Joseph N. Straus - Uniformity, Balance and Smoothness in Atonal Voice Leading

Uniformity, Balance, and Smoothness in Atonal Voice Leading

Joseph N. Straus

Music Theory Spectrum, Vol. 25, No. 2. (Autumn, 2003), pp. 305-352.

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Un zformity, Balance, and Smoothness in Atonal Yoice Leading JOSEPH N. STRAUS

This paper offers a broadly applicable model for atonal voice leading, a model of pitch-class counterpoint to connect any nvo harmonies. Voice leadings are evaluated by three criteria: (1) unfor-mity: the extent to which the voices move by the same interval distance and thus approach traditional transposition; (2) balance: the extent to which the voices move by the same index number and thus approach traditional inversion; and (3) smoothness: the extent to which the voices travel the shortest possible distance. The most uniform, most balanced, or smoothest way of moving from one set to another in pitch-class space, or from one set class to another in a proposed voice- leading space, provides a standpoint from which to assess any specific compositional realization in pitch space.

I. INTRODUCTION

Theories of atonal music have traditionally been better at describing harmonies-at devising schemes of classification and comparison-than at showing how one harmony moves to another.' In recent years, however, a number of studies have attempted to shift the theoretical focus from harmony

Earlier versions of this paper were presented at The Graduate Center of the City University of New York, New York University, the Summer 2000 meeting of the Princeton Theory Group, and the 2001 Annual Conference of the Societv for Music Theory. I benefited from com- ments from the audience on each of those occasions. I also received guidance from several friends and colleagues who read earlier drafts: Cynthia Folio, Ed Gollin, Henry Klumpenhouwer, Philip Lambert, and Shaugn O'Donnell. In creating the computer programsthat under- pin the methodology presented in this article, Christopher Ariza suggested numerous important refinements in my approach. Programs are available at Ariza 2002 As Maisel 1999 has observed: "For tonal music, we have two theories that stand in metatheoretical relation to each other: that is. harmonv and counterpoint. While we have in set theory a working theorv of harmony for post-tonal music, we hardly have the barest beginnings of a theory of counterpoint" (177-8).

to voice leading.2 Approaches to atonal voice leading have generally had one of three distinct theoretical orientations. First, theorists have sought to adapt or extend models of tonal voice leading, including particularly the Schenkerian model, to atonal music.3 A second approach has described the linear organization of atonal music in terms of lines of pitches linked by shared register, timbre, metrical placement, or other contextual means4 A third approach takes recent work by David Lewin as its point of departure and adopts

For a valuable survey of recent approaches, see Morris 1998. Straus 1997a c d s this approach "prolongational" and provides an extensive bibliography. More recent contributions to this literature in- clude Vaisaala 1999. From a non-Schenkerian point of view, Lerdahl 1989, 1999, 2001 derive a model of atonal prolongation from Lerdahl & Jackendoff 1983. A related perceptual perspective is adopted in Huron 2001. The prolongational approach is critiqued in Straus 1987. See further discussion in Larson 1997 and Straus 1997b. Straus 1997a calls this avvroach "associational" and provides an extensive bibliography. For exemplary instances of this approach, see Forte 1988 and Morris 1987, which presents a theory of compositional de- sign in which "lynes" of pitch-classes are projected musically in a variety of ways.

I

what Lewin calls a "transformational attitude."' This approach has flowered in a number of d i re~t ions .~

11. TRANSPOSITION AND INVERSION

The present study adopts a transformational approach, and identifies transposition and inversion as sources of atonal voice leading.7 As shown in Example 1,when Set X is trans- posed (a) or inverted (b) onto Set Y, each note in X maps onto a corresponding note in Y. These mappings take place in pitch-class space, and are thus independent of the registral order of the notes in X and Y. As such, they comprise a potential "pitch-class counterpoint" for atonal music.8 Indeed,

5 Straus 1997a calls this third approach "transformational" and provides an extensive bibliography. The phrase "transformational attitude" and much of the theoretical apparatus associated with it are found in Lewin 1987. Lambert 2000, 45-6, refers to "the basic reorientation that Lewin's approach requires-the shift of focus away from equivalence relations among individual pitch-class collections and toward transformational processes that relate them."

6 The two principal current offshoots that bear on atonal voice leading involve Klumpenhouwer networks and Neo-Riemannian Theory (see specific references later in this article).

7 The present study takes as its point of departure Straus 1997a, as well as Klumpenhouwer 1991, O'DonneU 1997, and especially Lewin 1998. In a more general sense, I am indebted to Morris 1995b and Morris 1998.

8 The phrase "pitch-class counterpoint" is taken from Benjamin 1981, 4: "As I understand it, a harmonic progression is not a succession of verti- cal complexes so much as it is a counterpoint of lines. What separates such a counterpoint from what we ordinarily call counterpoint is that the lines in a harmonic progression are PC lines. Whereas the structures of traditional counterpoint-such as Fuxian species counterpoint -are defined in pitch-specific terms, the structures of harmony are here, by definition, conceived of in PC terms, that is, without reference to the octave placement or irreducible elements-scale degrees-in an individual harmonic progression. Therefore, whereas traditional counterpoint distinguishes between the bass voice and upper voices, or between perfect fourths and perfect fifths, harmony as I conceive it makes no such distinctions and regards even the matter of melodic contour, so basic to counterpoint, as foreign."

I propose to understand such mappings as transformational voices within a transformational voice leading that arises from harmonic successions motivated by transposition or in~er s ion .~

The kind of pitch-class counterpoint I am describing has always been implicit in atonal set theory. Any time we say that two chords are related by transposition or inversion, we are also saying that each note in the first chord maps onto a corresponding note in the second, thus creating a network of linear connections between the chords. The approach I adopt here can thus be thought of as set theory in which a sense of linearity and movement has been brought to the fore.

The transformational voice leading I describe here distinguishes pitch-class voices from actual musical lines, realized in register. This basic distinction is analogous to the distinction in tonal theory between voice and part.'' Schenkerian

9 My sense of atonal voice leading as the pitch-class counterpoint in-

duced by certain transformations is derived from Klumpenhouwer 1991

and O'Donnell 1997, and explored at length in Straus 1997a. See also

Lewin 1998, Gollin 1998, and O'Donnell 1998. In some of these

sources, the transformations in question are not the traditional transpo-

sitions and inversions.

I:o As Wen 1999,277, observes: "I should like to establish a distinction between 'voice' and 'part.' Although these terms are often used inter- changeably (and indeed frequently overlap), it can be helpful to employ them in two different ways. In this divided usage, the term 'part' is defined by a work's performing forces or textural elements, and is usually bound by the constraints of a particular tessitura (e.g., SATB). A 'voice,' on the other hand, is not dependent upon the limitations of the medium expressing the musical idea. A part can usually be identified by simply viewing the score; a voice is a linear succession that might traverse two or more parts. Such an 'ideal' conceptual entity can be an object of musical experience because its continuity results from basic properties of the tonal system: the melodic fluency produced by step- wise succession, the need to prepare and-especially-resolve disso-nances, the affinity subsisting between members of the same chord, and the connection of high and low registers effected by octave equivalence." Similarly, Karpinski 2000, 125-6, states: "The distinction between part and .voice is an important one. Apar t is a feature of the musical surface, a line performed as a continuity. In ensemble music, apart

UNIFORMITY, BALANCE, AND SMOOTHNESS I N ATONAL VOICE LEADING

(a) (b) (n is an ordered pc interval) (n is an index number)

EXAMPLE I. VOice leadingfiom set X t o set Y v i a transposition (T,) or inversion (In).

theory, for example, is replete with melodic techniques that depend on this distinction, including substitution, unfolding, reaching over, and motion into and out of an inner voice. More elementary tonal phenomena, like the transfer of reso- lution of a dissonance, also suggest the conceptual indepen- dence of voice and part.

In tonal theory, generally speaking, as in the transformational voice-leading theory described here, voices are opera- tional and systematic whereas parts or lines are contextual. In tonal music, of course, the relevant transformations are not the Tnand In that are so characteristic of atonal music, but rather idiomatic tonal harmonic progressions that are modeled in recent literature as DOM, SUBD, M E D , REL, PAR, LT, and so on.'' These idiomatic transformations in- duce a leading of voices that may or may not correspond to

is just that-the notes played or sung from a single printed part (e.g., "Bassoon I"; "Altos"). In keyboard music, parts are those lines separated from one another by the musical texture (often pointed up by composers or editors through the use of markings such as stem direction, slurs, and dotted lines). A voice, however, is a theoretical construct. Upper voices are the abstract lines formed by the smoothest possible voice leading from one prevailing harmony to another." See Lewin 1987, 175-8, for definition of these transformations, which also figure prominently in the Neo-Riemannian literature.

the motion within any particular registral or instrumental part.

