Beat-Class Set Classes and the Power Group Enumeration Theorem

Abstract

The theory of beat-class sets originates in the work of Milton Babbitt, who demonstrates a correspondence between modular pitch-class spaces and metric spaces in the framework of total serialism. Later authors, particularly Richard Cohn, John Roeder, and Robert Morris, apply similar concepts to a variety of analytical situations, drawing on concepts and procedures from pitch-class set theory. In light of the correspondence between these theories, the universe of beat-class sets for a given modulus may be partitioned into equivalence classes similar to pitch-class set classes. This study investigates processes of enumerating these equivalence classes.We consider extensions to the theory of beat-class sets by including rhythms with more than one voice. Specifically, we examine equivalence classes of multiple-voiced beat-class sets using the Power Group Enumeration Theorem (PGET) of Frank Harary and Edgar M. Palmer. The PGET allows us to determine the numbers of equivalence classes of beat-class sets as determined by various groups of transformations: metric shift, retrogradation, and voice permutation, among others. Our results have implications for further applications in pitch-class set theory, serial theory, and transformational theory.