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In general, I shall use the terminology of the cited article. Briefly: X,Y,Z etc. will denote pcsets, that is, unordered collections of pitch sets. Individual pitch classes will be denoted by lower case letter names c, c#, d etc., and also by certain letters such as x,y and z for variable or unknown pcs in formulas and arguments. A,B, etc. will denote members of the group of canonical transformations, those operations on pcs which transform any pcset X into a pcset accepted as "equivalent" in a given abstract context. A pcset Xl is "a form of" a pcset X2 if there is some canonical transformation A such that Xl = A(X2); in this case X2 will also be a form of X1. The "chord type" of X, denoted by /X/, is the family of all forms of X. The number C(j,n) is the number of distinct non-ordered combinations of j things that can be extracted from a set containing n things.

Babbitt 1 has formulated the highly suggestive notion of considering a twelve-tone (or other species of) row as a family of what one might call "protocol pairs" (x,y). Each such pair might be interpreted as a rule: "do not state y before stating x." From such a point of view, one characteristic feature of serial structure is that, given a specified row and distinct objects x and y, either (x,y) or (y,x) will be a protocol pair for the row, but not both. Babbitt then goes further to discuss some other protocol systems, ones in which, given a distinct x and y, neither (x,y) nor (y,x) need necessarily be a protocol pair. Figure 1 below symbolizes one such system.

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.

A: Examples and discussion SEGMENTAL relations between different forms of a row can and often do play an important constructive role in much twelve-tone music. In any of Schtnberg's "hexachordal" pieces, for instance, many significant consequences of the hexachordal construction of the rows are apparent upon even the most cursory examination of the music. The row 00 of the Violin Concerto Op. 36 is of typical hexachordal structure (Ex. 1): there is an inverted form, Io, such that the segment consisting of the first six pitch-classes of 00 is also a segment (the last six pitch-classes) of Io; likewise, the segment consisting of the last six pitch-classes of 0o is also a segment (the first six pitch-classes) of Io. 00 I I I I

T H E D RAM A TI C idea of the work hinges on the paradoxical nature of God: the Unvorstellbares that commands itself to be vorgestellt. The musical metaphor that reflects (or better defines) the dramatic idea is the nature of the twelve-tone row and system as "musical idea" in Schoenberg's terminology. "The row" or "the musical idea" is not a concrete and specific musical subject or object to be presented for once and for all as referential in sounds and time; it is, rather, an abstraction that manifests itself everywhere ("allgegenwaertiger") in the work. And yet it can only be perceived, or realized, by means of an aggregation of specific Vorstellungen, even Darstellungen. Or, more exactly, the composer may perceive it as a sort of resonant abstraction, but it remains unrealized and unfulfilled until it is manifested through performance and communicated to an audience by means of material sounds, representing the idea in all its manifold potentialities. In this connection, the multiple proportion-God: Moses: Aron: Volk equals "the idea" (row): composer (Schoenberg): performer: audienceis suggestive. Moses, like Schoenberg, perceives directly and intuitively a sense of divine ("pre-compositional") order. He cannot communicate this sense directly, however. As he suggests in Act I, Scene 1, he would much prefer to spend his life in simple contemplation of this order. But God commands him to communicate it ("Verkuende!") and he is powerless to resist. God demands that His order be communicated to the Volk. Yet how

CLASSICAL twelve-tone practice involves the functional interrelationship of two abstractly independent structural principles. One is serialism, that is, the use of a fundamental succession of "things" (no matter what) through time, together with various derived successions. Elsewhere,' I have attempted to begin an investigation of the way in which this practice exerts a structuring influence through time. The other is what we might, following Babbitt,2 call permutational total chromaticism. One means, by "permutational," that the primary structural focus is on configurations of all twelve possible pitch-classes (in time), and on the relations among such configurations; this as opposed to primary structural focus on combinations of some of the twelve pitch-classes. Such combinations are to be regarded as taking structural significance from their relations to total configurations, rather than as possessing inherent structural meanings. The use of a twelve-tone row imposes a definite structure on the total pitch-class chromatic. But the total chromatic can also be somewhat structured intrinsically, in a number of different ways. In particular, there are twelve ways of inverting the total chromatic into itself; one can fix any one of these inversions and regard the total chromatic as "balanced" with respect to that inversion. Thus, e.g., the inversion that leaves D fixed also leaves Ab fixed, and balances C# with Eb, C with E, B with F, etc. The inversion that interchanges C with C# also interchanges F# with G; it balances B against D, Bb against Eb, etc. We may think of these pitchclass inversions as possessing "axes"; each may be regarded, also, as having a pair of antipodal "centers." (See Exx. 1 and 2.)