I do not want to insist too much on the analogy between tonal and atonal voice leading, however, as there are also significant differences between tonal voices as generally understood and the transformational voices I describe here. In this study, a voice is simply the mapping of pitch-classes induced by transposition or inversion. As I will use the terms here, voices are transformational (they result from pitch-class mappings in pitch-class space) while lines are contextual (they result from associations of register or timbre in pitch space).12 An underlying counterpoint of pitch classes, induced by transposition and inversion, is what I will call transformational voice leading.

Example 2 shows the transformational voice leading induced by transposition in two atonal passages. In (a), the first two chords are related by transposition at T8, and, in fact, each note in the first chord simply moves down four

IZ Gollin 1998 makes the same distinction between "transformational voice leading" (i.e., the linear connections induced by transformational mapping) and "registral voice leading" (i.e., what I am referring to as "registral lines"). Similarly, O'Donnell 1998 speaks of "transformational voices," by which he also refers to the mappings induced by some transformation.

11

MUSIC THEORY SPECTRUM 25 (2003)

Vn. 1,2

Va.

vc .

Va.

Vn. 2

Vc.

EXAMPLE 2. Voice-leading induced by transposition in (a) Webern, Movements for String Quartet, op. 5, no. 5, mm. 2-6 and (b) Webern, Movements for String Quartet, op. 5, no. 2, mm. 1-3.

UNIFORMITY, BALANCE, AND SMOOTHNESS I N ATONAL VOICE LEADING 3O9

semitones onto its counterpart in the second chord. The lines beneath the score show the note-to-note mapping induced by the transposition. In a similar fashion, the third, fourth, and fifth chords are also related by transposition, and again each instrumental line simply follows the transpositional path: T7,followed by T2, composing to T9.13For each of the transpositions in Example 2(a), the transpositional mappings are realized by the registral, instrumental lines- meaning that the voices and lines discussed above here coincide. I t is also worth noting that the quasi-ostinato melody in the cello includes frequent reiterations of a closely related transpositional sequence: T4 (the complement of T8) followed by T9.

In Example 2(b), the second and third chords are related by transposition at T4.14 But there is a voice crossing: the highest note in the second chord, A, can be heard as moving, via T, , onto the lowest note in the third chord, C#.Mean-while: the other two notes in the second chord (Eb and G) ascend four semitones to their destinations in the third chord (G and B). This transpositional gesture, and particularly the voice-leading from G to B, repeat the initial melodic leap in the viola.

The registral lines are also of interest: the highest descends 2 semitones (A-G); the middle descends 4 semitones (Eb-B); and the lowest descends 6 semitones (G-C#). These three intervals (2,4,and 6), heard descending in the progression of chords, are the intervals heard ascending in the viola's initial three-note figure (G-B-c#)." Furthermore, because the chords in mm. 2-3 are related by transposition to the initial motive and to each other (the final chord has the same

13 No connection is shown between the second and third chords, or between the fifth and sixth chords, because they are not related by transposition. This is an important theoretical problem to which we will return.

14 The first chord is left out of the discussion for now because it is not related to the others by transposition.

15 The three-note motive G-B-C# is identified by Lewin 1982-83 as a "Hauptmotiv" for this passage.

pitch-class content as the initial motive), the intervals between the chords duplicate the intervals within the chords. In this progression, the transformational voices (A-C#, Eb-G, and G-B) and the registral lines (A-G, Eb-B, and G-C#) do not coincide, but each provides a valuable perspective on the linear organization of the passage.

In both passages in Example 2, transposition maps each note in one set onto some corresponding note in another. The resulting pitch-class counterpoint comprises a transformational voice leading for each progression. Musical lines can be formed in many different ways-through association based on register or instrumentation, for example-but transfor-mational voices result from an operation of some kind. In this case, it is transposition that leads the voices from chord to chord.

Inversion works the same way: it maps notes in one harmony onto corresponding notes in the next and thus creates transformational voices. In Example 3(a), the first and second tetrachords are related at 19.16 The motion in the bass line, F-E, defines an axis of inversion around which the other voices flip. Each note in the first chord is mapped onto its inversional partner in the second: F-E, C#-G#,D-G, and F#-EL.'' In the lower two parts, these inversional mappings

16 The first two tetrachords in Example 3(a) can also be understood as related by transposition at T2-this lund of transpositionaVinversional ambiguity always obtains when the harmony in question is inversionally symmetrical. The voice leading induced by the transpositional hearing of this progression involves a complete registral inversion of the trichords in mm. 2 and 5 (F-G, C#-EL, and D-E) and supports the melodic motion F#-G# that comes between the trichords. My segmen- tation of the piano part into tetrachords is not the only; or the most ob- vious, possible parsing. Nonetheless, it is not difficult to hear the passage in this way, and this hearing provides the benefit of set-class recurrence.

17 Note that when two chords are related by inversion, the distance traversed by the voices is identified as an index number (sum), not an interval. In Example 3(a), each voice moves by index 9; in Example 3(b), each voice moves by index 7. The idea of moving by an index number may seem strange initially-we normally think of voices as moving by

mit Dampfer am S t e m

c# G# take place within a registral line; in the upper two parts, the F E voices cross from one registral line to another.

I I9 The F-E in the bass is simultaneously a transformational

voice (in which F moves onto E via 19) and a registral line, in which F descends a semitone to E. In that motivic, intervallic sense, the descending semitone is also featured prominently in the cello's melodic line, in mm. 2-3 and 9-10. The transformational voice leading and the registral lines thus both contribute to the linear organization of the progression.18

In Example 3(b), the soprano and bass lines are also transformational voices, moving by 17,but the tenor and alto

E D# cross. In addition, there is a voice exchange: E and D#in the (b) intervals, not by sums. In talking about inversion, the index numbers offer simply an arithmetic convenience. The actual motion should be B G# understood as a flip around an axis. In Example 3(a), to say that the

I7 voices move by index 9 is to say that the voices flip around a shared axis, which, in this case, is defined by the pitch classes E and F. Similarly, in

E X A M P L E 3. h'oice-leading induced by inversion in (a) Wedern, Example 3(b), the voices share an index number of 7, which is to say Three Pieces for Violoncello and Piano, op. 11, no. 3 and (6) they flip around a common axis defined, in this case, by D# and E. Ravel, Le Tombeau de Couperin, "Forlane," m. 1. 18 As in Examples 2(a) and (b), one chord has been left out of the discus-

sion temporarily because it is not related to the others by transposition

D#"

- -

311 UNIFORMITY, BALANCE, AND SMOOTHNESS I N ATONAL VOICE LEADING

first chord exchange with D# and E in the second.19 Of course, it is also true that in this progression the accompanying chord, an augmented triad, simply moves down 3 semi-tones, in contrary motion with the melody. Indeed that is probably one's first impression of the linear connection between these chords. At the same time, however, the inversional relationship, and the linear connections it induces, are also audibly present. As with transposition, the voices created by inversion interact in interesting ways with the registral and instrumental lines. Sometimes the lines and voices support each other; at others, there is tension between them, and the voices cross.

111. UNIFORMITY AND BALANCE

Transposition and inversion, however, are inherently limited in their ability to forge connections among chords. In the atonal literature, it is quite common to find apparent progressions of harmonies not related by transposition or inversion and thus not members of the same set class. I t is de- sirable, then, to have a theoretical model that accommodates progressions among diverse harmonies and describes the voice leading that results.

One possible solution would be simply to abandon transposition and inversion and look for other sources of voice- leading continuity.20 But I am reluctant to do so. Transpo-

19 In tonal music, a voice exchange normally suggests the prolongation of a harmony that contains the exchanged notes, but in atonal music, voice exchanges normally suggest pitch-class inversion around the axis created by the exchanged notes. See Straus 1997a for discussion.

20 This is the approach taken in Roeder 1994, which conceives of voice leading as "the intervals between registrally corresponding members of simultaneities, which are collections of simultaneously attacked pitches" (41). See also Roeder 1984 and 1989. Indeed, most recent approaches to atonal voice leading have seen only a limited role for transposition and inversion. This is doubtless due to the inherent limitation of these operations: they permit connections only between sets that are members of the same set class.

sition and inversion lie at the core of most of our theories of atonal harmony, and it is appealing to keep them at the core of a theory of atonal voice leading. Their musical binding power has been amply demonstrated. But clearly the con- cepts of transposition and inversion must be expanded and generalized to accommodate voice leading between non-equivalent harmonies.

