Musical Chords by the Numbers

TL;DR

This paper introduces a mathematical model for measuring the consonance of musical chords based on simple fractional ratios, emphasizing symmetry and invariance properties, and compares it with existing models and empirical data.

Contribution

It develops a novel, symmetry-invariant measure of chord consonance grounded in simple ratios, filling a gap in mathematical models of musical harmony.

Findings

The model aligns well with human perceptions of consonance.

It demonstrates invariance under chord transformations.

Comparison shows advantages over previous models.

Abstract

The mathematics of musical intervals and scales has been extensively studied. Vastly simplified, our ears seem to prefer intervals whose frequency ratios have small numerator and denominator, such as 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), and so on. While there also have been numerous studies on the mathematics of musical chords, very few of them consider a model that measures consonance/dissonance of a given chord in analogy with this simple-fractions perspective. Our aim is to develop a measure for the consonance of a chord with crucial symmetry features, including invariance under chord translation, inversion, and interval sets. We apply our model to chords in various musical scales and compare it to existing models and empirical studies.