Crystals of Sound: Applying the Physics of Phase Transitions to Musical Intonation
Ryan Buechele, Alex Cooke, Jesse Berezovsky · Empirical Musicology Review · 2025
We explore a physics-based model describing systems of musical intonation. Under the assumption that tuning systems seek to balance minimization of dissonance with maximization of compositional variety, we can use the same methods that are used to describe how transitions between the phases of matter are governed by a balance between minimizing energy and maximizing entropy. In both cases, a parameter – temperature – controls the balance between these factors. We show that as the temperature is raised from low to high, our model generates tuning systems that closely resemble those used throughout the history of Western music, from just intonation, to meantone temperaments, to equal temperament. We demonstrate how our model reflects the way in which tuning systems evolved along with trends in composition by comparing the results of our model vs. temperature to a corpus of 9620 pieces vs. year of composition from 1568-1968. By changing parameters in the model, we also find that other tuning systems, such as 31-tone temperaments, emerge. These results provide a new lens for understanding tuning systems, and a new approach for generating novel systems of intonation.