Set.

An unordered collection of associated musical elements, usually pitch classes.

JOHN ROEDER

1. Sets as representations of musical structures.

The concept is based on the representation of music as discrete events characterized by properties such as pitch, duration, timbre and onset time. This representation is consistent with musical notation and with practical descriptions such as ‘the tuba’s low E’ or ‘the loud harmonic in bar 6’. In this view, the similarity of events in some of their properties associates them into structures, while dissimilarity distinguishes events within and between those structures. For example, a chord is composed of events that have the same onset time but differ in pitch, and a melodic motif is composed of events that are played consecutively, usually by the same instrument. Chords and motifs are specific instances of more general types known as ‘segments’. Examples of these types are indicated by circles in exx.1 and 2, which present characteristic passages of tonal and atonal music respectively. The central columns of Tables 1 and 2 indicate the shared properties that associate the events in each segment.

Although events may be identical in some respects, they necessarily differ in at least one property. In a set, associated events are represented by distinct values that denote their differences in one specific property. These values are listed within set brackets, {}, to indicate their segmental coherence. There are as many set representations of a given segment as there are properties in which its events differ. For example, various set representations of the segments in exx.1 and 2 are given in the right column of Tables 1 and 2. If two events in a segment are identical in some property that otherwise distinguishes events in the same segment, then both events are denoted in the set by a single element. In segment 1(e), for instance, several distinct events have the same pitch class, E, so in the pitch-class-set representation of the segment, {A,E}, the element E denotes all of them.

Representing an ordered segment as an unordered set of values discloses a fundamental identity that the segment shares with all other segments of events represented by the same values. For example, the segments (a), (b) and (c) in ex.1 are represented by the same sets of pitch classes, despite their many differences; and segments (b) and (e) can be represented by the same sets of onset times and durations, despite their complete difference in pitch.

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2. Historical background of set theory.

Although a disregard for order seems antithetical to a temporal art, it is in fact central to many important theories of music. Both the modal classification system, which attributes the same mode to different orderings of the same pitches, and theories of tonal harmony and figured bass, which assert the functional equivalence of different registral orderings of the same pitch classes, can be framed as theories of musical sets. In such theories, the unordered representation of ordered segments supports the recognition of unity among apparently diverse musical structures, and leads to the discovery of processes – such as root progression – that cannot be defined in less abstract representations.

Musical developments of the 20th century, however, stimulated a more explicit development of set theories. Precedents may be found in compositional treatises by Hauer, Hába, Schillinger and others, which recast the harmonic definition of chord inversion as a registral permutation of scale degrees, making it possible to catalogue all possible combinations of notes from any scale, even chromatic or microtonal. (In contrast, Hindemith’s classification scheme asserts that the majority of chords cannot be inverted.) Composers such as Messiaen extended the concept of mode to embrace sets of durations and intensities. A crucial contribution was Babbitt’s introduction of algebra and number theory to model 12-note rows and operations. The idea that the structural relations central to the 12-note system could form a basis for analysing pitch structures in the pre-serial, atonal music of Schoenberg and his contemporaries was subsequently developed by Perle and Forte, leading to more rigorous theoretical formulations by Rahn, Morris and Lewin. The theory has influenced the practice of some composers, notably Elliott Carter.

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3. Synopsis.

Pitch-class set theory treats the properties and relations of unordered collections of pitch classes. The basic properties of any set are its content and its cardinality, which is the number of elements it contains. A dyad is a set of two pitch classes, a trichord a set of three, and so on through tetrachord, pentachord, hexachord, heptachord and octachord. The set of all 12 pitch classes is called the ‘aggregate’, and the ‘complement’ of a pitch class set S is the set of all members of the aggregate that are not members of S.

