hasty - 1987 - an intervallic definition of set class

Yale University Department of Music

An Intervallic Definition of Set ClassAuthor(s): Christopher F. HastySource: Journal of Music Theory, Vol. 31, No. 2 (Autumn, 1987), pp. 183-204Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843707 .

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AN INTERVALLIC DEFINITION

OF SET CLASS

Christopher F. Hasty

During the past two decades a large body of post-tonal composition has become increasingly intelligible from the theoretical point of view, largely through the work of Allen Forte. The concept of pitch-class set and the systematic investigation of the relations of sets have enabled the theory of post- tonal music to free itself from both the anachronism of tonal paradigms and the superficiality of motivic analysis. Doubtless, the success of the theory of unordered sets in providing a basis for the study of 20th century music stems in part from the explicitness and generality of its fundamental concepts. These virtues have permitted many scholars to contribute to the development of the theory both in its systematic elaboration and in its range of application.

In the present essay I shall examine critically the notion of set class (or collection class), a basic concept of post-tonal theory which I believe has yet to find adequate definition. I have chosen a brief excerpt from the first song, "Wiedereshen," of Milton Babbitt's cycle, Du, to provide the material for the examples in this essay. Since Babbitt is responsible for so much of the conceptual structure that the following discussion engages, it seems appropriate that a work of his should provide the starting point for our investigation.'

The first measure of "Wiedereshen" (Ex. la) may be taken to present

183



=4 8 mm3p 3 1. Ipp

Dein Schrei - ten bebt

p 3 3 mp pI

3-4 m 34

b.

8(+12) 7(+24) 4(+24) 11(+12A

Example 1

184



consecutively three forms of set-class 3-4; E6, E, Ab followed by the transpositionally related form G, A6, C followed by the inversion C, B, G. I wish to examine carefully what this assertion and others similar to it can be taken to mean.

Considering the first two instances of set-class 3-4, we might say that to indicate these structures implies that on the third beat of this example we can hear a new sonority, differentiated somehow from the first sonority, yet sounding in some way like it. Differentiation is necessary if a set is to be conceived of as an entity with distinct properties that may be transformed or compared with other entities. This determinate, thing-like view seems generally accorded to the notion of pitch-class set. Likeness here may be taken to mean "composed of the same or similar intervals." In the present case the interval classes found in each of the two trichords are the same while the registral intervals are similar (Ex. lb). Differentiation and likeness are reciprocal terms of structure that interact in very complex ways. For instance, could not the intervallic similarity we have noted incline us to hear a continuity in the intervallic realm rather than a segregation into two discrete entities? The interval class 4, which transposes the first trichord into the second, is an interval that appears within each trichord possibly making the distinction between them less plausible. Such questions are the province of segmentation or structural formation and are beyond the scope of the present inquiry which is not directed toward an analysis of this work. For our purposes it will be assumed that the pitch-class sets indicated in Example la are differentiated, autonomous structures. This assumption will allow our attention to be focused exclusively upon questions of similarity and equivalence.

The concept of set class has achieved its powerful generality through progressive abstraction. What defines a set class must surely be the intervallic relations among its pitch-class constituents, not the pitch classes themselves (already highly abstracted) nor their temporal order nor the myriad other possible relations and qualities which inhere in an actual musical structure. These abstractions, however, are not pure negations because the excluded qualities can be recovered to gauge the degree of transformation a set undergoes or the similarity of various forms of the same set class. Specific characteristics that are omitted from the definition of an equivalence class are not denied existence but rather take on the character of variable terms. In this way the concept of set class while existing on a different plane from any of its concrete musical manifestations holds all these as possibilities. Moreover, the notion of set class, since it rests on a series of abstractions, can to some extent organize these possibilities. For example, while pitch-class set has eliminated a determination of temporal ordering from its elements, such a determination may in fact intensify the relation of members of the same set class. Assuming that there is some sort of correspondence between the pitch classes of members of the same set

185



class and that this correspondence is essential to the "equivalence" of these sets, the temporal ordering of pitch classes may in varying degrees match this correspondence. Thus the category of temporal order can take its meaning from the more abstract set class. We could in principle even establish a scale of possible orderings from the maximum to minimum similarity based on set-class correspondence. Likewise many other properties abstracted out of a pitch-class set may, operating singly or in concert with one another, find a principle of organization in the relations of pitch-class sets. The generalizations we make to create the concept of set class are in this sense never completely detached from the particulars. Such generalization may in fact be a step toward a more systematic connection of these particulars. And, that such connections are possible clearly shows the productive- ness of the concept and its potential as an organizing principle.

An elementary question that I have tried to circumvent thus far must now be addressed. What is the principle of equivalence that allows the formation of the concept of set class? To begin answering this question let us return to the opening of DU. It was assumed that there are two instances of the single set class 3-4 found in the beginning of this passage. If we know in what way they are considered equivalent we will understand what is meant by set class. The customary answer to this question is that these pitch-class sets are transpositionally related. The operation of transposition (or possibly of inversion) applied exhaustively to each element of a given set yields a set of the same class. However, this account when pressed will not provide a definition of set class. We might say that in the case of transpositionally related sets each pitch in one of two so related sets is related by a specified interval (modulo 12) to a single pitch in the other set, and thus whatever structure characterizes or defines the one set is replicated in the second. But to understand the concept of set class, we must locate the fundamental similarity it seeks to establish. This characteristic structure will itself have some bearing on the way we interpret transformational operations? From the operational point of view one might argue, absurdly, that since all the elements of one set share precisely the property of being related by a single interval (mod 12) to the members of another set-a property not shared by any pitch outside of this set-they are unified as a structure. From this definition we must assume that the structural properties that define a set class come into being only when a set is compared to another set of the same class or perhaps that a single instance of a set can acquire this property by being transposed into itself at t = 0. I bring in this argument only to indicate the fundamental problem of circularity: set-class must be given before operations can be performed which can be claimed to determine membership in a set class.

