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The math behind musical harmony: phase transitions and balance between order and disorder

The math behind musical harmony: phase transitions and balance between order and disorder

Music, while allowing nearly unlimited creative expression, almost always conforms to a set of rigid rules at a fundamental level. Jesse Berezovsky, an associate professor of physics at Case Western Reserve University, defines the “emergent structures of musical harmony” inherent in the art, just as order comes from disorder in the physical world.

 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.

Recent work has attempted to expand these generalized theories in order to discover new possible patterns that might lead to new music theories. Instead, we see patterns that arise organically from a bottom-up paradigm in this case. We begin with two fundamental (and contradictory) principles: A music system is most successful when it (i) reduces discordant sounds and (ii) provides for enough complexity to allow for the desired creative expression.

Tone lattice simulation results.

The arrangement of pitch domains on the lattice shown above (B) reflects elements of musical harmony. The graph first identifies all pitch domains, defined as contiguous regions with at least five lattice sites having the same pitch, rounded to the nearest of the 12 pitch classes. Pitch domains that share a border are then identified as those that have at least two adjacent lattice sites in one domain bordering two adjacent lattice sites in the other domain. The resulting graph of all neighboring domains on the tone lattice (red lines in C) maps directly onto the Tonnetz. All possible fifths are present (for example, the ellipse in B traverses the circle of fifths), with many of the thirds present as well.

These concepts may be directly mapped into a conventional statistical mechanics framework based on their mathematical expression. As a result, we may use statistical mechanics methods to investigate the events that occur from this music model. We see ordered phases of music self-organizing from disordered sound, just as in physical systems where ordered phases with lower symmetry (e.g., crystals) arise through transitions from higher-symmetry disordered phases (e.g., liquids). These structured phases can reproduce features of conventional Western and non-Western music systems while also proposing new areas to investigate.

The theory also relates to why we like music: it is stuck between being too discordant and too complicated.

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A single note played constantly would have no dissonance (low “energy”) yet be utterly boring to the human ear, but an overly complicated piece of music (high entropy) is typically unappealing to the human ear. Most music, across time and civilizations, resides in the tension between the two extremes, according to Berezovsky.

The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory, Jesse Berezovsky

Published: May 2019
DOI: 10.1126/sciadv.aav8490

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