The simplest content relation of pitch-class sets is expressed by their intersection, a set that contains the pitch classes they have in common. For example, in ex.2 segments (b) and (h) intersect in the trichord {D,G,C}. This intersection creates continuity between successive segments, just as common notes link harmonies in tonal progressions. The greater the cardinality of the intersection, the more alike are the contents of the sets. The closest content relation arises when one set includes all the pitch classes of the other; in this circumstance the smaller set is called a ‘subset’ of the larger set, which is called the ‘superset’. Inclusion relations help account for the similarity of segments that use many but not all of the same pitch classes. For example, segments (j) in ex.2 includes the same trichord, {D,G,C}, that is the intersection of (b) and (h), and segment (e) includes a set, {B,F,E,B,D}, that is also a subset of the concluding segment, (i). Large sets, such as (e), can be understood as unions of smaller sets. In the music of the Second Viennese School, even before their 12-note works, the largest set of all – the aggregate – is formed regularly in this manner. As shown in ex.2, an aggregate results from the union of the first, essentially non-intersecting sets (a), (c), (d) and (g); at the end of the excerpt, the low strings also build up a larger set, creating closure by completing the aggregate at the last chord.

Another property of a pitch-class set is its interval-class content, which tallies how many of each interval class are formed by the set’s dyadic subsets. This property makes some sets quite distinctive. Augmented-triad sets, for example, contain only interval class 4, while all other trichords contain at least two different types of interval class. The diatonic-scale set is also distinctive: it contains more instances of interval class 5 than any other heptachord. Equality of interval-class content constitutes another basis, besides inclusion, for relating sets. For example, it accounts for the similarity we attribute to different inversions of C major and F minor triads, which include no common pitch classes. To quantify the relation of sets that have different interval-class content, various measures of similarity have been proposed. They show, for example, that segment (c) in ex.2 is very similar to the last cello chord, (k), because five out of the six interval classes in the corresponding sets are the same (both sets contain two interval-class 1s, two 5s, and one 6).

Other pitch-class-set relations of compositional and analytical interest arise from considering pitch-class transformations such as transposition and inversion. If pitch classes are represented by numbers, these transformations may be formulated as arithmetic operations that possess an algebraic group structure; another less familiar but formally similar transformation is multiplication by 5 and 7. The transformation of a given pitch-class set S is defined as the set of pitch classes that result from applying that transformation to each of the pitch classes in S. Of course, set transformations can represent the relations of segments that are clear pitch-transpositions or pitch-inversions of each other, such as the cello chords in the second bar of ex.2, but they can also reveal more abstract relations among segments with different registral or temporal orderings. For instance, the trichord {B,C,F} that represents the bassoon-clarinet chord (f) is an inversion of the trichord {D,G,G} that represents the preceding, identically orchestrated segment (a), even though the segments are not related by pitch inversion. Transformational set relations are reinforced by other relations; for instance, sets that are transpositions or inversions of each other always have the same interval-class content, although the converse is not always true.

Although two sets are equal only if they contain the same pitch classes, two different sets are considered ‘equivalent’ if one is a transformation of the other. According to the group structure of the mathematical model, each type of transformation – transposition, inversion and multiplication – induces an equivalence relation among sets. Two sets belong to the same transformational equivalence class (or ‘set class’) if they are equivalent under that type of transformation. For example, since all major triads are transpositionally related, they belong to the same transpositional equivalence class; all minor triads belong to another transpositional class; and the sets {C,D,F} and {D,E,G} belong to yet another transpositional class. Under transposition and inversion together, however, all major and minor triads belong to the same equivalence class, while the other two sets still belong to another class. Under transposition and inversion together with multiplication by 5, all these sets are equivalent. The formal definitions of set class make it possible to name the type of any set without reference to possibly inappropriate tonal-harmonic descriptions.

One branch of pitch-class-set theory distinguishes set classes by the degree to which their sets are invariant (do not change their content) under transformation. Common-tone theorems relate the interval-class content of a set, and the pairwise sums of its pitch classes, to the cardinality of the intersection of the set with its transpositions and inversions respectively. Some types of set, called ‘symmetric’, are completely invariant under certain transformations. However, many hexachords, as well as most smaller sets, can be transformed into sets with entirely different pitch classes. A set that can be combined with transformations of itself to form the 12-note aggregate is said to possess the property of ‘combinatoriality’.

Many of the connections linking inclusion, complementarity, interval-class content and transformation have been investigated in the literature on set theory. For example, the precise relation of the interval-class contents of complementary sets is expressed by the generalized hexachord theorem. ‘Z-equivalence’ denotes the relation of sets that have the same interval-class content but are not related by transposition or inversion. Forte has proposed ways of grouping set classes themselves into larger set complexes and set genera on the basis of abstract inclusion relations, which obtain between two sets if one includes a subset that is transformationally equivalent to the other. From another standpoint, Lewin has shown that inclusion and interval content are specific instances of more general functions of sets, and has proposed an ‘injection function’ that subsumes these relations as part of a generalized set theory.