To lay the groundwork for such a generalization, it is useful to think about voice leading in a basic way. Imagine pitch-class sets X and Y of cardinality n, their elements arranged in registral order from highest to lowest, and a voice leading that connects each note in X to some note in Y. If X and Y are singleton sets, as in Example 4(a), only one voice leading is possible: xl-yl. That is, xl (a pitch class in set X) moves onto yl (a pitch class in set Y). If X and Y are dyads, as at (b), two voice leadings are possible (xl-y ,x -

1 2.yz and xl-y2, x2-yl). If X and Y are trichords, (c), six voice leadings are possible; if X and Y are tetrachords, (d), 24 voice leadings are possible; and so on. In other words, there are n! (n factorial) voice leadings, where n is the cardinality of the sets in~olved.~'

Example 5 shows cases in which X and Y are of different cardinality. Here, a single note in the smaller set may split onto two or more notes in the larger one. Conversely, two or

-

more notes in the larger set may h s e into one note in the smaller set.22 Although the sets in Example 5 differ in size by only one, they may in principle differ by any amount. Any note in the smaller set may split onto any two or more notes in the larger set; any two or more notes in the larger set may fuse onto any note in the smaller set. As many splittings and fusings will occur as are necessary to map each note in one

21 For practical reasons, Example 4 considers only sets of cardinality 1 through 4.

22 I adopt the terms "split" and "fuse" from Callender 1998. But whereas Callender's splittings and fusings involve exclusively motion by semi-tone, I impose no such restriction. Also, unlike Callender, I permit a note to split onto more than two destinations and more than two notes to fuse into one.

(1)

xl- ~3

(a)X and Y are singletons (one voice-leading)

(1) (2) xl- Y l

x2- Y2 x2"'x;;

(b)X and Yare dyads (two voice-leadings)

(c)X and Y are trichords (six voice-leadings)

x3"4 Y3"X'Y4

(d)X and Y are tetrachords (twenty-four voice-leadings) EXAMPLE 4. VDice leading between set X and set Y (same cardinality).

(a) X i s a singleton and Y is a dyad (one voice-leading)

(6) X i s a dyad and Y is a trichord (six voice-leadings)

EXAMPLE 5 . VOice leading between set X andset Y (dflerent cardinality).

3I4 MUSIC THEORY SPECTRUM 25 (2003)

set onto some note in the other. For practical reasons, Example 5 is confined to sets of cardinalities 1through 4.

Strictly speaking, the voice leadings of Example 5 , involv-ing sets of different cardinality, are no longer transformations, as the mappings involved are not one-to-~ne.'~ None-theless, they can still be thought of as "transformational" in a looser sense to suggest some kind of dynamic action (as opposed to a static, "relational" mode of conception). In the approach adopted here, something happens to Set X that maps its pitch classes to the pitch classes of Set Y, with all the notes of X and Y covered in the process-no note is left

In the process, notes split or fuse to accommodate differences in cardinality between X and Y.

When X is a singleton and Y is a dyad, there will be only one voice leading, as the single note in X splits onto the two notes in Y. When X is a dyad and Y is a trichord, there will be six voice leadings, as either of the two notes of X may split onto any of three pairs of notes within Y. When X is a trichord and Y is a tetrachord, there will be thirty-six voice leadings, as any of the three notes in X may split onto any of six pairs of notes within Y, and the remaining notes of X have two ways of moving onto the remaining two notes of Y.~'The chart in Example 6 summarizes the number of voice leadings between any two sets X and Y of cardinality one through six.

Amid this profusion of possibilities, this study will evalu- ate voice leading by two criteria: unformity and balance. Uni-formity refers to the extent to which the voices move by the same intervallic distance. The more uniform the voice leading, the more closely it approximates traditional transposi-

See Gollin 2000,347-69, for definition and discussion of functions, operations, and transformations.

24 My approach thus differs from that taken in Lewin 1998, which does not insist that every note in X have a destination in Y.

25 The formula for calculating the number of voice-leadings (the number of "onto functions") from an n-element set to a k-element set, and its proof, may be found at: www.research.att. com/cgi-bin/access.cgi/as/ njas/sequences/eisA. cgi?Anum=019538.

23

1:

EXAMPLE 6. The number of voice leadings between any two sets X and Y of cardinalities 1-6.

tion. Balance refers to the extent to which the voices flip around the same axis of inversion, that is, the extent to which they move by the same index number. The more balanced the voice leading, the more closely it approximates traditional inversion. Uniformity and balance thus represent generalizations of transposition and inversion for the purpose of describing voice leading.26

IV. UNIFORMITY

Example 7 addresses the voice leading between two pitch-class sets of the same size: {F, F#, B) and {G,Bb, D}, two trichords that are related by neither transposition nor in~ersion.~'As with any two trichords, there are six distinct

26 To deal with the large numbers of voice leadings and to sort them accu- rately as to their relative uniformity or balance obviously requires extensive computational assistance. The initial computer work on this project was done by Dave Smey. More recently, Christopher Ariza, in the cross-platform Python language, has developed sophisticated, flexible, and fast software for all of the calculations described in this article. The program is free, runs on every platform, and is licensed under the GPL open-source license. Downloads are available at his website, Ariza 2002.

27 Note that these are pitch-class sets moving in pitch-class space. The sets are arranged on the staff purely for visual convenience. No voice should be understood as conceptually higher or lower than another.


Most uniform Least uniform Most transposition-like 4 >Least transposition-like Most parallel motion Least parallel motion

EXAMPLE 7. Voice-leading un@rmity in theprogressionfiom {E F#,B) to {G, Bb, Dl.

ways of leading the notes of the first set onto the notes of the second. Within each voice leading, each voice traverses an ordered pitch-class interval.

Example 7 arranges the six voice leadings in order according to their relative unformity, measured by the extent to which the voices move by the same or nearly-the-same interval. The first voice leading closely approximates a transposition at Ts The bass actually does move by T3.The soprano, however, moves by T2 (a semitone "too low") and the alto moves by T4(a semitone "too high"). This first voice leading is defined as oflset by 2 semitones from a transposition at T3 vindicated by the (2) at the bottom of the analytic diagram.

This paper relies on offset as the principal means for measuring degree of uniformity. But there are two other measures that seem equally plausible, or nearly so. The first judges voice leadings as relatively uniform depending on the number of voices that move by the same interval, a condition termed consistency. If all of the voices move consistently by

the same ordered pitch-class interval, then the voice leading is entirely uniform (and the two sets are related by transposition at that interval). If all but one of the voices move by the same ordered pitch-class interval, then the voice leading can be understood as nearly uniform because of the inconsistent behavior of the one voice.28 In general, the greater the number of voices that participate in an actual transposition, the more uniform the voice leading. In Example 7, the second voice leading is an instance of "near-transposition": two of the three voices move by the same interval. I t is nonetheless considered less uniform than the first because it has a higher offset, (3).

Another way of measuring voice-leading uniformity involves the extent to which the voice-leading intervals diverge.

28 When all but one of the voices move by the same interval, the two sets are related by what Straus 1997a calls "near-transposition" and Lewin 1998 calls "pseudo-transposition."

316 MUSIC THEORY SPECTRUM 25 (2003)

If we imagine such intervals in some voice leading displayed progressively higher offset values. The last of the six is offset on a twelve-place vector, with entries available for each of by 6 semitones from a transposition at TI1-this is thus the the twelve ordered pitch-class intervals, then that voice lead- least transposition-like of these six voice leadings. ing whose entries are clustered together within the smallest Example 8 assesses the uniformity of the voice leading span on the vector would be judged most uniform. The between two sets of different size: (D, G] and (A, C, C#]. In smaller the range of difference among the voice leading in- the first, fifth, and sixth voice leadings, the G in the first set tervals, the more uniform the voice leading. Conversely, the splits as it moves onto two notes in the second set. In the more widely dispersed the voice-leading intervals among other three voice leadings, the D in the first set splits onto the possibilities, the less uniform the voice leading. two notes in the second set. If the order of the two sets were

All three measures of uniformity I have proposed-offset, reversed, two notes in the larger set would be understood to consistency, and span-capture at least part of our intuitive fuse onto either G or D in the smaller. sense of what transposition is, namely, the uniform move- The first voice leading in Example 8 is the most uniform ment of all the voices. But they do not always produce the because its three voice leading intervals scarcely differ at all same results. Offset provides certain systematic advantages, -the voice leading is offset from T6by only two semitones. particularly in its relationship to the notion of displacement The disparity among the voice leading intervals increases as developed later in this paper. But I will also invoke consis- one moves through the example from left to right, with the tency and span as the analytical situation warrants.29 sixth voice leading the least uniform: its three voice leading

The offset is the total number of semitones by which intervals are maximally dispersed among the twelve ordered the voices would have to be adjusted to bring them all into pitch class intervals, and a large adjustment of 8 semitones conformity with each other.30 For the first voice leading in would be required to repair the de~ia t ion.~ ' Example 7, I call T3the convergencepoint, which is the trans- In the six progressions of Example 8, if the order of the positional level from which there is the least possible offset. two sets were reversed (that is, if {A, C , C#] preceded {D, A voice that moves by the transpositional interval that de- GI), each transposition number would be replaced by its fines the convergence point is indicated with a solid line; complement mod 12. But the offset numbers and the rank- voices that diverge are indicated with a dotted line. The as- ing of relative uniformity would be unchanged. When the terisk attached the T3label indicates that the transposi- mapping from X to Y has a certain offset, and thus a certain tion is not exact, with the offset number in parentheses mea- degree of uniformity, the retrogression of that mapping will suring the extent of the deviation. O f the six voice leadings, have the same offset and thus the same degree of uniformity. the first is the one that is most nearly transposition-like Y can move to X simply by reversing the pathways from X to according to the offset. The remaining voice leadings have

29 Ariza 2002 permits exploration of the different results produced by 31 In the sixth voice-leading o f Example 8 , there is thus ambiguity as to these three different approaches to voice-leading uniformity. the convergence point: *T2, *T6, and *Tlowill all produce the same off-

3 0 Different voice leadings may produce the same offset. In Example 7 , set ( 8 ) . As with the ambiguities previously discussed (transposition or the third and fourth voice leadings have the same offset (4), but pro- inversion [see note 161 and voice leadings sharing the same offset num- duce it in different ways. I will not try to establish any mechanism for ber [see note 30]), I will not create any mechanism for resolving this breaking ties o f this kind. Rather, I will simply point them out as they ambiguity. It is simply a fact of life, to be identified and explored as it occur. occurs.

