A group-theoretical classification of three-tone and four-tone harmonic chords

TL;DR

This paper introduces a group-theoretical framework to classify three-tone and four-tone chords within the twelve-tone scale, linking mathematical symmetries to musical harmony concepts.

Contribution

It provides a novel classification method based on subgroup actions of symmetric groups, connecting mathematical structures with musical harmony theory.

Findings

Classifies chords using symmetric group subgroups

Establishes a graph relating harmonic actions and chord relationships

Suggests a new concept of distance in harmony theory

Abstract

We classify three-tone and four-tone chords based on subgroups of the symmetric group acting on chords contained within a twelve-tone scale. The actions are inversion, major-minor duality, and augmented-diminished duality. These actions correspond to elements of symmetric groups, and also correspond directly to intuitive concepts in the harmony theory of music. We produce a graph of how these actions relate different seventh chords that suggests a concept of distance in the theory of harmony.