In attempting to define set class it will be useful to pursue further problems involving transposition and operations in general. It is stated without dispute throughout the theoretical literature that transposition and inversion

186



are relations, that sets are "related by" these operations or mappings. We may reasonably askl what sort of relationship is implied here. Operations may be viewed from the logical, axiomatic perspective as eternal relationships existing independently of any particular (musical) manifestation. The mathematical operation of addition may be regarded as a universal property of number, a relation completely detached from any activity. On the other hand, addition may be regarded as a real operation or human activity which mathematics symbolizes-the linguistic development of the concept of number certainly points to this interpretation. However this issue might be argued for mathematics, a theory of music which hopes to comprehend the structure of sounding music will have to consider whether its operations or relations are divorced from acts. If we say that two sets are related by transposition defined as a one-to-one mapping, we may be asserting, for example, that there is a functional relationship established between the interval of transposition and the corresponding pitches and that this can in some way be detected in the sensuous medium. Or we may simply be asserting a formal connection between transposition (whatever range of musical possibilities this may encompass) and arithmetic. One might point to a musical instance in which a demonstrable operation strongly contradicts hearing or intuition, but this would in no way impugn the logical validity of the operation. Such a divergence of the experiential and the logical will never cast doubt on the integrity of the group structure governing ideal transformations.3

If we wish to bring these two quite properly distinct worlds together, as we are compelled to do in attempting a musical analysis, then relations or operations must give up some of their universality and timelessness to take on the character of actual constructions, that is, they will come to be viewed as structuring rather than as immanently structured. Following this path we are led to ask what sort of act is, for example, transposition. In response we should first recognize that there may be many results of such an operation. Thus in transposing a 12-tone set we can point to various patterns of change, such as pitch-class invariance or order inversions, and even systematically describe such phenomena, but these results of the operation are not the operation itself-it might in fact be possible to perform a different operation yielding the same sequence of pitch classes from which perspective we could view the pattern of transposition relations as a result. The difference between such "equivalent" operations will have significant methodological consequences for any theory which attempts to apply its operations to the analysis of music. (Note for example cases where identical results can be reached by the different operations of transposition or inversion applied to inversionally symmetrical sets. Such possibilities may en- rich the structure of the mathematical group but pose serious problems for musical analysis if the notion of equivalence is strictly maintained.)

Since operations are customarily given the form of one-to-one mappings

187



the act of relating corresponding pairs of pitches is suggested. Presumably, each pitch of one set is somehow connected to a single pitch of another set by a fixed interval. In order to sense the set-class equivalence of such collections we must therefore sense this connection; that is, we must hear this fixed interval. In Example 2a two trichords from Example la are extracted and presented in a simplified form retaining only the qualities of pitch and temporal order. (In Examples 2 and 3 intervals are labeled by the number of semitones measuring the "distance" modulo 12 from the first note to the second. Plus and minus signs indicate direction. In parentheses are shown octave displacements.) Here the pitches of the second trichord are arranged (in a perhaps artificial interpretation of Ex. la) to expose an identity of order among corresponding transpositionally related

elements." Even with

the help of this ordering it is quite difficult to imagine that our sense of the similarity of these two sonorities must arise from the perception of the three bracketed intervals, such that hearing the three instances of the same interval type-E-AL, Ak-C, EL-G-we arrive at an intuition of equivalence. My questioning of the aural validity of these one-to-one correspondences is not meant as an indictment of the operational model of transposition as a mapping but rather as a test of its limits. In Example 2c the replication of set class is perceptually very immediate and one easily senses the tranposi- tional relationship. This perception may be strengthened in part because the interval of transposition, 3, is not confused with any interval within the trichords and because the interval of three semitones, being the smallest interval presented, is not challenged by other intervals for relation by registral proximity. Yet, the interval, 8 (modulo 12), appearing both between the sets (EL-G) and within them may to some extent weaken the relationship in question. Example 2d rectifies this problem. Because of the necessary complexity of musical phenomena, these examples do not satis-

factorily isolate the specific factors which contribute to the transpositional relationship; if one feature is changed all relationships are changed. This is of course the mark of wholeness. Nevertheless, as one listens to Exam-

ples 2a, b, c, and d, a particular sort of connection emerges, absent in

Example 2a and quite strong in Example 2d. If inquiring very briefly and superficially after our experience has not

settled this issue it surely has illuminated it to some extent. Even the few observations we have made in connection with Example 2 indicate that transposition as an actual operation connecting pitches is a special and therefore "abstractable" attribute of the relations of sets of the same class. Like temporal order, duration pattern, and so forth, the operation of trans-

position is a structural possibility held by the concept of set class. The scalar arrangement of Examples 2a-d represents a small selection from the whole range of such possibilities.