UNIFORMITY, BALANCE, A N D SMOOTHNESS IN ATONAL VOICE LEADING 3I7

Most uniform Least uniform Most transposition-like + Least transposition-like Most parallel motion Least parallel motion

EXAMPLE 8. Kite-leading unformity in theprogression from {D, G] t o {A,C, C#].

Y. All of the voice leadings discussed in this study are re- versible in this sense.32

There have been a number of recent theoretical attempts to measure degrees of divergence from true transposition. Most of these have been concerned with situations in which all but one of the voices move by the same interval.33 Of par-

32 Strictly speaking, my voice leadings are not retrogradable because their mappings are not necessarily one-to-one. As a result, the functions I am describing can have no well-defined inverse. Nonetheless, it is perfectly reasonable, and in conformance with musical intuitions, to welcome the possibility that if, for example, the voice leading from X to Y has a certain quality (such as a certain degree of uniformity), that the reversal of that voice leading in the progression from Y to X will have precisely the same quality.

33 Forte 1988 has described what he calls a "unary transform": "A unary voice-leading transformation results in the mutation of one pitch-class set into another by a change of a single element." For Forte, then, two

ticular importance to the present inquiry, Lewin develops a notion of "maximally uniform voice-leading": "Given pitch- class sets X and Y, a voice-leading V from X into Y will be

sets may be related by unary voice-leading transformation if all but one of their pitch classes are held in common (that is, are related by Toor whatever In maps them onto themselves). Straus 1997a designates "near-transposition" and "near-inversion" to generalize this idea to other intervals of transposition and indexes of inversion: "Two harmonies are related by near-transposition or near-inversion if all but one of their notes are related by actual transposition or actual inversion." The possibility of relating sets by the simultaneous application of two or more operations (e.g., some notes move by Tn,some by T,+x)is explored in detail in O'Donnell 1997 and 1998. See also Roig-Francoli 2001, which develops a notion of "pitch-class-set extension" that includes the possibility of intervallic expansion or contraction, and Vishio 2002, which develops a concept of "skew" to measure deviation from transposition and inversion.

called maximally unform if it differs as little as possible from a straight t r a n ~ ~ o s i t i o n . " ~ ~ also presents H e the related concept of an "offset number": "Given a maximally uniform voice leading V from pcset X to pcset Y, the offset number of V is the positive (absolute) real number of semitones (either up or down or both) by which Y differs from T,(X), in a pertinent pitch manifestation of V.'j3'

More broadly, my notion of voice-leading uniformity, with its attendant generalization of the concept of transposition, has its roots in Quinn 1996. Quinn imagines actual transposition as a "crisp" extreme at one end of a continuum of transpositional effects. For Quinn (and in the present study), any two sets are related by transposition in some measurable degree. If that degree is sufficiently low, it may not be musically advantageous to invoke the relationship. But there will be instances where two sets are not related by traditional, crisp transposition but are still related by transposition to some significant degree. I will say that the relationship between such sets is highly transposition-like. The connections created by such fuzzy transpositions may serve to link harmonies that would be judged as incomparable by traditional, crisp atonal set theory. And once two harmonies are understood as linked by fuzzy transposition, we can trace the voice leading that results.

V. BALANCE

Just as transposition can be generalized into a concept of voice-leading uniformity, inversion can be generalized into a concept of voice-leading balance, that is, the extent to which the voices can be understood to flip symmetrically around some common axis. Uniformity refers to the transposition- like quality of the voice leading; balance refers to its inversion-

34 Lewin 1988,31. 35 Lewin 1988,33. As noted earlier, the approach taken in this paper dif-

fers from Lewin 1998 in my insistence that every note in X have a destination in Y, that is, that every note in both sets be covered by, or ac- counted for by, the voice leading. Neither X nor Y can be incomplete.

like quality. Example 9 returns to the progression of Example 7 but reorders the six voice-leadings in order of their relative balance. Each transformational voice now describes an index of inversion.

The first voice leading is the most balanced, the most inversion-like, because two of its voices describe index 1 (they flip around a shared axis) and the axis of the remaining voice is offset by only two semitones. The sixth voice leading is the least balanced, the least inversion-like, because its three voices describe axes of inversion that are widely dispersed among the possibilities and would require an adjustment of 6 semitones to repair the deviation.

Example 10 returns to the progression from Example 8, and reorders the six voice leadings in order of their relative balance. The offsets range from a minimum of (2) to a maximum of (8). Offset numbers work for voice-leading balance just as they do for voice-leading uniformity: they measure the degree of deviation from normal, crisp inversion or transposition.

Throughout this discussion of uniformity and balance, I am imagining that each transformational voice may be heard as though it were seeking to do, to the greatest extent possible, just what the other voices are doing at the same time. In discussing voice-leading uniformity, I invoke a potential urge toward parallel motion-in moving from harmony to harmony, each voice may be heard to imitate the behavior of the other voices and move by roughly the same distance. The result is a kind of pitch-class organum, where each voice de- rives its sense of direction from the parallel (or nearly parallel) activity of the others in pitch-class space.36 In discussing voice-leading balance, I imagine that each voice may be heard to move so as to create a sense of symmetry and balance, as each note in the first chord appears to seek an inversional partner in the second. If the first chord is asymmetri- cal, the voices may be heard to create a sense of symmetry by

3,6 The phrase "pitch-class organum" is Michael Friedmann's (private communication).

UNIFORMITY, BALANCE, AND SMOOTHNESS IN ATONAL VOICE LEADING

F... G F o..........G F !'..........G F 7. ,...-G

FtJBL F+-BL B-1 D B B- D B

Most balanced Least balanced

Most inversion-like 4 Least inversion-like

Most symmetrical Least symmetrical

EXAMPLE 9. Voice-leading balance in the progression porn {& F#, B) to {G, Bb, D).

moving to a chord related by inversion to the first. If the first chord is symmetrical, the voices may be heard to maintain the sense of symmetry. Depending upon the context, transformational voices may be heard as though seeking to move around a shared axis of inversion.

The idea that a pitch class may be understood as desiring to emulate the behavior of the pitch classes around it is ex- pressed with particular vividness by Lewin, who speaks of the "urge" and the "lust" that a note in one set might feel to become the transpositional (or inversional) partner of a corresponding note in another set: "I find it suggestive to think of these generative lusts as musical tensions and/or poten- tialities which later events of the piece will resolve and/or realize to greater or lesser extents."37

37 Lewin 1982-83,341. To imagine that notes have desires, urges, lusts, or volitions of any kind-even to imagine that notes are capable of moving -is to indulge in a familiar music-theoretical metaphor. Our discourse often anthropomorphizes notes, endowing them with a limited degree of human agency-we permit them to move from place to place, even

Traditional atonal set theory is based on the transpositional and inversional equivalence of pitch class sets. Two sets are either equivalent or non-equivalent-a stark either/or- and the relationships are crisp. But the methodology I have presented here makes transposition and inversion fuzzy, not crisp. Now, it appears, any pair of sets is related by transposition in some degree; any pair of sets is related by inversion in some degree. If the degree of equivalence is sufficiently low, it may

to desire partnerships with other notes. If all music-theoretical dis-course involves what Guck 1994 refers to as an "analytical fiction," then the fiction of the tones having a will strikes me as a relatively attractive one, enlivening the discourse with an appealing sense of motion and purposeful activity. The danger of this metaphor, or any such metaphor, is that one will be seduced into believing it too literally, and thus be led into various lunds of conceptual distortions-Harrison 2001 discusses this issue in interesting detail. In the present paper, however, my ascrip- tion of will to tones is always anchored in substantial, technically trans- parent descriptions. Behind every metaphorical urge and lust lies a fully theorized musical relationship.


