An Application of Burnside's Theorem to Music Theory

Introduction

Isihara and Knapp [1993] introduce fundamental ideas in an area of music theory that musicians refer to as set theory. At least one reader of that UMAP Module was inspired to do a more advanced treatise that enumerates mosaics [Fripertinger 1999]. Also, a very active community investigating the connections between mathematics and music has developed. In fact, there is a newly formed Society for Mathematics and Computation in Music, which publishes the Journal of Mathematics and Music devoted to exploring those connections; and recently, Science published for the first time an article about music theory [Tymoczko 2006].

We discuss an aspect of this theory that is not discussed by Isihara and Knapp but that is a topic of continuing interest to music theorists [Hook 2007], namely, how to count the number of distinct set classes. We emphasize using Burnside's Lemma (Neumann [1979] recounts its history) for doing the counting. This approach easily extends the results to include counting set classes of pitch class multisets (pitch class sets with repeated entries). Since pitch class sets that are members of the same set class sound similar [Morris 1991], composers want to know how many distinct set classes there are for a given size of pitch class set, since this number would describe the number of different sound combinations available (for more on composing withpitchclasssets, see Morris[1987]). Thereaderfamiliarwithelementary group theory and combinatorics should be well-prepared for this paper.

First, we give a brief introduction to some musical terms and operations to set the stage. Next is a brief introduction to group actions and Burnside's Theorem. Following this material is a development of the invariance properties of transformations introduced earlier. Once the invariance properties are established, we devote some attention to some examples of Burnside's Theorem in action, and an extension to orbits of multisets. The last section has some concluding thoughts.