Even if the possibility for transposition is an attribute of the set rather than the foundation or ground of the set, it is nonetheless a necessary

188



a. b. +4

intervals: -8(-12), +7(+24)1-7(-24)I +4(+24), -5 -8(-12), +7(+24)1 -7 1 -8(-12), +7(+24)

.+3 d. +2

+3 +3 +2 +2

-8(-12) +7(+24)-8 -8(-12), +7(+24) -8(-12), +7(+24) -10 -8(-12), +7(+24)

e. f.+3

V-+3 -3

-8 +7 -8 +5 3-4 3-11 3-4 3-1

Example 2

a. x y b. x y

S11

7 3 a

(+24) 8 (+12) C+ 12)

11 11 11 1 S7(+24) 7(+24)

17(+24) 7(+12) 8(+12) 8(+12 8(+12) 8

y C. X d. x

3(+12)

3-3

S7(+24) 7(+24) 8(+24) 11

7(+24) 11(+12)7(+48) 8(+12) 7(+24) 8(+12) 8(+24) "

Example 3 189



attribute; every set that we call a member of a set-class can be transformed into any other member of the class by transposition (or transposition cou- pled with inversion). From a different point of view let us now inquire why the operation of transposition-even when conceived axiomatically, apart from an actual event- is incapable of serving as a basis for the definition of set class. We must begin with a brief excursus by distinguishing three types of pitch-intervalic equivalence: J-Pitch intervals i and i' are J-equivalent if and only if i = i' in exact semitones; hence the interval -3 semitones is not equivalent to +15 semitones or to -9 semitones. K-Pitch intervals i and i' are K-equivalent if and only if the following conditions a) and b) are both satisfied: a) the difference of i and i', measured in exact semitones, is divisible by 12 (numbers "divisible by 12" are 0, ? 12, ? 24, and so forth), and b) i and i' have the same sign; that is, both are positive or both are negative. Here +3 semitones becomes equivalent to +15 or +27 semitones; thus we speak, for example, of parallel fifths rarely finding it necessary to make a distinction between fifths and twelfths. L-Pitch intervals i and i' are L-equivalent if and only if condition a) above is satisfied. Through the registral inversion of two pitches (i.e., inverting the relations of "above" and "below") in addition to allowing displacement by octave or octave multiple, the interval +3 semitones can be regarded as equivalent to -9 semitones. While severely restricting its application to the bass, clas- sical tonal music allows this sort of variability to occur quite freely. Another sort of equivalence, "interval-class" (M), will be considered later when we investigate interval in more detail.

In each of the Examples 3a-d the second trichord, Y, is "related to" or "transformed from" the first, X, by the operation of transposition where the value of the operator t is 3, or more succinctly, Y = T (X,3). Since the operation is performed on pitch classes and results in pitch classes, there are a vast number of possible pitch realizations. If we endow the transposition operator with more intervallic selectivity so that it can discriminate among the three types of interval listed above, we find that unless the operator has a fixed numerical value the degree and type of intervallic change effected by the operation cannot be predicted. Consider Examples 3a, b, and c. For the transpositions +3 on the lower stave in Example 3a a transposition +15 (+3 displaced upward by an octave) has been substituted in Example 3b. The resulting change in the intervallic structure of the set is shown by numerals below each trichord. Comparing the two sets in Exam- ple 3b we see that one interval (+11) remains the same and two intervals are altered by octave displacement. By applying these same intervals of transposition to different pitches of X as in Example 3c we can generate a set of more dissimilar intervallic structure. Compared to set X, set Y of Ex- ample 3c exhibits each of the three different types of intervallic relationships listed above. Conversely, Example 3d applies the three types of interval as transposition operators to produce in Y a structure very similar to that of X.

190



Now it may be objected that the operation of transposition as employed in the customary notion of set-class equivalence avoids any involvement with such issues since the operation deals only with abstract pitch classes. Certainly this is true. There is no question of contradiction in this usage, but only a question of the relation of this model to the phenomenon it is modeling. The above demonstration can shed some light on this relationship. It should be recalled that in Example 3 a-d the lack of correspondence between intervals of transposition and the structure of the transformed set did not occur because a too abstract or too general kind of interval-type was unfairly asked to be responsible for finer discriminations-the distinction of 3 interval types was given both to the transposition operator and to the comparisons of sets. The reason for the failed correspondence is that the operation of transposition takes place from outside the context of the set. It is a mapping of single pairs of pitch classes (or possibly of pitches) without any regard for the relations to pitches outside the pair. This atomism produces the analytic difficulties seen in Examples 3 a-d since in order to describe the intervallic structure of a set we have to regard the set as a whole.

Perhaps a fundamental problem here arises from the distinction between pitch and interval. We may regard a set as a collection or aggregate of pitches or pitch classes, but to view a set as a whole we turn our attention from the collection of objects (or pitches) to the totality of relations among those objects. Traditionally these relations are expressed as intervals (though it should be remembered from the list above that this is a complex sort of relationship involving several types of interval). Moving from set to set class, we assert some similarity among all the members of the class, something common to all such sets and only such sets. Surely, this common property has nothing directly to do with the pitches (or pitch classes) as such but rather with the similarity of intervallic relations among the pitches of sets belonging to the same equivalence class. Nevertheless, the customary means of expressing this similarity is through the operations of transposition and inversion applied to individual pitch classes mapping each element of a set into the corresponding pitch class of an equivalent set. As suggested earlier, the possibility of performing such mappings presupposes an already formed set; it does not create or found the set itself. Since mapping can replicate a founded set we might therefore be inclined to say that this replication creates the equivalence class; however, such an explanation leaves the nature of the set itself undetermined; from this perspective the set could be described simply as an aggregate of pitch classes.

