath

R P R L

? . I

" " I J I

3-11 ,j '. I

3-11 etc. " 3-11 etc. ' -.

pc. So on the two-partition graph, the two lines corresponding to the two triangles on the tonnetz are connected to the same single node, while their other ends are connected to nodes containing dyads. Both spaces model the same basic situation: two members of a trichordal SC intersect in one or two pcs under a TnI operation.40 That is, under some TnI oper-

40Lewin, "Cohn Functions," shows how SCs whose members have some or all of Cohn's properties of the 3-11 set class described above can generate

ation, a subset of a trichord remains invariant, while the rest of the trichord changes into another member of the same SC.

Example 13 makes the connection of the two spaces explicit with respect to the LPR transformations. Thick arrows label mappings of LPR transformations on two portions of the wreath. For instance, the thick arrows labeled with R

graphs isomorphic to two-partition graphs relating pcsets under the twelve T.I operations.


Example 13. LPR on the Riemann Wreath

L

R

O 09 19~~~~~

P


show that the R transformation preserves the contents of the node including the dyad {19} and exchanges the contents of the singleton nodes including pc 6 and 4. Thus a move over these nodes,41 say from the nodes 6 through 19 to 4, in effect performs the R transformation on the trichord {169} (represented by the line connecting nodes 6 and 19) into the trichord {149} (represented by the line connecting nodes 19 and 4).

In sum, while the tonnetz space and the two-partition space can both model the LPR transforms on trichords, the two-partition space need not be limited to TnI operations in order to connect pcsets, nor need the SC represented by lines connecting nodes on the graph be a trichord; and there may be any number of connecting SCs.42 Two-partition graphs therefore offer an entry into the study of voice-leading of any kind among members of any group of SCs.

Example 14 displays a graph having the same basic property as the Riemann Wreath: nodes connected by lines may be combined to form members of the same set class. Because of the graph's complexity it uses letters to connect the nodes on its perimeter; otherwise it is too hard to read. Thus, the node 27A at the top of the space is connected via the letter "b" to the node 169 on the bottom right and via "g" to the node 38B at the very bottom. The nodes of Example 14 contain trichords from SCs 3-9 [027], 3-10 [036], and 3-11 [037], and their pair-wise combination by a line always forms a member of set-class 6-19. But in addition, the pairs of connected trichords also connect via proximate voice-leading. If two nodes are connected by a thick line, the connection be-

41I shall henceforth identify nodes by their content rather than using a

longer but more precise description such as "the node containing pcs 19." 42Another advantage of the two-partition graph over the tonnetz is that

pcsets in the latter are related by intersection resulting from invariance of subsets rather than the direct mapping of pcs under TnI; thus proximate voice-leading and transformational connection are opposed (as illustrated in Example 4).

tween the two implies that two pcs of one trichord move to pcs in the other by ic 1 while the other pc moves via ic 2; if the line is of ordinary thickness, all three pcs move by ic 1.

Examples 15a and b are generated by two cyclic paths traced on the graph. The path for Example 15a starts at the node 16A near the bottom of Example 14, then proceeds up to node 059 near the top. The rest of the path continues to node 168, to 259, then via letter "f" to 36A (on the bottom left), to 25B, and returns to 16A. Example 15a sequences the node's pcsets as chords. Each chord is a member of either 3-9, 3-10, or 3-11 and leads to the next in proximate voice-leading; adjacent chords form a member of 6-19 [013478]. The chord sequence in Example 15b has the same properties as the one in Example 15a, but it is also a round; the last chord in Example 15b is a permutation of the voices of the first, so that each voice leads to another if one performs the sequence three times.43 There are many other paths on the spaces, some of which are transpositions, inversions, retrogrades, and/or rotations of each other. The smallest closed cycle of nodes is four nodes long, but there is no upper limit to the length of a path, since one can repeat a sequence of nodes as many times as desired.44

Example 16 shows the abstract space that generates the literal one of Example 14. Abstract graphs have nodes and lines that represent SCs rather than pcsets as in literal graphs. The lines on Example 16 represent SC 6-19. Here the connection of the three types of trichords to each other is plainly illustrated. For instance, we learn from this diagram that a combination of a member of 3-9 and a member of 3-10 cannot form a member of 6-19 (since no line connects their nodes on Example 16) and that only 3-11 can combine two of its

43The study of these canonic situations and their connection to rotational arrays is but one topic raised by these kinds of compositional spaces.