Most balanced Least balanced Most inversion-like 4 Least inversion-like Most symmetrical Least symmetrical

EXAMPLE 10. Voice-leading balance in theprogressionfiom {D, G] to {A, C, C#].

not be musically advantageous to invoke the relationship. But there will be many instances where two sets are not related by traditional, crisp transposition or inversion, but are related by transposition or inversion to some significant degree. We might say that the relationship between such sets is highly transposition-like or highly inversion-like. The connections created by such fuzzy transpositions and inversions may link harmonies that would be judged as incomparable by traditional, crisp atonal set theory. Thus fuzzified, transposition and inversion can serve as an impetus for voice leading, creating a counterpoint of pitch classes bound together by their shared, or nearly-shared, intervals of transposition or indices of inversion.

VI. SMOOTHNESS

There is one additional factor to consider. It can be useful, as I have suggested, to imagine the pitch classes in set X

driven onto the pitch-classes in set Y by transposition or inversion (crisp or fuzzy). But it can also be useful to think about the total distance the pitch classes in set X have to travel to get to their destinations in Y. Example 11illustrates with respect to the sets from Examples 7 through 10, with the voice leadings now arranged in the order of their relative smoothness.38

Voice-leading smoothness is measured by the total displacement, the sum of the intervals traversed by each note from its origin in X to its destination in Y. The intervals in

38 Wha t I call voice-leading smoothness is also referred to in the literature as parsimoniousness, efficiency, nearness, closeness, and proximity Notice that I use the term here to refer to proximity in pitch-class space, not in registral, pitch space. The voice leadings that I am describing in pitch- class space as relatively smooth could thus be realized in relatively un- smooth ways in pitch space, and vice versa. The same is true, of course, of the voice leadings I describe as relatively uniform or relatively balanced.


(a) ;#:5ib B D

displacement: 5

Smoothest Least smooth Most efficient 4 )Least efficient Closest, nearest Farthest, most distant

'c# displacement: 5

Smoothest Least smooth Most efficient Least efficient Closest, nearest Farthest, most distant

EXAMPLE 11. Eice-leading smoothness in theprogressionj+om (a) {E F#,B] t o {G, Bh, D) and (b) {D, GG) to @, A,, C#].


question are unorderedpitch-class intervals (interval classes).39 Given two voice leadings, the one that has the lower total displacement is the smoother.40 Unlike transposition and inversion, smoothness is a quality, not an operation. I t cannot, therefore, propel the voices, but it can usefully characterize the voice leading that results from transposition or in~ersion.~'

39 Cohn 1998b has proposed a related definition of what he calls "total voice leading distance" or "directed voice leading sum," but these are based on the orderedpitch-class interval within each voice. This has real advantages for systematization, as Cohn shows, but is not an effective measure of the actual exertion or intervallic "work" involved. For example, if three voices move respectively by 3, 4, and 6 semitones, I would call that a considerable total exertion of 13 semitones, near the theoretical maximum for trichords, which is 18. Cohn's "sum," however, which involves arithmetic modulo 12, would consider the total voice leading distance to be only 1, and thus equivalent to the voice leading between two trichords where two of the notes are retained as common tones and the third moves by semitone. Furthermore, in Cohn's approach, ordered pitch-class intervals can effectively cancel each other, so if one voice moves up a semitone and another moves down a semitone, Cohn considers the voice leading displacement to be 0; I think it should be 2. I follow Lewin 1998 and Alegant 2001 in measuring voice leading proximity according to the sum of the interval classes involved. In the measure employed here, total displacement values are not to be taken modl2. The total displacement values for all trichords, for example, range from 0 (when set X maps onto set Y at Toand all three notes of the first chord are retained as common tones) to 18 (when set X maps onto set Y at T6and all three voices move by tritone).

40 It is possible for two or more voice leadings to share the same total displacement, as in the third and fourth voice leadings in Example l l (a) . It would be possible to devise an algorithm for breaking ties (for example, by seeing which voice leading has the greatest number of small intervals or the smallest number of large intervals), but I will not do that here. Instead, I will simply note such ties as they occur.

41 There is a close relationship between smoothness (measured by displacement) and uniformity or balance (measured by offset), but it is one that becomes apparent only at the level of the set class. Both offset and displacement measure deviation from an ideal state, either transposition/ inversion or pitch-class identity (that is, either Toor a self-mapping inversion). The convergence of smoothness with uniformity and balance will be discussed later in this paper.

Voice-leading smoothness has long been of interest to theorists.42 In discussing it, I invoke an implicit desire of tones to conserve their energy, to move by the shortest possible distances. Notes exert a gravitational pull on each other, and the attraction increases as the distance between them shrinks. Notes move only through an expenditure of effort, and it requires greater effort to move a greater distance. In moving from Chord X to Chord Y, there are many routes available, some smooth, short, and direct; others that are distant and more difficult. There is at least potential expressive meaning in this contrast, as composers plot their voice leading courses from harmony to harmony, either following a path of relatively little resistance by moving smoothly, or pushing their harmonies along a harder path, forcing them to move relatively great distances.

42 An interest in voice-leading smoothness apparently goes back as far as Marchetto of Padua's Lucidarium of 1317-18; see Cohen 2001. The classic formulation is "the law of the shortest way" described in Schoen- berg 1987: "The first of these directions [for connecting chords] requires that in the voice leading, at first, only that be done which is ab- solutely necessary for connecting the chords. This means each voice will move only when it must; each voice will take the smallest possible step or leap, and then, moreover, just that smallest step which will allow the other voices also to take small steps. Thus, the voices will follow (as I once heard Bruckner say) the 'law ofthe shortest way'" (39).An interest in smooth voice leading is a central aspect of the recent outpouring of Neo-Riemannian theory-see Cohn 1996,1997,1998a, 1998b, and the extensive bibliography provided in the Journal of Music Theory 42 (1998): 335-41. The Neo-Riemannian enterprise is particularly relevant to the present project in that it detaches voice leading from acoustical or tonal/functional issues. Lewin 1998 is an even more im- mediate source in its discussion of "maximally close voice leading," which depends on a concept of "total shiftn-a measure of the total distance traveled by all of the voices added together: "the total number of semitones traversed by the three voices, as they move from their notes in one chord to their notes in the next chord" (24). Lewin engages familiar metaphors of musical exertion in referring to "melodic hus- bandry" (30, 31), and the possibility of moving between sets "with as little overall strain as possible" (38).

- -

UNIFORMITY, BALANCE, AND SMOOTHNESS IN ATONAL VOICE LEADING 3 23

The methodology set forth here is designed to establish voice-leading connections between any two pitch-class sets. The sets may in principle be of any size and they need not be the same size. Transposition (generalized as uniformity) and inversion (generalized as balance) impel the voice leading- they lead the voices. I imagine that each note in one chord seeks a path in pitch-class space onto some note in another chord, and that these paths are distinguished by their uniformity, their balance, and their smoothness. These paths comprise a potential counterpoint of pitch classes. An actual musical passage may realize a particular counterpoint in a variety ofways in pitch space. A kind of dialectical relationship between pitch-class counterpoint and compositional realization characterizes the analyses that follow.

VII. ANALYSES

Example 12 revisits a passage from Example 2(a).43 Above a quasi-ostinato in the cello, the upper three parts play a progression of six chords representing three different set classes. As we observed previously, the first and second chords are related by crisp transposition, as are the third, fourth, and fifth chords. The analytical problem is to connect the second chord to the third and the fifth chord to the sixth. In the progression from the second to the third, three of the voices deviate from the prevailing T2, but they do so as little as possible, by only a semitone each (Example 12[a]). In moving from the fifth chord to the sixth, the deviation from an actual transposition is even smaller-only one voice is off, and it's off by only one semitone.

The individual transpositional moves combine to create a single, larger transpositional move shown in Example 12(b): from the first chord to the last is a fuzzy-transposition at *T5,with an extremely small offset of only two semitones. Notice that the offsets are as likely to cancel each other out

43 This passage is discussed in related ways in Lewin 1998 and Roig- Francoli 2001.

as they are to accumulate. In the second-lowest voice, for example, the motion from F#to A is a semitone "too h igh for the prevailing *T2while the motion from F#to D#is a semitone "too low" for the prevailing *TloAs a result, that line conforms in its entirety to the larger *T5, even though it contains two smaller deviant moves. The progression as a whole, embracing members of three different set classes, can thus be heard as a single, unified gesture. Furthermore, the progression-spanning *T5can be heard as a large-scale state- ment of two smaller T5's at the opening of the cello melody: the motion from F#to B, and from (G, G#, B] to (C, C# , E].

The Stravinsky passages in Examples 13 and 14 involve a similar confluence of fuzzy-transpositional voices with registral lines.44 In Example 13, from the first chord to the third is an actual transposition at T9, diagrammed at (b), with each of the three registral lines moving down 3 semitones. But, as Example 13(a) shows, they move down at different rates: the soprano and alto move first by semitone and then by whole- tone, while the bass moves first by whole-tone and then by semitone. This difference accounts for the small offsets in the chord-to-chord succession.