In an attempt to understand the internal structure of the pitch-class set we are led to consider the totality of intervals formed among the pitch classes of the set. The concept of interval-class vector represents this holistic understanding of set structure. In the case of set-class 3-4 the interval vector, [100110], shows in the form of an array the unique intervallic structure of this

191



set class: there is one instance each of interval classes 1, 4, and 5 formed among whatever pitch classes might constitute members of this set class. It is these intervallic relations that form the set as a structural whole, and it is the identity of these relations

among.various collections of pitches

which prompts us to regard these collections as similar or in some sense "equivalent." While the interval vector displays the internal structure of the set, it makes no reference to the pitch classes among which the intervallic relations are distributed. In this sense the pitch-class representation and the interval vector representation complement one another. But this is not a productive complementarity, since the two principles exclude one another so completely; instead of creating a universe through their mutuality, each lays claim to the whole.

The intervallic relations, then, are what make a set out of the constituent pitches: prior to the set there are no "constituent" pitches. Once formed as constituents, these pitches may then be subjected to transformations. Apply- ing the same transformation to each of the pitches of the set presumably guarantees similarity of the original to the transformed product? The interval vector proves its usefulness when we wish to consider the set as a whole; for example, when we wish to determine the invariants between two transpositionally related sets or when we wish to compare sets of different classes or the relation of a set class to the set class of its complement. On the other hand, when we attend to transformations of set classes our understanding of whole changes in an attempt to comprehend the totality of the transformations -hence the attention given the structure of the operations themselves. Since the operations are translated into a modular arithmetic their systematic inter-relationship can be very powerfully described using the notion of the mathematical group.

The ideal reciprocity of the two perspectives outlined above is seriously compromised by the anomaly of the Z-related pairs of set classes.0 In these cases sets which cannot be related by transposition or inversion perversely present the same interval vector, and as yet there has been no "canonical transformation" found which can map each of these thirty-eight set-classes into its correspondent intervallically equivalent mate.' Even if such a transformation were to be found, there is reason to doubt that this would rein- state the interval vector to the role of determining set-class equivalence. Such a reinterpretation could only be possible if the new transformations could be systematically reconciled with the operations of transposition and inversion. Even if such a solution were found there might still be some objection to the. perceptual obstacles presented by this special sort of equivalence. It is doubtless a consideration of these two factors, the systematic and the perceptual, which has, in the face of the anomoly caused by the Z relation, resulted in the abandonment of interval vector rather than the abandonment of operations in the definition of set class. This is an unfortu- nate loss; particularly if, as I have argued above, the intervallic structure

192



of a set is the essential property compared to which the operations that relate a set to others of its class can be viewed as epiphenomena. This is not to deny that since transformations are necessarily intimately tied to the structures they transform, the operational perspective can reveal a vast range of properties of the twelve-tone universe. The susceptibility of twelve-tone operations to systematization has enabled many scholars to contribute to the creation of a very potent theoretical instrument. And yet, divorced from its complementary perspective the operational view becomes misleading when it demands to be taken for the whole. This view, because it is atomistic and takes the elements of a set to be points, simple quantities, will tend toward reification, toward the thing-like, the quantitative and away from the relational, the qualitative and perhaps the less sharply determined. In order to begin bridging this difference I shall attempt to establish an intervallic definition of set class.

A set may be understood not as a given collection or aggregate of pitches or pitch classes, but as a genuine whole defined by the totality of intervallic relations among its constituent tones. There is a reciprocity or dialectic uniting pitch and interval: musical interval is the relation of pitches and so is clearly dependent upon pitch; conversely, pitches are dependent upon interval or relation if they are to be regarded as constituents or tones. (As we begin considering the relation of pitches it may be appropriate to use the word "tone." This usage has the advantage of shifting our attention from the acoustical toward the musical and allows us to circumvent the distinction marked by pitch and pitch class whenever this distinction is not useful.) As noted earlier, "interval" is a complex concept comprising several types. We distinguished four types, arranged in order of increasing generality or ab- stractness: J, K, L, M. While there is apparently nothing to prevent us from taking any of these intervallic relations to define the set, there is reason to begin our investigation by giving preference to the most general or inclu- sive; thus if the relation L holds so must the relation K, but the reverse is not true.

On the basis of the foregoing, the interval vector would seem to be an excellent representation of set structure-it lists all the intervals formed among the set's constituents, using the most general interpretation of interval, the "interval class" (our M). The usefulness of this theoretical con- struction is, however, limited by its suppression of the importance of the constituent elements of the set over against the relation of these elements. This difficulty becomes apparent when we look more closely at the concept of "interval class."

