Wednesday, May 22, 2019

The statistical mechanics of musical harmony

Jesse Berezovsky, The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory, Science Advances 17 May 2019: Vol. 5, no. 5, eaav8490 DOI: 10.1126/sciadv.aav8490

Music, while allowing nearly unlimited creative expression, almost always conforms to a set of rigid rules at a fundamental level. The description and study of these rules, and the ordered structures that arise from them, is the basis of the field of music theory. Here, I present a theoretical formalism that aims to explain why basic ordered patterns emerge in music, using the same statistical mechanics framework that describes emergent order across phase transitions in physical systems. I first apply the mean field approximation to demonstrate that phase transitions occur in this model from disordered sound to discrete sets of pitches, including the 12-fold octave division used in Western music. Beyond the mean field model, I use numerical simulation to uncover emergent structures of musical harmony. These results provide a new lens through which to view the fundamental structures of music and to discover new musical ideas to explore.


The ubiquity of music throughout history and across cultures raises a fundamental question: Why is this way of arranging sounds such a powerful medium for human artistic expression? Although there are myriad musical systems and styles, certain characteristics are nearly universal, including emergent symmetries such as a restriction to a discrete set of sound frequencies (pitches). Historically, the theory of music has followed an empirical top-down approach: Patterns are observed in music and generalized into theories. Recent work has aimed to generalize these generalized theories to uncover new potential patterns that can lead to new theories of music (1–3). Here, instead, we observe patterns that emerge naturally from a bottom-up theory. We start from two basic (and conflicting) principles: A system of music is most effective when it (i) minimizes dissonant sounds and (ii) allows sufficient complexity to allow the desired artistic expression. Mathematical statement of these principles allows a direct mapping onto a standard statistical mechanics framework. We can thereby apply the tools of statistical mechanics to explore the phenomena that emerge from this model of music. Just as in physical systems where ordered phases with lower symmetry (e.g., crystals) emerge across transitions from higher-symmetry disordered phases (e.g., liquids), we observe ordered phases of music self-organizing from disordered sound. These ordered phases can replicate elements of traditional Western and non-Western systems of music, as well as suggesting new directions to be explored.

The basis for a bottom-up approach was provided by discoveries in the field of psychoacoustics originating with Helmholtz (4) and further developed in the 20th century, which established a quantitative understanding of how sound is perceived. This leads to the idea that the structure of music is related to a minimization of dissonance D, as explored by Plomp and Levelt (5), Sethares (6, 7), and others. Minimization of D cannot be the only criterion for an effective musical system, however, or we would all listen to “music” composed from just a single pitch. Instead, an effective system of music must have some degree of complexity to provide a sufficiently rich palette from which to compose. A recognition of this idea has led to work on quantifying complexity in music, including by computing the entropy S of music in the context of information theory (8) or by considering musical systems to be self-organizing via an evolutionary process (9).

The model I present here combines both the minimization of D and the maximization of S. I draw an analogy to thermodynamic systems with energy U and entropy S, whose macrostate is determined by minimizing Helmholtz free energy F = U − TS. The fixed temperature T is a parameter that specifies the trade-off between decreasing U and increasing S. Here, I similarly introduce a parameter T that specifies the trade-off between decreasing D and increasing S. A musical system in equilibrium will then be found by minimizing F = D − TS, allowing us to exploit the powerful array of tools developed for studying physical systems in statistical mechanics.

The remainder of this paper is organized as follows: I next describe the general model presented here, including how dissonance is quantified. Then, we study the behavior of the model in the mean field approximation and observe phase transitions between disordered sound and ordered distributions of pitches that reproduce commonly used musical systems. Last, we turn to a more realistic model with fewer assumptions and use numerical simulation to explore the patterns that emerge on a lattice of interacting tones.

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