Pitch-class Sets Research Articles

Use of the Exclusion Relation to Profile Pitch-Class Sets

In the development of pitch-class set (pc set) theory, there is no concept more central than that of inclusion. It is the basis of Forte's setcomplex and plays an important role in recent investigations by Lewin, Morris, and Rahn.1 Taking a different (or complementary) approach to the problem of characterizing and comparing pc sets, I shall explore here the exclusion relation, which in many cases yields sharp profiling of pc sets and provides an alternative perspective for judging the similarity of pc sets of the same or different cardinality.2 My discussion uses the following notational conventions:

Set Groups, Complementation, and Mappings among Pitch-Class Sets

Set Groups, Complementation, and Mappings among Pitch-Class Sets

Relationships of Symmetrical Pitch-Class Sets and Stravinsky's Metaphor of Polarity

Stravinsky's well-known remarks on polar in Poetics of Music delivered in French in 1939-40 at Harvard University concern both music in general and his own works: Our chief concern is not so much what is known as as what one might term polar of sound, of an interval, or even of a complex of tones... All music being nothing but a succession of impulses and repose, it is easy to see that drawing together and separation of poles of in a way determine respiration of music. In view of fact that our poles of are no longer within closed system was diatonic system, we can bring poles together without being compelled to conform to exigencies of tonality.' Stravinsky's metaphors of are suggestive, yet puzzling, for according to Berger, polarity is a word which cannot accurately be applied (as [Stravinsky] applies it) to one thing without its opposite.2 However, in context of his works this idea of poles of attraction, or polarity, may be correlated with relationships of paired, transpositionally-related pitch-class structures. Moreover, his metaphor of the drawing together and separation of poles of attraction corresponds to changes in transposition relationships of paired sets or of paired intervals within a set. These changes often feature a developmental expansion and contraction of transposition distance of paired elements. Implicit in idea of a pole is dynamic interaction of a pair of equivalent structures (e.g., either extremity of an axis of a sphere, or either of two terminals of an electric cell or ends of a magnet). The idea of has suggested a gravitational pull or a tonality by assertion to several writers3 as well as an equilibrium often discussed as a harmonic principle (e.g., Pousseur's idea of maintained tension4). In characterizing tritone relationships of Petrouchka chord Berger has proposed that a necessary condition of polarity is the denial of priority to a single pitch-class precisely for purpose of not deflecting from priority of a

Some Intervallic Aspects of Pitch-Class Set Relations

Pitch-class sets are a necessary tool for the analysis of non-tonal music. They provide an organized means of classifying harmonic structures (or any other pitch combinations), but they also pose a problem, one which can most easily be exposed through a tonal analogy. In Example 1 a, the tonal analyst would recognize the structural connection between the first and third chords and describe the second chord as a secondary pitch structure which is linear in origin. This judgment is both facilitated and supported by the lack of a tonal name (for example, triad or seventh chord) for the second chord. In Example ib, the third chord, taken out of context, has a name (minor triad), but in context this is not its meaning. It too is linear in origin and to place the Roman numeral iii beneath it would be erroneous. In the analysis of non-tonal music, pitch-class sets allow us to assign a name to every conceivable pitch combination. Cases like Example l a, therefore, do not arise. Every non-tonal chord succession is comparable to Example ib, but because we lack the structural criteria of functional harmony, we run the risk of considering only the name of a pitch-class set at the expense of its contextual role. How, then, can we evaluate the structural significance of different pitch-class sets in a composition? Such evaluations are invariably based

Ideas of Order in Motivic Music

It seems as if we've finally begun to figure out how to get into those mysterious houses built just after the turn of the century by Schoenberg and Webern. To say that we've found the keys does not, however, represent an apt way of narrowing the metaphor, since, for better or for worse, they built these houses without doors. At best they may be said to have painted door-like signs on their exteriors, or, in some cases, to have graced their facades with doors grouped in threes and fours, leading not to the inside but instead to one another and back out. The use of keys is not recommended in connection with attempts to penetrate doorless, lockless houses, even if these be the well-worn master keys with which much success has been had in the past. Entry is gained only by dismantling part of the structure, for which task keys are not the implement of choice. New tools are needed to do this kind of work: powerful precision tools which can be modified to meet the demands of particular jobs. The tools developed in recent years by people who care deeply about motivic or, as it is often called, atonal music do appear to be tools of the required sort. With respect to power, precision, and flexibility, the set-theoretic concepts developed by Babbitt, Lewin, Perle, Martino, Howe, and, most notably, by Forte constitute an avenue of approach to this music which is impressive in its potential. At the same time, the nascent overview of what this music is all about, one emerging as a result of initial attempts to apply these concepts, is not an encouraging one and tends to undermine the positions of those who regard this repertoire as a musical legacy of the first rank. The problem as I see it resides not in the tools but in the manner of their application. Precision equipment for penetrating facades it not designed to be hurled like a demolition ball at the objects it is meant to transform. One is better off, on the whole, with crude tools employed with subtlety than with a refined technology for which one lacks the appropriate strategies. In my view, misuse of the new technology for dealing with motivic music has taken the form of attempts to apply all of the relations defined in the theory of unordered pitch-class sets to each an every analytical context. Instead of regarding the theory as a universe of possible relations and trying to determine criteria for deciding which relations are actually germane to