The passage in Example 14 contains four different chords representing three different set classes. The second and fourth chords are related by transposition at TIo,but the first and third are related neither to the second and fourth nor to each other. The voice leading that connects all four chords, however, is highly transposition-like.45

Consulting Example 14(a), we note that in moving from the first chord to the second, three of the four instrumental lines descend by semitone-only the viola does not participate. This voice leading thus diverges from TI1with an offset of only (3). This is also the maximally smooth voice leading between these chords. The progression from the second

44 This passage has been extensively discussed in the literature, beginning with Schenker 1926/1996. See relevant citations in Straus 1997a, which discusses it in similar terms. More recently, see Traut 2000.

45 Straus 1997a studies this passage in a related way.

MUSIC THEORY SPECTRUM 25 ( 2 0 0 3 )

Vn. 1 , 2

Va.

vc.

EXAMPLE 12. Wedern, String Quartet, op. 5,no. 5,mm. 1-6.


EXAMPLE 13. Stravinsky, Concerto for Piano and Winds,fist movement, 4 mm. at R11.

chord to the third is also maximally smooth, with a total displacement of only 4 (all four instrumental lines move up or down by semitone). In terms of uniformity, the progression might be described either as *TIor *Tll, with an offset of (4) in either case.46 The progression from the third chord to the fourth is not maximally smooth, but once again the most

46 In other words, the convergence point is ambiguous-either TI1or TI produces the same offset. In addition, the offset of (4) can be produced by several other voice leadings, but these all involve intervals that diverge more widely (that is, they occupy a larger span among the twelve possible ordered pc intervals).

uniform voice leading follows the instrumental lines. As with -

the progression from the second chord to the third, two of the voices move by Tll while the other two are "off" by two semitones, resulting in a total offset of (4).47

If one takes the first violin as the leading part, then the succession of chords is *Tll-*TI-*T9. If one takes the cello as the leading part, then the succession of chords is *TI,-*Tll-*TI,. In either case, the three fuzzy-T's combine to create a single fuzzy-*T9 that spans the progression from

47 Again, there is an ambiguity regarding the convergence and again the offset of (4) can be produced by other voice leadings.

MUSIC THEORY SPECTRUM 25 ( 2 003 )

EXAMPLE 14. Stravinsky, Pieces for String Quartet, no. 3, mm. 3-5.

327 UNIFORMITY, BALANCE, AND SMOOTHNESS IN ATONAL VOICE LEADING

the first chord to the fourth (Example 14[b]). Stravinsky's registration and instrumentation of these chords reflects the -

uniformity of their progression. Each instrumental line thus acts also as a transformational voice.

Example 15 revisits a passage previously discussed in connection with Example 2(b).48 The passage consists of a lyrical melody in the viola accompanied by three different chords in the second violin and cello. The second and third chords are related by transposition at T,, as we noted earlier. The analytical problem is how to handle the first chord, which has four notes, not three, and is not related in any ob- vious harmonic or intervallic way to the second and third chords.

But, as seen in Example 15(a), the first chord is related to the second by fuzzy transposition at *TI. The tritone in the first chord, Ab-D, moves up by T1to the tritone in the second chord, A-Eb. The other two voices are slightly off. The note A moves onto A at To(a semitone "too low"), and F moves onto G at T2(a semitone "too high). The result is a fuzzy-transposition with an extremely small offset of (2).

The fuzzy-*TI that connects the first chord to the second combines with the crisp-T, between the second chord and the third to create a fuzzy-*T5 that spans the entire three- chord progression, represented in Example 15(b). In this larger fuzzy-*T5, as in the smaller fuzzy-*TI it contains, two voices move by five semitones, while a third is a semitone "too high" and the fourth is a semitone "too low." So, fuzzy- *TIplus crisp-T, equals fuzzy-*T5, and the first chord is thus integrated into a coherent three-chord progression.

The transpositions of the tritone, shared among all three chords, can be heard to guide the larger progression. In the fuzzy-*TI that connects the first two chords, the tritone Ab- D moves by crisp-T1 onto the tritone A-EL, and the other two voices tag along as closely as they can. Similarly, in the fuzzy-*T, that connects the first chord with the third, the

49 48 This passage is discussed in related ways in Lewin 1982-83, Roeder

1994, and Straus 1997a.

tritone AC-D moves by crisp-T5 to the tritone C#-G, and the other two voices tag along with minimal deviations.

Furthermore, the idea of combining TI with T4to make T5is reflected in the viola melody also, once in the overall melodic shape (G-Ab-C) and twice in closely related intervallic cells within the melody (C-C#-Ab and F#-B-G), which involve the complements of these transpositional intervals. But neither the harmonic progression itself nor its relationship to the melody can be understood without ac-cepting the progression from the first chord to the second as a fuzzy-*TI.

Offset measures the degree to which some set Y can be heard as a deformed X-that is, the amount of work that would be required to repair the deformation, to transform X into Y. It is thus a measure of the distance between two sets and, by extension, between the set-classes to which they belong. In the passage in Example 15, the first chord is within a two-semitone adjustment of being a member of 3-8[026]. If the A were an Ab and the F an F#,the first chord would be Tn-equivalent (at TI) to the second. Conversely, the second chord is within a two-semitone adjustment of being a member of 4-18[0147]. If the A in the second chord were both an A and a Bb, and the G an F#, the second chord would be Tn-equivalent to the first at TI. So, offset measures both the relationship between two sets and between the set class to which they belong. The relationship between set and set class is one to which we shall return.

Voice-leading smoothness provides an additional interesting perspective on this progression. In moving from the first chord to the second, there are three voice leadings that produce a minimum total displacement of (4). Webern has arranged the chords to project one of these in register, with the bass moving Ab-G, the tenor F-Eb, the alto A-A, and the soprano D moving to an implied ~ b . , ~ In moving from

Actually, the voice-leading model shows the F and D in the first chord fusing onto Eb in the second. But it is easier to grasp that relationship, in this particular context, if the pitch class Eb is imagined as represented

MUSIC THEORY SPECTRUM 25 (2003)

Vn.

vc

EXAMPLE 15. Wedern, Movementsfor String Quartet, op. 5, no. 2, mm. 1-3.

the second chord to the third, Webern could have arranged leading that is minimally smooth, with a maximum total them to project the same minimal displacement of 4 (G-G, displacement of (12).~' Chords 1and 2 are harmonically dis- Eb-C#, A-B). Instead, his registral lines project a voice parate, and Webern connects them smoothly; chords 2 and 3

50 As noted earlier, the registral lines (G-Cb, Eb-B, A-G) state the inter- by two different pitches: Eb, and, an octave higher, an implied Eb,. The vals (6,4, and 2) found in 3-8[026], the set class of the first three notes other two voice leadings that produce the same total displacement of 4 in the viola melody as well as the second and third accompanying are D-Eb, A-A, F-G, Ab-A and D-Eb, A-A, F-G, Ab-G. chords.


are harmonically uniform (they are related by transposition), and Webern connects them non-smoothly. In this way, the registral lines, moving with varying degrees of smoothness, interact in interesting ways with the voice-leading voices, motivated by transposition.

Example 16 revisits an entire short movement previously discussed in connection with Example 3(a). The first two chords are related by Iy and, as observed earlier, the bass line (F-E) defines an axis of inversion around which the remaining three voices Aip (see Example 16[a]).'' The third chord is not related to the second by crisp-inversion, but it is related by a fuzzy-inversion at *I3, with an offset of (3). That operation induces a complete registral inversion: bass and soprano exchange, as do tenor and alto. Those two inversional moves-crisp I9followed by fuzzy-*I3-compose to a fuzzy-*T6 that connects the first chord with the third, shown at (b).

That fuzzy-*T6 can also be understood in transpositional terms, as a crisp-T2 followed by a fuzzy-'T,, which is shown at (c). In moving from the first chord to the second, each note moves onto a transpositional partner two pitch-class semitones higher, a gesture that is reinforced by the melodic motion F#-G# in mm. 3-4. In moving from the second chord to the third, the transformational voices, induced by fuzzy-*T4, correspond to the registral lines. That is, each note in the second chord moves up by four pitch-class semitones (more or less) onto a note in the same registral position in the third chord. In moving from the second chord to the

-

third, then, the inversional hearing (via *I3) induces a registral inversion (all of the voices cross), while the transpositional hearing (via *T,) induces registral invariance (all of the voices move in parallel). Lambert 2000 observes that all three pieces ofVVebern's op. 11are "sat- urated with [0145]sn and constructs relevant transformational networks. Among the [0145]s identified by Lambert in the third piece, shown in Example 16, are the first four notes in the cello part, (B, C, Eb, Fb) and the four upper notes in the piano right hand in mm. 2-5, {D,Eb,Fb,GI.