44It is an open research question as to how one would search efficiently for all and only the unique (basic) cycles in any cyclic graph.


Example 14. A voice-leading space

b

i

e

j

1 h

1) The content of two connected nodes is a member of 6-19[013478]. 2) The content of two connected nodes progress to each other by ic 1 (thinner

lines) or icl and ic2 (thicker lines). 3) To make the graph more readable, lines that lead to the same letter are

connected.

f

d

k

c


Example 15a. Realization of a cyclic path on Example 14

3-11 3-11 3-9 3-11 3-11 3-10 3-11

'Sj l0 _ a o- L-0

'' u

rv-e o Gio ml ho vo

Example 16. Abstract compositional space underlying Example 14

TnI

TnI

I ! I I 6-19 6-19

I 1 6-19

Example 15b. Realization of another cyclic path on Example 14: a round

3-11 3-11 3-11 3-10 3-11 3-9 3-11

a , a

. . .|

n o r0-o Io t, 't * .to

I I I 6-19 6-19 etc.

619 6-19

members to form 6-19 (shown by the line that connects the 3-11 node to itself). As in abstract two-partition graphs, the short thick lines indicate the invariances of the trichords. It is these invariances that lead to the complexity of the literal space.45

The compositional spaces of Example 14 and 16 are special indeed. How many other graphs of the same types are available? The answer is given by the abstract graph in Example 17. Here trichordal SCs are connected if and only if their members can guarantee voice-leading by semitone and also produce a hexachord. The connecting lines are labeled with the pertinent hexachordal set class. This (abstract) trichordal voice-leading space has two connected components, one associating eight trichordal SCs and the other associating three; the latter is a subnetwork formed exclusively by the whole- tone trichords. In all, only seven hexachordal SCs and eleven of the trichordal SCs are invoked. Note that Example 16 is a subset of the present space. Note also that 3-8 [026] and

45See Morris, "Compositional Spaces" for more about the relation of abstract to literal two-partition graphs.


Example 17. Abstract compositional space underlying all voice-leading spaces of the type of Example 14

6-20

6-3 ,6-3

6-13

6-38 6-5

6-20

All Trichordal Set Classes:

except 3-1[012]

3-12 [048] can form members of 6-19 in proximate voice-

leading. This was not registered in Example 16, because that abstract space is not a subset of the whole-tone subnetwork in Example 17. Finally, there are also other voice-leading spaces of the type of Example 16 that are subsets of Example 17.

Hexachordal Set Classes:

6-1 [012345] 6-3[012356] 6-5[012367] 6-13[013467] 6-19[013478] 6-20[014589] 6-38[012378]

Examples 16 and 17 show that the content of connected nodes in a voice-leading space need not form members of the same set class. In addition, voice-leading spaces are not limited to connecting nodes whose contents do not intersect. In fact, any specified voice motions between nodes can be

stipulated. Example 18 provides a voice-leading space in


Example 18. Another abstract voice-leading space

Set Classes:

4-17 <133>

3-3[014] 3-5[016] 3-8[026] 3-11 [037]

4-17[0347] 4-29[0137]

5-7[01267] 5-16[01347] 5-32[01469]

5-32 <333>

which voice motion between selected trichords is by ic 0, ic 1, and/or ic 3. A voice may therefore move by one or three semitones up or down or remain stationary (to within octave equivalence). This space shows all the SCs that fulfill this voice-leading requirement between 3-3, 3-5, 3-8, and 3-11. The four trichords form in pairs either 4-17[0347], 4-29[0137], or one of three other pentachordal SCs. The lines on the space are not only labeled by the SCs that are formed by the members of the connected nodes, but also by voice-leading lists that indicate how the voice-leading proceeds. For example, 3-5 and 3-3 are connected by a line labeled "5-16 <033>"; this means that a member of 5-16 is produced and

NB: three integers in angle brackets are voice-leading lists.

that the voice-leading will proceed by a unison and two moves by ic 3. Likewise, the voice-leading list <013> seen elsewhere in the space indicates that one voice will sustain, another will move by ic 1 and another will move via ic 3.

