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 from an analogous trade-off. 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. Numerical simulation of quenched tones on a 3D lattice shows a transition with a divergence of correlation length and relaxation time, consistent with the expected critical exponents for a 3D XY model. Furthermore, the resultant topological defects, frozen vortex strings, are 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