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4. Meaning.

Pitch-class set theory can be a compositional resource. A knowledge of the properties of a given set type – its interval-class content, symmetry, combinatorial potential, invariance and its subsets and supersets – may suggest specific processes and forms for which the set is well suited. For example, invariance properties such as combinatoriality are especially pertinent to 12-note serial composition and to non-serial textures that consistently feature the aggregate.

On the other hand, much of the theory has been designed to support analysis, especially of the problematic atonal repertory. The set relations remarked above in connection with ex.2 exemplify some of the basic types of analytical observations enabled by the theory: equivalence relations may be brought to bear in demonstrating the motivic unity of a piece that presents various orderings, transpositions and inversions of a few pitch-class sets, and the more abstract similarity and inclusion relations may be cited in demonstrating the coherence of a piece that exhibits a diversity of set classes.

Critiques of pitch-class-set analysis focus on the fundamental identity of sets, on segmentation and on the meaning of the more abstract set relations. In theory, the same set, characterized only by its pitch-class and interval-class content, can represent segments that differ greatly in rhythm, in registration and in the specific ordered intervals that are emphasized. The motivation for such an identification, and its benefits, are most evident in analyses of music that shuns overt repetition of pitch and interval structures. In cases where this abstract representation is appropriate, the analyst must strike a balance between identifying segments purely on the basis of rhythmic and textural cues and minimizing the number of set types across the entire composition; the task is clouded by the difficulty of determining how events associate in the highly variable textures of non-repetitive atonal music. Lastly, as with most abstractions, there is disagreement among scholars about the degree to which equivalence, inclusion and similarity relations are audible or analytically pertinent.

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5. Other types of set.

One sign that these difficulties are not insurmountable is the continuing development of new set theories for composition and analysis. For example, set models of the diatonic system as a seven-pitch-class aggregate have revealed the special structural properties of its triadic subsets. An exploration of the properties of the diatonic heptachord with respect to the 12-note aggregate has stimulated the discovery of similar sets embedded within aggregates with greater and lesser cardinalities. In the domain of rhythm, as well as that of pitch, the work of Babbitt has been seminal. Many of his compositions rhythmicize events in a way that is analogous to 12-note composition: for example, by placing every attack on one of the beats of a 12-beat mensural unit, so that the series of attack time-points in each rhythmic segment constitutes an ordering of all 12 possible beats. This system has fostered a theory of beat-class sets for analysing the content and form of Steve Reich’s ‘phase-shifting’ music; each repeated rhythmic pattern is represented by a set of beats that are attacked in the mensural unit, and the shifting of a pattern with respect to the notated bar is modelled as a temporal transposition of the set. Future directions in set theory may take into account Lewin’s model of intervals and transformations, which provides a very general framework for theorizing sets of time-points, durations, pitches and more complex elements.

See also Analysis.

Set

BIBLIOGRAPHY

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A. Hába: Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel-, und Zwölftel-Tonsystems (Leipzig, 1927)

P. Hindemith: Unterweisung im Tonsatz, i: Theoretischer Teil (Mainz, 1937, 2/1940; Eng. trans., 1942, 2/1948)

J. Schillinger: The Schillinger System of Musical Composition (New York, 1941, 3/1946/R)

O. Messiaen: Technique de mon langage musical (Paris, 1944)

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M. Babbitt: ‘Twelve-Tone Invariants as Compositional Determinants’, MQ, xlvi (1960), 246–59; repr. in P.H. Lang, ed.: Problems in Modern Music (New York, 1962), 108–21

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M. Castrén: RECREL: a Similarity Measure for Set-Classes (diss., Sibelius Academy, Helsinki, 1994)

J.W. Bernard: ‘Chord, Collection, and Set in Twentieth-Century Theory’, Music Theory in Concept and Practice, ed. J.M. Baker, D.W. Beach and J.W. Bernard (Rochester, NY, 1997), 11–51