A Similarity Index for Pitch-Class Sets

The last twenty-five years have seen the rise of several systems of classification of unordered sets of pcs (henceforth, sets) for both compositional and analytical use in music.2 Most of these systems involve the explicit or implicit notion of the set-class (SC) which breaks up the entire corpus of 4,096 sets into a smaller repertory of set-types, each containing many sets. The criterion of membership in a particular settype or SC is often that all included sets are related to one another by Tn, I, or both. One reason for using such an inclusion rule is that all the sets within a SC will be identical from the point of view of interval-class content since any ic (interval-class) is invariant under Tn and/or I. Presumably, sets that are so related will be aurally as well as logically related somewhat independently of their compositional realization. It is therefore rational to assume that compositions will derive sonic and syntactic unity from the use of sets selected from the same SC.

A Response to a Response: On PCSet Relatedness

Morris, R. 1979. A Similarity Index for Pitch-Class Sets. Paper read at the National Conference of the Society for Music Theory, 1-4 November 1979, New York, N.Y. Perle, G. 1977. Twelve-Tone Tonality. Berkeley: University of California Press. Rahn, J. 1975. Gentle Reminder #1: Two Common-Tone Theorems. In Theory Only 1, no. 2: 10-12. 1980. Basic Atonal Theory. New York: Longman, Inc. Regener, E. 1974. On Allen Forte's Theory of Chords. Perspectives of New Music 13, no. 1: 191-212.

Sets, Invariance and Partitions

This paper examines certain fundamentals of twelve-tone theory and stems from previous attempts to deal with its higher-level aspects. I develop here a calculus of unordered pitch-class sets. While unordered sets have been studied before, notably by Hanson,' Martino,2 and Forte,3 the present study considers different aspects of the problem and emphasizes modular arithmetic as a basis of twelve-tone operations. Much recent whether it is twelve-tone, less-thantwelve-tone, collection music, or something similar, has been based on a small, conceptually simple set of operations, which have been used differently from work to work. Such operations can be considered axiomatic, whereas a particular set or row occurring in a specific work is the result of a unique compositional process, and acquires significance specifically from its context. It is perhaps meaningful to say that such an only possesses attributes which are demonstrated by the transformations applied to it in those contexts. I find it both fruitful and intuitive to conceive of an operation-object duality, in which operation is a concept subject to general discussion, while object arises from the discussion of specific

Sets and Nonsets in Schoenberg's Atonal Music

A consideration of Schoenberg's atonal music ought to refer to a theoretical framework for atonal music in general. The present article, however, will not set forth such a framework, but will focus, instead, on an elementary problem in Schoenberg's atonal oeuvre,1 that of distinguishing compositional units or sets from secondary formations or nonsets.2 In order to concentrate on this aspect of Schoenberg's music and to avoid digressions into other interesting aspects, no effort will be made to show the function of sets in the music. No complete analyses will be presented, although every musical example quoted is extracted from a complete analysis-and, indeed, could not be convincingly and confidently presented to the reader otherwise. There will also be no comparison of Schoenberg's atonal and twelve-tone procedures. Schoenberg's atonal compositions exhibit a remarkable and innovational kind of structuring, one that cannot be characterized simply, although a matrix of two or more dimensions might serve as an adequate model. With one exception, this aspect of Schoenberg's music has not been documented in the literature.3 The components of these musical structures are pitch-class sets containing from four to eight elements, usually. For any given composition the stock of sets that have structural significance (to be discussed below) is relatively small. As some indication of number, Herzgewiichse, Op. 20, is constructed of exactly nineteen sets and their complements.4 An extreme case is the fourth of the Five

Pitch-Class Sets in Fourteen Measures of Mozart's Jupiter Symphony

C. pitch-class (pc)-all pitches whose frequencies are related by powers of 2 belong to one pc, and only those pitches whose frequencies are so related belong to that pc. There are twelve distinct pc's: c, ct, d, d, e, f, f#, g, g#, a, a#, b or their enharmonic equivalents. These may be represented by the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Note that integers which belong to the same equivalence class, mod 12, represent octave-related pitches. This shows the principle of octave equivalence. If pc's are not the same, they are distinct.