Sets, Invariance and Partitions

Abstract

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