The space of Example 18 generates sequences of three- voice counterpoint where chords from (up to) four SCs will in pairs form only (up to) five other SCs. The first-species passage in Example 19 was generated from Example 18 to illustrate these properties. The space also shows how the same two SCs may produce the same composite set class in more than one way. The 3-3 and 3-11 nodes are connected by two distinct lines, both labeled 4-17 but with different

5-7

<011>


Example 19. Realization of a path on set classes traced on Example 18

3-3 3-3 3-11 3-3 3-3 3-8 3-5 3-5 3-8 3-11 3-11 3-8

IIKEthr r, o , o -,. o o o *^ #" tl" n- vo vo? v, o 0v q

? o "

4-17 5I7 ______ | I5-7 1 1

5-16 4-29 5-16 5-32

5-32

Example 20. Elaboration of Example 19

J=88MM A

5 5 7

piano 0) #4 5 b solo fnbrawt ra

conbrawira I 6 6

6 6 -- -

ir~ 4AJi_~~f_ - , -Y-; - 1 q,I "1 se ̂ _ J j< J J J ,

W "I m

v> /

_^ -

I zK601 ....-

I l el 6 I . I \? |J~- PIr-

A r-

^; _ ' /-L?.-lw-k? ^J^^*= ^ r

^^-1~~~~~~~~~ I

r.. 5 ---- 3 k

3 5

."O,

6 n


voice-leading lists (<003> and <013>). Thus two voice-

leadings are possible involving the same three pcsets, one by sustaining two notes and moving the other by ic 3, the other by sustaining one note, moving another by ic 1, and moving another by ic 3. The second and third chords in Example 19 illustrate the first voice-leading; the third and fourth chords show the second.

I should point out that Example 19, as well as any of the other examples using staff notation, should be regarded as frameworks for compositions, not compositions in their own right. The brief piano passage of Example 20, on the other hand, is music, based on the first-species counterpoint of Example 19, embellishing it via techniques of repetition, overlapping, and octave displacement.

CONSTRUCTING VOICE-LEADING SPACES

Generating voice-leading spaces ad hoc is often difficult, tedious, and error-prone. In this section I outline some methods using t-matrices that help one to construct voice-leading spaces of any degree of complexity or closure. Although these methods can easily be executed by hand with paper and pen- cil, they are probably best implemented in the context of a computer program.

First I will describe how to build a t-matrix. The t-matrix constructed from two pcsets A and B lists all the intervals from A to B. It has #A rows and #B columns and #A x #B positions.46 Specifically for t-matrix E, position E(i j) = Bj- Ai. Bj-Ai is the interval from the ith pc of A to the jth pc of B.47 We construct the matrix by writing A vertically as the row-heads of the matrix and B horizontally as the column heads. Then we in fill the matrix according to the rule just

46# is the cardinality operator. If A is a set, #A is the cardinality of A. 47We number the rows and columns of the matrix starting with 1.

presented: the number in the position of the ith row and jth column of the matrix is the interval from the ith pc of A to the jth pc of B. The pertinent pcs are in the row and columns heads of the position. The t-matrix constructed for the pcsets A = {05} and B = {256} is found in Example 21a.

