Set-theoretic solution for the tuning problem

TL;DR

This paper introduces a set-theoretic framework for musical tuning, generalizing Just Intonation to inharmonic sounds and unifying spectral and harmonic factors in consonance, with quantifiable measures.

Contribution

It presents a novel mathematical approach to quantify musical consonance using set theory, enabling dynamic tuning systems based on affinity and harmonicity measures.

Findings

Sets of intervals generated for tuning systems

Quantification of consonance through affinity and harmonicity

Unified framework for harmonic and inharmonic tuning

Abstract

In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity contributions to consonance within a single framework. The main achievement of the work is the ability to mathematically quantify the phenomenon of musical consonance using set theory. That quantification is done by defining two measures of consonance: affinity and harmonicity. These measures naturally generate sets of intervals that can be used as dynamic tuning systems. The paper is aimed at a broad audience of people who may not be skilled in music and tuning theory or mathematics. Thus, I attempt to give as much details and explanations as I can, while keeping the number of pages as low as possible.