Musical rhythm and meter as an ordered phase of sound

Rhythm and meter appear in the music of virtually all human societies. The temporal patterns of events associated with rhythm and meter can be viewed as a reduced symmetry state as compared to non-musical sound. Motivated by this, we apply methods from statistical mechanics to explore how musical rhythm and meter can emerge from fundamental assumptions about human psychology. Following a similar approach to our previous work on musical harmony [1], we posit that the temporal ordering of events in music is governed by two conflicting factors: (1) a desire to perceive repeating patterns, and (2) a desire for novelty and variation. We map these factors to the (negative of) internal energy and entropy of a system, respectively, where the balance between them is set by the temperature, via a free energy. As a function of temperature, as well as a chemical potential that governs the average concentration of events in time, we observe phase transitions occurring between disordered, Poissonian events, and ordered patterns of events that closely reproduce familiar musical meter. As an initial demonstration, we use our model to predict the distribution of note lengths as a function of temperature and chemical potential, and find good agreement with the range of distributions found in the 36 movements of J. S. Bach’s Suites for Solo Cello.

[1] Berezovsky, J., 2019. The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory. Science advances, 5(5), p.eaav8490.