Let us begin by inquiring further into what the notion of interval in- volves. In Example 4a two tones are heard in relation to one another. We are not concerned now with their temporal order of presentation-A could precede or follow C, they could be played simultaneously, or other tones might intervene. Likewise we are not concerned with their relative durations,

193



a. b. -Register C. -Registral order d. -Registral order A -Register -Register

A= -3 A= -15 -3 A--+9 -3 A= 3 C=+3 C=+15= +3 C= -9 +3 C=3

J K L M

Example 4

C (11 86)[146] .--C (11 5)[135]

C# (1I" ) [135] -*

C# (1 10 6) [126] E (10 4)[234] -------- D# (2 8) [234] F# (2 5 6)[256] G (4 7 6) [456]

4-Z15 [111111] 4-Z29 [111111]

b. set class 4-Z15 (10 3 4)

0 10 3 4

0 il i2 ... in-1 -0 0 10 3 4 -10 2 0 5 6

-0. 0-0 i-0 i2-0 .. in-1-0 -3 9 7 0 1 O-i 0-i1 il -il i2-i1

. in-1-i -4 8 6 11 0

-i2 0-i2 il-i2 i2-i2 ?

* in--i2

-in-1 0-in-1 i -in-1 i2 -in-1 ?

in-1 -in-1

c. D (1 4 6)

. . .. C# (11 3 5) SA#(2 9 8)

4-Z15 G# (10 7 6)

d. 1 9 1(46) (2)9(8)

9 7 (10)7(6)

Example 5 194



loudness or timbre. Their relation as tones is restricted to the sole domain of interval. We may call the interval they form "3 semitones" or "minor third," but should not forget that in naming a single interval we are still presented with two tones, C sounding three semitones above A and A sounding three semitones below C. C as a tone has acquired the relationship or, we might say, the quality +3. It is this quality that makes of the pitch C (c.524cps) a tone. This same quality can be heard in any isolated pair of tones three semitones apart. A similar quality can be heard when the interval is expanded by an octave as in Example 4b. In this example "minus register" means that expansion or contraction by an octave or octave multiple is not a distinctive feature, so long as the A remains below the C. Our sense of the similarity of an interval and its inversion is represented in Example 4c. Here the category of registral order or the distinction above/ below is mitigated so that the quality of A, -3, is equated with the quality +9. Notice that -3 is not equated with +3. This distinction is eliminated in Example 4d, an interpretation corresponding to the notion of interval class and based on a type of inversional equivalence. The operational perspective doubtless informs this view of intervallic equivalence-the action of moving by three semitones can be carried out in an upward or downward direction; or, more abstractly, the pure arithmetic operation of subtraction yields a quantity which does not owe its existence as number to this operation. While this understanding of interval proves very valuable in some con- texts, it creates difficulties for the definition of set by renouncing the complementarity of pitch and interval in favor of the hegemony of the latter. As Example 4 illustrates, the advance in abstraction marked by Example 4d is the elimination of the distinction between the two pitch classes-A and C are assigned the same quality and are thus undifferentiated.s The interval vector heightens this separation of pitch and interval by tallying the interval- class content of a set in such a way as to make it impossible to infer from this list a pitch-class representation of the set. That is to say, interval vector can be derived from a set of pitch classes but not vice versa.

It is, however, possible to represent the intervallic structure of a set while preserving the relation pitch/interval or tone. For this two alterations to the interval-vector model are needed. First, our representation of interval must distinguish between the two tones. While each of the representations in Examples 4a-c do this, we will for the present take the most general, Exam- ple 4c, for a model. (For the sake of economy we will eliminate plus and minus signs from representations of interval type L, substituting for negative integers the complementary positive integers. Thus, in Example 4c, the intervallic attribute of the tone A will be called 9 and that of C will be called 3.) Second, our list of intervals will refer in turn to each of the tones of the set. Example 5a shows such a representation applied to two set classes, the Z-related tetrachords 4-Z15 and 4-Z29. After the letter name of each tone (which could of course be rewritten as an integer) are listed all the intervals

195



L that tone forms with the remaining tones of the set. (These numerals are rather arbitrarily listed in the ascending order of their corresponding interval class simply to facilitate certain comparisons.) Thus C forms the interval 11 with CI, 8 with E, and 6 with FR. (Recall from Ex. 4 that 11 here means 11 semitones above, 1 semitone below or any octave multiple of these intervals.) Again for economy, in this representation the intervallic comparison of a pitch class with itself has been omitted. Interval-class equivalents of these relations are listed in brackets in Example 5a to indicate a similarity between the Z-related sets. Note that two tones of 4-Z29 (C and DI) are constituted of the same interval-class relations as are two tones of 4-Z15 (C# and E). But since interval types are mixed in each case there is no more reason for positing an equivalence relation here than there would be in the parallel "operational" situations shown in Examples 2e and f.

While such a representation of set class is rather cumbersome, it con- tains a great deal of structural information and can for certain purposes be abbreviated. Since each tone is defined in relation to the whole, the quality of any tone can define the set. For example, since C# in relation to the other members of the set assumes the intervallic qualities 1, 9, 7, we can, from the perspective of Ct, generate the set by supplying the complements of these intervals: 11, 3, and 5 (C, E and F#). Similarly, the set may be transposed by transposing any of its constituents and reconstituting the intervallic relations of that constituent. From the intervallic associations of any tone of a set the intervallic associations of the remaining tones can be calculated by subtracting in turn each member of the given string of intervallic associations from every member of the string. This process can be represented in the form of a matrix in which columns and rows contain unordered collections as in Example 5b. Here the variable in stands for an intervallic asso- ciation of an unspecified pitch O representing the interval that pitch forms with itself. The cardinality of the set is represented by the variable n. Thus since any of the sets of intervallic associations can serve to generate all the others, every set class can be represented by a single string of n-1 integers (O being understood) where n is the cardinality of the set class.