We use the t-matrix for two related purposes. First, since it contains all of the possible intervals from A to B, it registers the total voice-leading from A to B. In general, the total voice-leading from A to B is partitioned into as many classes as there are subsets of the body of the matrix-in this case, sixty-four voice-leading classes in all, including no movement at all from A to B (the null set). In traditional counterpoint however, each rest or pc in a voice progresses by one and only one route to the next rest or note. We have called this restriction R1. Consider the voice-leading from pcset A to B. Even if the two verticalities that represent them are of the same cardinality, the motion of pcs from the first to the second may not be one-to-one and onto; the mappings may exclude some notes of A and/or B. If we adopt restriction R2, then pcs may be missing from A or B, but not both. We can easily find the mappings for voice-leadings limited by R2 on the t-matrix for A and B by taking subsets from the body of the matrix such that no number occupies the same row or column as another and all rows or columns contribute a number. Example 21b shows the six voice-leadings from A = {05} to B = {256} that can satisfy R2. Each is headed with a ver- sion of the original t-matrix (Example 21a) that has all numbers blanked out except the ones that specify the intervals in a voice-leading. Below that is a musical realization. Since pcset A has fewer pcs than B, in each case either a pc in B is doubled, or the verticality representing A has a rest in one voice.48

48The reader should remember that these realizations are only a small portion of the set of all the voice-leadings derived from these six t-matrix subsets.


Example 21. A portion of the total voice-leading from {05} to {256} as shown by a t-matrix

a) t-matrix: A = {05}; B = {256}

256 2 5 6 o 2 5 6 5 9 o 1

b) total voice-leading limited by R2:

256 256 256 256 256 256 0 6 0 6 0 2 02 0 5 0 5 5 9 5 0 5 1 5 0 5 1 5 9

int = 6

int = 9

int = 6

1.t =

2.

int = 0

int = 2 1.

t 1 2.

int = 1

int = 2

1.

f -

J _ 3. int = 0

int = 5

1. I I

2. I

3. int = 1

int = 5

1.

2.

int = 9

The second purpose for the t-matrix is its use for determining if any members of set-class X have a specified voice- leading connection to a member of set-class Y. This depends on a well-known property of the t-matrix. The interval n from the ith note of A to the jth note of B is found in the position in the ith row and jth column of the matrix. Now, if we transpose A by n, then the interval from the ith note of TnA to the jth note of B will be 0, and A and B will intersect. Thus, the number of ns in the body of the matrix gives the cardinality of the intersection of TnA and B. This is known as the "common-tone theorem."49

49In older or introductory accounts of atonal theory, the theorem is limited to comparing transpositionally related members of one SC in the context of interval (class) vectors.

Now let us specify a list Z of y permitted intervals by which pcset A can be voice-led to pcset B. So Z = {z,, z,2 ...Zy}. We construct another list W derived from Z by transposing all the members in Z by n. Then W contains the y integers {wl =

zi + n, w2 = z2 + n, ...Wy = Zy + n}. If we find members of W in positions on the t-matrix of A and B so that no position is in the same row or column as any other, then we can lead T,A to B by the specified intervals in set Z under R1, R2, or R3.

Let us look at some examples. Suppose the set class of A is 3-4 [015] and the set class of B is 3-3 [014]. The permissible intervals are 1, 2, A, or B. The voice-leading will be proximate, but without common tones. By constructing and in- specting the t-matrix, we can determine if there are one or more members of 3-4 that can move to a member of 3-3 in definitive voice-leading under restriction R3.


Example 22a gives sets A and B and the Z-set intervals [1,2,A,B], and 22b shows the t-matrix for A and B. The transposition number n is set to A; thus W is [B,0,8,9]. Example 22c uses boldface to show that some members of W, namely 0 and B, occur in the t-matrix in the no-row, no-column intersection pattern demanded by the commit- ment to R3 and definitive voice-leading. Thus pcset TAA will voice-lead to pcset B by the permitted intervals, in this case, 2 and 1. This is verified in 22c with a t-matrix constructed from pcsets TAA and B, and, at the far right, the actual voice- leading.

For members of the set class of A that are (pc) inversions of A, the technique is exactly the same except that we make a new t-matrix from IA and B. The numbers in this t-matrix give the total voice-leading (intervals) from pcs in IA to those of B. Example 22d gives the sets and permisible intervals, and 22e the t-matrix for IA and B. Three examples of voice- leading by the same Z set from TnIA to B follow. In the first, Example 22f, n = 3; W is [4,5,1,2]. We find the numbers 1 and 5 in the requisite pattern in the matrix (shown in boldface). The example goes on to show that T3IA leads to B by the permitted intervals 2 and A.

