Phase transitions, critical behavior, and emergent order in systems of musical harmony*

The emergence of order in a thermodynamic system can be understood as a temperature dependent trade-off between minimizing energy and maximizing entropy of the system. We posit that the ordered arrangement of musical pitches that constitute a system of musical harmony arises analogously from a trade-off between minimizing the perception of dissonance and maximizing the variety of compositions possible within the system. We show that by quantifying these factors, a system of harmony can be formulated as an XY model governed by an effective free energy. Methods from statistical mechanics can then be applied that serve to minimize free energy, reproducing familiar structures of Western and non-Western harmony. In the mean field approximation, phase transitions occur from disordered sound to ordered distributions of pitches, including 12-fold octave divisions used in modern and historical Western music, as well as those used in several other musical traditions [1]. Numerical simulation of quenched tones on a 3D lattice shows a transition from disordered sound to the 12-fold octave division with frozen vortex strings as predicted by the Kibble-Zurek mechanism. These topological defects can be interpreted as musical chords, and the branching network of strings as chord progressions. These results provide a new approach for understanding, appreciating, and composing music from first principles.

References:

[1] Berezovsky J. "The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory." Science Advances. 2019; 5(5). doi: 10.1126/sciadv.aav8490.