This view of set class distinguishes inversionally related forms but

clearly allows for a special equivalence relation based on the principle of intervallic inversion. Inversion from this perspective appears as a rather Escheresque dialectic of "inside" and "outside." Comparing the set 4-Z15 of Example 5a with the inverted form shown in Example 5c, note that all intervallic relations are inverted. Example 5d illustrates this situation from the point of view of the individual tone: the intervals by which the other elements of the set constitute a particular tone become in the inverted form the intervals through which the "corresponding" tone contributes to the con- stitution of the others. This "inversion" of intervallic relationships, not sim-

ply of interval, is also illustrated in the matrices of Example 5b where inversion of the set results in the exchange of columns and rows. From

196



this perspective we can abstract from inversion the concommitant features of "directionality" (up/down) and axis which, like transposition, may or may not have a structural function in an actual musical instance.

In Example 5 intervals were calculated according to our most general model, L. We can discriminate more specific intervallic qualities by em- ploying the relations K and J to represent sets. In Example 6 a series of six comparisons is made. In each case a different form Y of set class 3-4 is compared to the fixed form X . While all the trichords in Example 6b display the same L-intervallic associations, only Examples 6a, b and c display the same K-intervallic associations, and, of these, only Example 6a maintains the same J intervals. K and J may be considered sub-classes of set equivalence, and indeed the qualitative differences posited by these distinc- tions are, I believe, perceptually quite clear (that is, to the extent that Exam- ples 6b and c as a group can be heard to differ from Example 6a and that Examples 6d, e and f as a group can be heard to differ from Examples 6a, b and c as a group). The various forms belonging to any of these equivalence classes are of course undifferentiated as members of that class. But since we are dealing here with three levels of abstraction it may be possible for the members of one of the classes to be ordered according to their degree of similarity as measured by a less abstract interval type. Thus, for example, while Examples 6b and c are undifferentiated in their K-intervallic associations, trichord Y in Example 6b shares two J entries with X whereas Example 6c shares none.

It is not clear how similarity should be defined in all cases. Presumably the "order of octave displacement" (the number of octaves by which an interval is expanded or contracted) would be needed for comparisons. Even with this refinement we cannot order, for example, Examples 6fY and 6gY in their similarity to X, nor does this distinction appear to have much aural justification. While such a ranking of similarity does not seem capable of exhausting all the possible registral forms of a set and may in some cases be of uncertain analytic value, nevertheless it does point to an important feature of our definition of set class-that rather than simply abstracting a single characteristic from an actual set of tones, the concept is capable of organizing to some extent the particular instances it generalizes. This capa- bility is of considerable methodological value since it creates a systematic whole from among the possible forms which a set class can assume. The analytic result is that we can compare different forms of a set class and judge to some extent their degree of relatedness, and, most importantly, that this judgment is based on the concept that unites them as a class.

For an example let us return to the first measure of DU Again we will assume that this music can be segmented as three instances of set class 3-4 labelled X, Y and Z in Example 7. In this example the three interval types are listed together in the form e ((L) (K) (J)) where e is an element of a set. In comparing sets, intervallic correspondences can be considered

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L: E (11 7) G(11 7) E(1 8) AS4(1 8) Ab(4 5) C(4 5) S

EI(ll 31) G(11 31) J: G(11 19*) J: G (23* 55*) 7 E (-11 20) Ab(-ll 20) Ab(-ll 8*) Ab(-23* 32*)

f A?(-20-31) C(-20-31) j b C (-8*-19*) C (-3 2 *-5 5 a - Ib

. C.

x y x y x y

K: E6(11 7) G(11 7) K: G(11 7) K: G (11 7) BE(-11 8) Ab(-ll 8) AbI(-1 8) Ab(-11 8) A (-8 -7) C (-8 -7) C (-8 -7) C (-8 -7)

J: G(-1* 31) J: G(23*"-5*) J: G(-1* -5*)J:G(-13*-29*) Ab(l* 32*) Al(-23* -28*) Ab(l* -4*) Ab(13* -16*) C (-32* -31) C (28* 5*) C (4* 5*) C (16* 29*)

Sd. e. f. g.

x y x y x y y

K: G (-1* 7) K: G(11 -5*) K: G(-1* -5*) K: G(-1* -5*) A6(1* 8) Ab(-ll -4*) A,(l* -4*) Ab(l* -4*) C(-8 -7) C (4* 5*) C (4* 5*) C (4* 5*)

*indicates lack of correspondence to X

Example 6

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"stronger" or more specific or concrete as they appear toward the left of the array. As we saw in Example 6e, X and Y are not closely related. If we recognize inversional equivalence of sets then, while Y and X share no J- intervallic associations, Z has one such interval in common with each. The structural significance of this sort of analytic observation will of course depend upon a much broader context than we have exposed here, but since the various domains represent discrete structural realms, it is advantageous to describe their "internal" organization before attempting to describe their interaction.10 The structural possibilities of any domain will of course be realized (or suppressed) in varying degrees by interaction with other domains. For example, notice that the J-intervallic connection of set Z which is least similar to the intervals of sets X and Y-G (-16), B (16)-is rhyth- mically and timbrally most clearly exposed in the second half of measure 1 of DU (See Ex. 1). Indeed there are many factors which in this first measure subvert the relationships shown in Example 7.

