Sets and Nonsets in Schoenberg's Atonal Music

Abstract

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