Using Group Theory to Explain Symmetries in Classical Music

Abstract

This project explores the symmetries present in classical music and expresses them using group theory, with a focus on group actions. Mathematical objects and techniques from post-tonal music theory––such as transformation groups, pitch classes, and pitch-class sets––are applied to describe aspects of tonal music theory. While post-tonal theory traditionally uses these tools in a twelve-tone context, this project adapts them to the tonal framework of classical music, demonstrating how similar mathematical structures can capture different kinds of musical organization. This paper defines group actions relevant to symmetries in classical music and analyzes them using concepts such as orbits, stabilizers, and subgroup actions. Examples from classical pieces illustrate these concepts. Rather than the twelve-tone model commonly used in post-tonal theory, this project develops and applies a diatonic model for classical music, along with its corresponding diatonic symmetry group.