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Statistical mechanics meets music theory

14 June 2019

Techniques borrowed from physics explain how order arises to form the structure of musical harmony.

Virtually every known musical system is based on a discrete set of pitches rather than on a continuous spectrum of frequencies. To Jesse Berezovsky, a Case Western Reserve University condensed-matter physicist and a viola player, that seemingly innate development of structure from a continuum echoed the statistical mechanics framework that describes how physical systems undergo phase transitions.

Minimizing the free energy of a thermodynamic system requires balancing energy minimization and entropy maximization. According to Berezovsky, a similar tension can explain the structure of musical systems. Sound entropy—the number of available tones—should be maximized to increase the number of ways notes can be arranged. But dissonance, the perception of roughness or harshness when two of the tones are played together, should be minimized to ensure that combinations of notes are aurally pleasing.

In a thermodynamic system, the optimal balance between energy and entropy depends on temperature. At low temperatures, energetic interactions dominate and order emerges; increasing the temperature favors entropy and causes a transition from order to disorder. Berezovsky found a similar transition for pitches by varying an effective temperature, T, that indicates the degree of importance placed on sound entropy relative to dissonance. At low T, minimizing dissonance took precedence, and the musical system emerged with only one note (bottom row in the figure). At high T (top row), sound entropy dominated, so the entire continuous spectrum of tones remains available.

Mean field results.

When the temperature was decreased from high T, a familiar structure emerged: the 12-tone octave of Western music. A second phase transition occurred at still lower temperature to reveal 12 unequally spaced peaks with different heights, which could describe a musical system that favors a particular key.

Music theory has typically followed a top-down approach to explain patterns in existing music. But Berezovsky’s bottom-up method parallels the one that led to the development of statistical mechanics. With his technique, Berezovsky hopes to uncover new systems of harmony and tuning, which could lead to new kinds of music, the construction of unique instruments, and a better understanding of historical music systems across different cultures. (J. Berezovsky, Sci. Adv. 5, eaav8490, 2019.)

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