The *T,-induced voice leading from the second chord to the third is the least smooth way of connecting these two chords. As shown at (d), it involves a displacement of (17). Webern's registral lines, however, describe the smoothest way of connecting the first chord to the second, with a displacement of only (8).52 The linear organization of the progression cannot be attributed to transposition, inversion, or smoothness alone. Rather, it resides in some interactive combination of them. All are present, and all shape the linear flow in some degree.

Example 17 revisits a passage from Example 3(b) and places it in a more complete context.j3 This passage is obviously in E minor in some sense, and it ends with a strong, modal V-I cadence. But it contains elements that resist the pull of E minor and that create their own, distinctively non- tonal musical logic. At (a), eight strong-beat harmonies are connected by inversion and transposition. I t is worthwhile to follow a single voice through the progression. As the first chord moves to the second by 17, E moves to D# . The movement of the highest registral line thus defines an inversional axis, and the sense of inversion is confirmed by the voice exchange: E-D# and D#-E. The second chord moves to the third by Ty, and again that gesture is reflected in the movement of the highest registral line: D# to B#. At the same time, of course, the progression from the second chord to the third could be understood as a fuzzy-*TI, with the augmented triad in the left hand moving up 1 semitone and the melody moving against it in contrary motion. This fuzzy-*TI portrays the accompanying chords as being in control of the motion with the melody deviating from them. The crisp-Ty portrays the melody as in control, bringing about a crossing

52 There are four other voice leadings between the first and second chords that share the same displacement of (8). One of these is the T,-induced voice leading: all four voices move by 2 semitones.

53 This passage is discussed from a Neo-Riemannian perspective in Gollin 2001. See also Hyde 1996, which describes the voice leading in tonal terms (using figured bass, with numerous registral transfers and dis- placements).

51

MUSIC THEORY SPECTRUM 25 ( 2003 )

F# D

fa) C# F E

--.

.......

-" -" -(4 F ' E Bb (8) (17)

Displacement: (minimum) (maximum)

EXAMPLE 16. Webern, Three Little Pieces for Cello andpiano, op. 11, no. 3.


EXAMPLE 17. Ravel, "Forlanel'fiom Tombeau de Couperin, mm. 1-5.


of the voices within the accompanying chords. Both inter- pretations are easily audible and useful in characterizing the linear relationship of the harmonies.

As the progression continues, the soprano moves into an inner voice, first to G# (via *T8) and then to A (via *I5). In the motion from the fourth chord to the fifth, as from the first chord to the second, the inversion is confirmed by a voice-exchange: G#-A and A-G#. Example 17(b) shows the composite motion from the first chord to the fifth, in which the soprano E has moved 5 pitch-class semitones to the bass A, confirming the motion from the first chord onto the fifth by T5.In the remainder of the progression, three fuzzy- transpositions, largely coincident with the registral lines, compose to a larger fuzzy-*T7. The cumulative result of these motions, shown at (c), is a fuzzy-*To with an offset of only (1) that spans the whole progression. At this level, we are able to hear the final E-minor triad as a minimally dis- torted version of the initial 4-19[0148].~~ The reverse is also true: the initial 4-19[0148] can be heard as a minimally dis- torted E-minor triad, and one can imagine the progression as an extended effort to rectify the distortion, to transform the 4-19[0148] into an E-minor triad.

Example 18 involves a progression of four chords representing four different set classes.55 It is hard to generalize about them as harmonies: the first three are octatonic, but fourth is not; the first and third are symmetrical, but second and fourth are not; the first, second, and fourth embed tri- ads, but third does not. The analytical task is to describe the voice leading that binds these disparate chords.

Ruggles has connected the first two chords and the last two as smoothly as possible, as shown at (a). The smoothest

54 It is important to note again that, over the course of an extended progression, the offset numbers are as likely to cancel each other out as they are to accumulate. In the chord-to-chord successions, the offset numbers range from (0)to (3) and most of the successions involve some degree of offset. But in the progression from the first chord to the last, the offset is only (1).

jj See Slottow 2001 for a related discussion of this passage.

connection between the second and third chords involves a registral inversion, as does the connection from the first chord to the fourth that is shown at (b). Notice that displacement numbers, like offset numbers, do not necessarily accumulate; they may cancel each other out instead, as they do here. The progressions from chord to chord involve dis- placements of 6, 10, and 7 semitones, but the progression from the first chord to the last involves a displacement of only 1; the first and last chords share three out of four pitch classes, and the one note that is different differs by only one emito one.'^ The transformational voice leading induced by transposi-

tion and inversion, shown at (c), follows the same smooth path. In the motion from the first chord to the second, three of the registral lines move down two semitones, while the remaining line sustains a note. Something very similar happens in the motion from the third chord to the fourth, where three of the four lines move by an index of 5, while the remaining line sustains a note. In moving from the second chord to the third, *I3induces a registral inversion, as does the *Tothat connects the first chord with the fourth, shown at (d).57

Notice that, as with displacement numbers, the offset numbers do not necessarily accumulate. I n the chord-to- chord succession, the offsets are (2), (4), and (5 ) . But the overall offset as the first chord moves to the last is only (1). That is because offsets are as likely to cancel each other out

56 For each of the chord-to-chord successions, there is another voice leading with the same minimum displacement value, but the connection shown from the first chord to the fourth is uniquely smooth.

57 For each of the chord-to-chord successions, there is at least one other with the same minimal offset value. The same is true for the connection from the first chord to the fourth, which could be modeled as *I3 with an offset of only (1).The voice leading induced by *I3follows the registral lines (Db to D , F to Bb, and D to C#by crisp-I,, with Bb-F# deviating by semitone) and involves a voice-exchange (Db to D in the soprano and D to C#in the bass). Interestingly, Igis also the transformation that maps the inversionally-symmetrical first chord onto itself.


, A

Vn. 1

Vn. 2

Va.

vc.

I

Cb.

4-17[0347] 4-18[0147] 4-3[0134] 4-19[0148] 0 0 0 0

2 ' DL- Eb D 2 F#

(4 F k 3 Bb E c#

Total displacement: 6 10 7

Total displacement: 1

EXAMPLE 18. Ruggles, Lilacs, mm. 1-2.


as they are to accumulate. In this example, and in all of the

examples in this paper, each voice leading involves the small-

est possible offset. The moves from chord to chord are as

transposition-like or inversion-like as they can be. And the

larger gestures, the middlegrounds, are also as transposition-

like or inversion-like as they can be. Indeed, these examples

are chosen precisely because the chord-to-chord successions,

with their minimal offsets, combine into larger middle-

ground gestures that also involve minimal offsets.

Schoenberg's Little Piano Piece, op. 19, no. 2, is punctu-

ated by the three big chords shown in Example 19.'~ Differ-

ent though they are in size and internal structure, these

chords can be connected by transformational voice leading

resulting from fuzzy-transposition. As shown at (a), the first

chord is connected to the second by *TI1, which sends the

diminished triad at the bottom of the first chord to the top

of the second, and also sends the top three notes of the first

chord fuzzily to the bottom three notes of the second (the

Gb in the middle of the first chord splits up to F and down

to F#).The second chord is connected to the third by *T,,

which exchanges the upper four and bottom two notes in the

second chord with the top two and bottom four in the third.

In both moves, there is an offset of (4), the same as in the

motion from the first chord to the third by *T,, shown at

(b).j9 This voice leading is guided by the motion of the

minor third (C-Eb) at the bottom of the first chord, which

moves at TI1to the minor third (B-D) at the top of the sec-

ond chord, and then by T4to the minor third (Eb-F#) in the

middle of the third chord, where it acts as a center of inver-

sional symmetry There is a strong motivic aspect to these

transpositions: they effectively compose out the intervals of

58 See Vaisaala 1999 for a prolongational interpretation of the voice leading in this piece and among these chords.

59 The progression among these three chords can also be understood in terms of inversion, with *I6and *I5 composing to *TI1, with an offset of EXAMPLE 19. Schoenberg, Little Piano Piece, op. 19, no. 2: (4) in each case. But the transpositional interpretation has a significant punctuating chords.

motivic aspect and produces avoice leading that conforms more closely

with the registral arrangement of the chords.


3-3[014], such a prominent feature of the motivic surface of the piece, as in its opening six-note melody, some interval classes ofwhich are shown in Example 19(c).

VIII. SETS AND SET CLASSES

The discussion up to this point has focused on the voice leading between actual sets of pitch classes. I t has probably been apparent, however, that the voice leading between sets is highly constrained by the structure of the set classes to which these sets belong. The nature of the constraints is best illustrated with an example. Example 20 shows the voice leading between a set X {C, C#, E) and all transpositions and inversions of a set Y {C, E, F#).For each of the twenty-four pairs of sets, Example 20 shows the voice leading that produces the smallest offset, whether by *Tnor *I, In every case, the smallest offset is the same: (2). In other words, one can get from X (or any of its transpositions or inversions) to Y (or any of its transpositions or inversion) by a voice leading that approximates, to within two semitones, either transposition or inversion. The voice-leading patterns shown in Example 20 and the offset of (2) constitute a kind of finger- print for the set classes that contain sets X and Y.