In the above examples we have explored the concept of set class through comparisons of different pitch sets. In this light, set class appears as a useful analytic instrument for disclosing similarities, both obvious and hidden, among different moments of a composition. The comparisons may be inter- preted to reveal transformations of sets or to indicate degrees of relatedness among sets of the same class. There is, however, another and perhaps primary meaning of the concept which has emerged in this inquiry and which we should now consider more explicitly.

The formation of a class may be thought of "extensively" as the aggrega- tion of objects which share some characteristic. The comparison of such objects reveals the unifying characteristic which encompasses all particular representations. This interpretation seems to be favored by the operational view. Here there is a tendency to require a comparison of sets for the estab- lishment of the class. Thus to speak of any representative of the class reference must be made to a particular form ("prime form" or "representative form") rather than to a general form which includes all the members of the class. In published analyses one often encounters various sorts of "priority," "referential" collections - terms which lacking systematic definition nevertheless show an attempt to establish some privileged form outside and above the proliferation of particulars. The need for comparisons and thus the need for replication of forms through which comparisons can be made raises questions concerning the structural significance (and perhaps even the existence) of a single instance of a set class. The position outlined above and its atten- dant difficulties stem from the absence of a prior definition of that "common characteristic" which unifies the particular representations and thus defines set class. Ernst Cassirer states this general problem very clearly:

. . it is evident that before one can proceed to group the elements of class and indicate them extensively by enumeration, a decision must be made as

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(RI) (RI)

x y z

L K J

Eb (111 7) ( 7) (1 31) ) x E (1 8) ( 8) (! 20) )

AM,(

(4 5) ( -8 -7)" i( -20-31))

G ((117) ( 11 -5)

.••

(23 9) y Ab( (1 8) ( -1 -4) (-23 28))

C ( (4 5) (4 5) , \(28 ) ) mI)I C ( (1 5) (-1 (?5) ( )

z B ((11 4) ( 4) ( 16) G ( (8 7) -4 5) ( -16 5))

(dotted lines indicate inverse relations)

Example 7

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to which elements are to be regarded as belonging to the class: and this question can be answered only on the basis of a class concept in the "intensional" sense of the word. What seems to hold together the members united in the class is that they all meet a certain condition which can be formulated in general terms. And now the aggregate itself no longer appears as a mere sum of individuals, but is defined by this very condition, whose meaning we can grasp and state by itself, without having to ask in how many individuals it is realized, or even whether it is realized in any individual at all."

The representation of set 4-Z15 shown in Example 5a is intended to provide such a general formulation. The four strings of integers are common to all (non-inversionally related) forms of 4-Z15. (Recall that this set class could be represented by any one of these strings since the remaining strings can be derived from it.) Here there is no privileged form, no replica with which to compare the set. While the term set class implies in its extension a limitless number of forms, it is misleading to view the primary function of the concept as uniting all these possible particulars as a collective. Set class may be viewed perhaps more significantly as an interpretation of the totality of intervallic relationships presented by a group of tones. Strictly speaking, a pitch (or pitch-class) set is a purely negative abstraction. It is a set of pitches from which all other determinations-order, duration, and so forth-have been removed (including register if a pitch-class set). It is the concept of set class which relates the set of pitches, and it is such relationship that makes tones of the pitches. A set considered in the light of the structure provided by set class may be regarded as the "tonal" context of the pitches. Such tones may enter into all sorts of other relationships, but their definition as tones is dependent exclusively upon the intervallic relationships which characterize the set. The function of set class thus appears as a means of penetrating the intervallic structure (however this may be defined) of a particular group of tones.

That there may be other groups which possess the same characteristics is from this perspective not of primary interest. Where our interest is directed toward the structure of a particular set we can use the term "set" to mean "representative of a set class."•12 Viewed in this way, the abstract concept of set class defines the range of possible (intervallic) relations from which a particular set is realized. Set class is thus a theoretical instrument for describing "tonal" relationships.

This formulation opens our investigation to the crucial analytic questions of what, in a particular musical context, these relationships might be, how they are constituted, and what rhythmic and formal functions they may serve. Here I hope to have laid a more secure groundwork for the investigation of these highly complex issues which far exceed the scope of the present essay.3 The analytic (and also systematic) implications of the descrip- tion of set class I have proposed suggest numerous avenues for future study.

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Familiar problems of similarity, inclusion, and complementation could be reconsidered in light of an intervallic definition of set class. Comparisons of sets representing different classes will reveal varying degrees of similarity among constituent pitches. Sets of quite different total intervallic composition may present close similarities among some of their constituents. Similarly, a consideration of the specific intervallic structure of pitch collections might show considerable disparity between sets of the same class. While the above perspective will put into question the significance of the complement relation, it could provide a measure of chromatic saturation- the chromatic would be completed or closed to the extent all twelve pitch classes assume identical intervallic associations. I believe one of the more interesting ramifications of the concept of set class sketched above is the emphasis given to individual tones in the context of sets. This change of perspective might in fact permit some degree of rapprochement between theories of post-tonal music and theories of tonal music, properly so called.

ko?t % %Ise*

202



NOTES

1. I wish to thank Allen Forte and David Lewin for their sensitive reading of an early draft of this manuscript and for their many valuable suggestions.

