The broken symmetry of music: applying statistical physics to understand the structure of musical harmony

Despite myriad musical systems and styles, certain characteristics are nearly universal across cultures and throughout history, including a restriction to a discrete set of sound frequencies (pitches). In this talk, I will present a bottom-up approach to a theory of musical harmony, starting from two basic (and conflicting) principles: a system of music is most effective when it 1. minimizes dissonant sounds, and 2. permits sufficient complexity to allow the desired artistic expression. By quantifying these principles and assuming a parameter (temperature) that specifies the balance between them, the problem directly maps onto standard statistical mechanics [1]. A mean field treatment reveals phase transitions from disordered sound to ordered phases with distributions of pitches that closely match musical tuning systems used throughout the history of both western and non-western music. A numerical model with nearest-neighbor interactions displays the behavior of an XY system, including the appearance of topological defects following a quench. These defects, arising from the Kibble-Zurek mechanism, are interpreted as chords, with their arrangement reflecting a system of harmony.

[1] J. Berezovsky, Science Advances, 17 May 2019: Vol. 5, no. 5, eaav8490