It is thus possible to speak of a destjt, or what I will call an optimal offset, between two set classes. In Example 20, X belongs to set-class 3-3[014] and Y belongs to set-class 3- 8[026]. We can say that the best possible fit between these two set classes involves an offset of (2). That is, any member of 3-3[014] is related to any member of 3-8[026] by either transposition or inversion with an offset of (2). Optimal offset is a convenient way of measuring the distance between two set classes.

At this level, smoothness converges with uniformity and balance. The optimal offset is also the minimum displacement value associated with the smoothest way of moving between two sets. Thus it is possible to talk about smoothness to wi thinf izq-Tn orfuzzy-jz. In the case of the two sets in Example 20, the voice leading from X to 15(Y) (circled in

the Example) involves a total displacement of 2 semitones. That is, it deviates by only 2 semitones from the absolute smoothness of a crisp-To-the displacement value, which measures smoothness, is thus the same as the offset value, which measures uniformity and balance. Both measure basi- cally the same thing, namely the amount of deviation from the perfect smoothness of crisp-To.

It is possible to calculate the optimal offset, the best and smoothest fit, between any two set classes. Example 21 provides the optimal offset between any two tri~hords.~' The information is provided in the form of a chart: the optimal offset value for any two trichord-classes can be found at the intersection of the relevant row and column. (The customary parentheses around the offset numbers have been removed for the sake of legibility.) The first row of the chart, for example, shows 3-1[012] in relation to all twelve trichord- types. The optimal offset between 3-1[012] and itself is of course (0), since it is possible to map any member of 3- 1[012] onto any other member of the set class by crisp transposition or inversion. It is not so easy, however, to get from 3-1[012] to 3-8[026]. From any member of 3-1[012], moving by transposition or inversion, it will require an offset of at least (4) to map onto a member of 3-8[026]. I t is even far- ther from 3-1[012] to 3-12[048], with an optimal offset of (6). In fact, an optimal offset of (6) is as far apart as two trichords can get.

The contents of the chart in Example 21 can also be presented as the spatial map of Example 22.61 Offset values thus define what Morris (1998) calls a "voice leading space."62

60 The same information for sets of any size, cardinality 2 through 6, and for sets of different size, is provided in Ariza 2002.

61 See Cdender 2002 for a similar trichordal map. 62 See also Morris 1995b, which describes more general "compositional

spaces," which he defines as "out-of-time networks of pcs that can un- derlie compositional or improvisational action." The creation of such compositional spaces is a traditional activity of music theorists (see, for example, the Riemannian Tonnetz, described in many of the sources listed in note 42 and the "pitch space" described in Lerdahl2001).


X +T,(Y) E ..?.o F# C # L A # C -.O ........C

X +T,(Y) E ..A'....... G C # L B C ..A ........C#

X +T,(Y) ........E ..-Q G#

C # L C ..--.......C ? D

X +T,(Y) E ..-!........A C#--C#

-....-..C ...? D#

X ---+TlO(Y) ........E ..A A#

C # L D ........C --.! E

X +TI1(Y) .......E -." B

C # k D # ........C ...5 F

EXAMPLE 20. VOice leadingporn X {C, C#,E) to all transpositions and inversions of Y {C, E, F#).

Lines link trichord classes connected by an optimal offset of (1). 3-2[013], for example, is offset by (1) in relation to 3- 1[012], 3-6[024], and 3-3[014]. Each move on the map thus represents an offset of (I), and the moves are cumulative. From 3-2[013], for example, it requires two steps on the map, representing an optimal offset of (2), to get to 3-4[015] and 3-7[027]. The two set classes most widely separated on

the map (3-1[012] and 3-12[048]) are separated by 6 steps, representing an optimal offset of (6). 3-1[012] and 3-12[048] are the most isolated set classes on the map-each is in direct communication with only one other set class. 3-8[026] and 3-7[025] lie right at the center of the map and are relatively promiscuous in their relations-they communicate directly with 6 and 5 other set classes respectively.

UNIFORMITY, BALANCE, A N D SMOOTHNESS I N ATONAL VOICE LEADING

EXAMPLE 21. Optimaloffsets for trichords (as a chart).

Example 23 provides the same information for tetrachords in chart format, and Example 24 presents the same information as the map of a voice-leading space.63 Like Ex-amples 21 and 22, these show the smallest amount of adjustment (offset, displacement) that is required to get from one set-class to another via transposition or inversion. As with the trichords, the tetrachords on the periphery of the map are relatively isolated from the others. 4-1[0123], 4-9[0167], and 4-28[0369], for example, have an optimal offset of (1) with only one other set class, while those at the center are relatively promiscuous in their relationships (4-22[0247] and 4-27[0258] are in direct communication with the largest number of other set classes). Example 25 shows trichords in relation to tetrachords in chart format. Unfortunately it is difficult to present the contents of this chart in the form of a spatial map, in the manner of Examples 22 and 24, since the voice-leading connections among the set classes are too numerous and intertwined. The same is true, to an even greater extent, for pentachords and hexachords, both within them-

63 Cohn 2003 elegantly displays the tetrachordal space in three dimen- sions as a tetrahedron. See Quinn 2003 for additional discussion.

EXAMPLE 22. Optimal offsets for trichords (as the map of a voice leading space).

selves and in relation to set classes of different size. For set classes with more than six notes, the situation becomes sim- pler again: the maps for octachords, nonachords, and deca- chords will be identical to those for tetrachords, trichords, and dyads-each set class in the map is simply replaced by its complement. In the absence of a legible two-dimensional

338 MUSIC THEORY SPECTRUM 25 ( 2 003 )

EXAMPLE 23. Optimal oflets for tetrachords (as a chart).


EXAMPLE 24. Optimal ofsets for tetrachords (as the map of a voice-leading space).

34O MUSIC THEORY SPECTRUM 25 (2003)

EXAMPLE 25. Optimal ofsets for trichords in relation to tetrachords (as a chart).

representation, one needs to imagine the 220 set classes hov- ering in a multi-dimensional space and linked together by the bonds of parsimonious voice leading.

Within such a space, optimal offset numbers provide a good measure of the degree of proximity or distance between any two set classes. The set-class geography that emerges

from a focus on voice leading has interesting points of contact with that produced by two rather different theoretical enterprises: similarity relations and set complexes. Similarity relations have traditionally been concerned with assessing the resemblance between two set classes, usually based on shared intervallic or subset content.64 Theories of set complexes have attempted to group set classes into families, often based on some combination of inclusion and comple- men ta t i~n .~ 'Yet despite the variety of approaches to similarity and set complexes, they offer strikingly similar maps of the set-class terrain. That is, there is broad agreement as to which set classes are relatively similar that transcends individual methodologies for showing such similarity.66 What is more, the voice-leading spaces described here show striking resemblances to the topographies produced by similarity relations and set complexes. As Quinn observes, "the system of pc set classes has a natural topography that presents itself as the elusive intuitions of similarity-relation creators and that

64 For good, critical summaries of earlier work on pitch-class set similarity and important original contributions, see Buchler 2000, Scott & Isaacson 1998, Castren 1994, and Isaacson 1990. Ariza 2000 has over twenty similarity measures available for comparison to each other, and for comparison with the atonal voice-leading measures discussed in this paper. Similarity measures are usually based on intervallic or subset content.

65 For discussions of set complexes and their latter-day descendants, pitch-class set genera, see Forte 1973 and 1989, Morris 1997, Kaplan 1990, and Doerksen 1999.

66 This is the central contention of Quinn 2001: "One of the more significant contributions of the present research is the observation and char- acterization of the heretofore unnoticed connections between similarity relations advertised as widely divergent . . . It should be quite clear by now that all aspects of similarity, together with many topics of pc set- class theory, are deeply and inextricably interrelated. It is not a weak- ness, but a strength of these similarity relations that (when we listen to them!) they speak with a single extensional voice, regardless of what they ~rofess to measure" (155). See also Scott & Isaacson 1998, 117- 18, for a discussion of the high degree of correlation among a variety of similarity measures.

UNIFORMITY, BALANCE, AND SMOOTHNESS I N ATONAL VOICE LEADING 34I

lies behind many of the tools of pc set-class the~ry."~' From different points of view, then, both the voice-leading oriented methodology presented here and the harmony-oriented methodologies of similarity relations and set complexes appear to be engaged in mapping the same terrain.

Wha t this suggests is the deep connection between harmony and voice leading in atonal music.@ I previously observed that harmony constrains voice leading-that is, that the internal structure of a set shapes the kinds of voice leading connections it can create with other sets. And the reverse is also true: Voice leading constrains harmony in the sense that the kinds of voice leading connections a set is capable of making with other sets will go a long way toward defining its internal structure. Similarity relations and set-complex relations ask: How much like set-class X is set-class Y? The vo

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Joseph N. Straus - Uniformity, Balance and Smoothness in Atonal Voice Leading