2. Lewin explicitly avoids this problem in the following definition: "By 'canonical transformation' I shall mean an operation on pc's which is understood in a given theoretical context, to transform (the total content of) any pcset into a pcset which is accepted as 'similar'in that context." (my emphasis). ("Forte's Interval Vector, My Interval Func- tion, and Regener's Common-Note Function," Journal ofMusic Theory 21 (1977): 195.)

3. I shall frequently refer the discussion of set class to aspects of 12-tone theory, particularly the notion of closure under the operations of transposition and inversion. While a treatment of unordered sets does not necessarily imply a connection with 12-tone theory, there is some methodological advantage to uniting these two theoretical domains in the present discussion, especially since many of the basic concepts of the theory of unordered sets are derived from serial theory. I believe some of the conclu- sions of this essay have special implications for the analysis of 12-tone music (considered as a "combinational system"), and I intend to pursue this topic in a future study.

4. The simplification of Example la shown in example 2a may be worth considering in more detail. Our ordering of the second trichord might be justified on the ground that Ab sounds before C and the G follows C at the dynamic level of mp. In the case of

Example la the transpositional operation E - Ab refers only to the second attack of Ab and yet the first Ab must be heard as the second if we are to maintain that order is preserved between these two "instances of 3-4." I draw the reader's attention to this problem only because the irony of this sort of situation has not been sufficiently appre- ciated in many analyses.

5. An analogy can be drawn to geometry if we substitute triangle for trichord. Given two angles we can construct any similar replica-of any size and anywhere in space. Or we can start from three points of a Cartesian metric and operate on the resultant figure (moving it about in space, changing its size, and so forth) by performing arithmetic operations upon the three point coordinates, except that these operations must be so limited that the resultant figure maintains the same angles; hence, a squaring of quantities is, for example, disallowed.

6. It should be noted here that Forte, in "A Theory of Set-Complexes for Music" Journal of Music Theory 8 (1964): 136-83, initially attempted to base the notion of set class on intervallic relations using the interval-class vector devised by Martino ("The Source-Set and Its Aggregate Formations," Journal of Music Theory 5 (1961): 224- 73.). Because of the problems introduced by the Z relation Forte subsequently aban- doned interval as the determinant of set-class membership. A different intervallic representation of set class was proposed by Richard Chrisman in "A Theory of Axis- Tonality for Twentieth-Century Music" (Ph.D. diss., Yale University, 1969). This representation which Chrisman calls the "successive interval array," later endorsed by Eric Regener under the name "interval notation" ("On Allen Forte's Theory of Chords," Perspectives of New Music 13/1 (1974-5): 191-212), succeeds in distinguishing Z-related sets (as does Forte's "prime form" from which it is derived) but does not explain the correspondence of interval vectors. I believe these difficulties stem from the notion of interval class (or, alternatively, "directed interval") and shall later propose a refinement to the interval-class vector which excludes the equivalence of Z- related sets.

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7. In an important article concerning set-class operations and intervallic relations, Robert Morris discusses transformations which map many of these Z-related sets. Through an illuminating investigation of the systematic difficulties of the Z-relation, Morris arrives at a set-group system, SG(vz), that collapses all Z-related pairs through a set of transformational operations. As Morris points out, this accommodation results in an extraordinary degree of abstraction, reducing, for example, Forte's fifty distinct hexachord classes to only three groups. (Morris, "Set Groups, Complementation, and Mappings Among Pitch-Class Sets," Journal of Music Theory 26 (1982): 101-44.)

8. Example 4d represents a different sort of relation than that of the series in Exs. a-c. Since this new perspective applied to Ex. c results in Ex. d, we could apply it also to Exs. a and b to bring the total number of interval types to 6 (2 x 3)-two interpreta- tions of three fundamental types. However, since it annihilates the distinction between the pitches or pitch classes which "form" the interval, interval class (our M) does not strictly belong to the set representation I am proposing. Note also that "directed interval," while it closely resembles interval-type L, defines interval in one "direction" rather then in both simultaneously as does type L.

9. The selection of the tones G, Ab and C to compose Y is essentially arbitrary, since we are concerned only with the autonomous intervallic structures of the sets X and Y. The possible intervallic relations between the sets of each pair and the correspondence of common subsets will not concern us here since these are really questions of alternative segmentations of the examples. Thus if we admit intervallic relations across the boundaries of the trichords or consider subsets within these boundaries we will have to consider more than one set class. This could of course be done, but it is preferable to avoid such complications at this stage of our investigation. I point this out to assure the reader that the artificial simplicity of these examples does not reflect a necessary limitation of the concept set class.

10. For a more detailed treatment of this issue see my "Segmentation and Process in Post- Tonal Music," Music Theory Spectrum 3 (1981): 54-73.

11. Ernst Cassirer, The Philosophy of Symbolic Forms, vol. 3, The Phenomenology of Knowledge, trans. Ralph Manheim (New Haven: Yale University Press, 1957), pp. 294-295.

12. This use of the term "set" corresponds to Allen Forte's practice in The Structure of Atonal Music (New Haven: Yale University Press, 1973). In his criticism of this usage, Regener (op. cit., p. 194) ignores the rather intricate relation of set and set class I have tried to sketch above.

13. Elsewhere I have attempted to confront these issues in some detail. I refer the reader in particular to '"A Theory of Segmentation Developed from Late Works of Stefan Wolpe" (Ph.D. diss., Yale University, 1978) in which the concept of intervallic asso- ciation is applied analytically.

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hasty - 1987 - an intervallic definition of set class

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