Harmony and Duality: An introduction to Music Theory

TL;DR

This paper introduces a combinatorial framework for music theory, deriving scales and chords from constraints and dualities, offering a principled foundation for understanding harmony and improvisation.

Contribution

It presents a novel combinatorial approach to music theory, deriving scales and chords from constraints and duality principles rather than memorization.

Findings

Scales constrained by voice separation relate to commonly used musical scales.

A duality exists between scales constrained by two-voice and three-voice conditions.

The framework classifies chords based on combinatorial constraints.

Abstract

We develop aspects of music theory related to harmony, such as scales, chord formation and improvisation from a combinatorial perspective. The goal is to provide a foundation for this subject by deriving the basic structure from a few assumptions, rather than writing down long lists of chords/scales to memorize without an underlying principle. Our approach involves introducing constraints that limit the possible scales we can consider. For example, we may impose the constraint that two voices cannot be only a semitone apart as this is too dissonant. We can then study scales that do not contain notes that are a semitone apart. A more refined constraint avoids three voices colliding by studying scales that do not have three notes separated only by semitones. Additionally, we require that our scales are complete, which roughly means that they are the maximal sets of tones that satisfy these constraints. As it turns out, completeness as applied to these simple two/three voice constraints characterizes the types of scales that are commonly used in music composition. Surprisingly, there is a correspondence between scales subject to the two-voice constraint and those subject to the three-voice constraint. We formulate this correspondence as a duality statement that provides a way to understand scales subject to one type of constraint in terms of scales subject to the other. Finally, we combine these constraint ideas to provide a classification of chords.