Mathematical Harmony Analysis

TL;DR

This paper introduces a mathematical framework for analyzing musical harmony in Just Intonation using invariant functions, GCD, LCM, and lattice structures to classify consonant chords and scales.

Contribution

It develops a novel mathematical approach employing invariant functions and lattice theory to analyze and classify harmonic structures in Just Intonation.

Findings

Invariant functions characterize harmonic structure.

Algorithms classify consonant chords and scales.

Lattice structures relate to harmony complexity.

Abstract

Musical chords, harmonies or melodies in Just Intonation have note frequencies which are described by a base frequency multiplied by rational numbers. For any local section, these notes can be converted to some base frequency multiplied by whole positive numbers. The structure of the chord can be analysed mathematically by finding functions which are unchanged upon chord transposition. These functions are are denoted invariant, and are important for understanding the structure of harmony. Each chord described by whole numbers has a greatest common divisor, GCD, and a lowest common multiple, LCM. The ratio of these is denoted Complexity which is a positive whole number. The set of divisors of Complexity give a subset of a p limit tone lattice and have both a natural ordering and a multiplicative structure. The position and orientation of the original chord, on the ordered set or on the lattice, give rise to many other invariant functions including measures for otonality and utonality. Other invariant functions can be constructed from: ratios between note pairs, prime projections, weighted chords which incorporate loudness. Given a set of conditions described by invariant functions, algorithms can be developed to find all scales or chords meeting those conditions, allowing the classification of consonant harmonies up to specified limits.