Example 22g uses n = 3 again but finds members of W in a requisite pattern that is different from the previous example. Thus Examples 22f and g show two independent R3 and definitive voice-leadings from T3IA to B.

Example 22h shows a voice-leading from TBIA to B. Following is an algorithm for finding all the voice-leadings

(with or without Rl, R2, or R3) between any two SCs by a set Z of any y permissible intervals.

(1) A is a member of one SC and B is a member of the other. (2) Construct two t-matrices, one from A and B, and the other from

IA and B. (3) For each matrix, let n vary from 0 to B. For each n, construct

the set W of permissible sums, wk = n + zk (k ranges from 1 to Y).

(4) Determine if a subset of the t-matrix contains only permissible sums. If so, the subset defines a voice-leading from TnA (or TnIA) to B, but only by members of the set of permissible intervals.

The algorithm can be performed on a set of SCs taken in all possible unordered pairs. If the algorithm is successful for a pair of SCs, indicating that there is at least one n that allows a voice-leading from TnA or TnIA to B by the permissible intervals, then the SCs can be drawn as connected nodes on an abstract voice-leading space for the set of SCs. A literal voice-leading space can be constructed from the various values of n that produce voice-leadings between different members of different SCs.

SET CLASSES AND VOICE-LEADING

When voice-leading is constrained by R3, having n voices without rests, successive verticalities will have n pitches each. If the pcsets that model or generate the verticalities are of different cardinalities, then some of their pcs will have to be doubled or omitted. And if the voice-leading is to be definitive as well, then the number of voices will have to be no fewer than the cardinality of the largest pcset, and all pcsets smaller than this maximum will have to be articulated in the voice-leading with doublings.50 This combination of definitive and R3 may be in fact impossible when some of the voice motions in Example 3 are disallowed. For instance, in tonal three- or four-voice first-species part writing, where rests may not be inserted in voices at will, four-note chords such as dominant sevenths, or even triads, may have to omit pcs to accomplish smooth voice-leading and avoid parallels or other

50Omitting pcs or employing rests contradicts the stipulation of R3 and definitive voice-leading.


Example 22. t-matrices; voice leading from SC 3-4 [015] to SC 3-3 [014]

a) A= {015}; B = {014}. Permissible intervals: 1, 2, A, B R3 restriction

b) 0 1 4 0 1 4

1 B 0 3 5 7 8 B

c) t-matrix(A,B) t-matrix(TAA,B) 0 1 4 0 1 4

0 0 1 4 A 2 3 6 1 B 0 3 B 1 2 5 5 7 8 B 3 9 A 1

A+2=0 A+ 1 =B

voice-leading

A -- 0 int.=2 B - 1 int. =2 3 - 4 int. =1

Example 22 [continued]

g) t-matrix(IA,B) t-matrix(T3IA,B) 0 1 4 0 1 4

0 0 1 4 3 9 A 1 B 1 2 5 2AB2 75 69 A23 6

3+1=4 3+B=2 3+2=5

h) t-matrix(IA,B) t-matrix(TBIA,B) 0 1 4 0 1 4 0 1 4 B 1 2 5

B 1 2 5 A2 36 7 5 6 9 6 6 7 A

B+2=1 B+A=9

d) IA = {OB7}; B = {014}. Permissible intervals: 1, 2, A, B R3 restriction

e) 0 1 4

0 0 1 4 B 1 2 5 7 5 6 9

f) t-matrix(IA,B) t-matrix(T3IA,B) 0 1 4 0 1 4

O 0 1 4 3 9 A 1 B 1 2 5 2 A B 2 7569 A2 3 6

3+A=1 3+2=5

voice-leading

3 - 1 2 - 4 A - 0

int. = A int. = 2 int. = 2

forbidden contrapuntal situations.51 The problem of defini-

tively representing a given sonority in an n-voice framework is less pressing in modal counterpoint, where the verticalities are not instances of types of chords, but collections of in-

tervals, related to the lowest pitch. On the other hand, when set-class identity is of prime

importance, as in many sorts of twentieth-century music, omitting pitch-classes from pcsets will undermine such iden-

tification, and doubling will be required. The question is how to regard or regulate doubling in voice-leading. While un- restrained voice-leading of a kind many degrees knottier than in Example 2(a) is found in many recent and not-so-recent

scores, control of doubling can be implemented by defining pcsets with duplicate pcs. Such sets are termed multisets. Set classes can be defined on multisets, so for instance, the twelve

51Of course, in tonal music, there are rules or practices for determining which tones of a chord may be omitted and in which contexts.

voice-leading

3 - 4 2 - 1 A - 0

int. = 1 int. = B int. = 2

voice-leading

B - 1 A - 0 6 - 4

int. = 2 int. = 2 int. = A


conventional trichordal SCs are augmented to nineteen if two or three instances of a pc are permitted. The term multiset class can denote a set class that contains multisets related by Tn and TnI.52 Multisets have different interval vectors and invariances from ordinary pcsets and thus different cardinalities. For instance, {0112} is invariant under T2I, whereas {0012} is not; the multiset class containing {0112} has twelve members, whereas the multiset class containing {0012} has

twenty-four.

CONCLUSION

In the construction of voice-leading spaces it became nec-

essary to examine the nature of some aspects of twentieth century voice-leading. The categories of voice-leading motions, restrictions on total voice-leading, and the specification of sets of permissible voice-leading intervals help us better to understand and categorize some of the diversity in the musical and theoretical literature. These constraints help one construct voice-leading spaces of many different kinds, not only

52David Lewin has proposed the core of such a system in "Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function," Journal of Music Theory 21/2 (1977): 194-237.

offsprings of the tonnetz and two-partition spaces. Although many of my results are speculative, more useful for composers than analysts, the tools and concepts I have provided should prove useful in analysis and in the general theory of pcsets and contour. The spaces themselves suggest further research in musical cognition.

An important topic for future investigation is the role of voice-leading spaces in a theory of structural embellishment -prolongation, if you will-that is based in part on voice- leading, but does not have all (or in limited cases even any) of the deficiencies enumerated by Straus.53 Wayne Slawson has already taken steps in this direction but within the context of two-partition graphs.54

53Straus, in "The Problem of Prolongation," suggests that the voice-

leading intervals need to be different from the intervals within the simultaneities being connected. For instance, major and minor triads contain no seconds, the primary intervals for tonal voice-leading. In the present definition of voice-leading however, the set class formed by the union of the two successive simultaneities is specified and different from the simultaneities. Thus, the possible confusion of the ics of the voice-leading and the ics that make

up the simultaneities is no longer at issue. SC identification will distinguish the "vertical" from the "horizontal."

54See Wayne Slawson, "Connectivity and Completeness in Pcset Partition Graphs with Applications to Subaggregate Chains," Perspectives of New Mu- sic, forthcoming.


ABSTRACT This paper studies networks of pitch-classes that model voice- leading-the progression of pitches to form horizontal "voices" or "lines." While tonal voice-leading depends on distinctions between step and leap, consonance and dissonance, chord-tone and embellishment, voice-leading in post-tonal music simply and non- normatively describes the motions of pitches, one to another.

First, I stipulate some limitations on totally free voice-leading and construct a complete taxonomy of voice-leading motions. Then I examine Richard Cohn's explicit coupling of transformation and voice-leading of major and minor triads as represented on Hugo Riemann's tonnetz. The tonnetz is a traditional example of what I

have previously called a compositional space; such spaces are out- of-time networks of pcs that can underlie compositional or improvisational action. After studying various transformations of the tonnetz, voice-leading is implemented by the use of another type of compositional space, two-partition graphs. These graphs are generalized so that pairs of pcsets connected in the graph need not be disjoint nor form members of only one set class. Any specified voice motions can be stipulated. The result is a collection of voice-leading spaces, a new category of compositional space. Finally, I describe an algorithm that allows one to construct voice-leading spaces of any degree of complexity or closure.

morris - voice-leading spaces

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