Unified Music Theories for General Equal-Temperament Systems

TL;DR

This paper explores the mathematical and musical foundations of equal-temperament systems beyond the traditional 12-tone system, proposing axioms and sequences to unify and extend music theory concepts.

Contribution

It introduces a set of axioms for general equal-temperament systems and redefines classical music theory ideas within this broader mathematical framework.

Findings

Proposes axioms for general equal-temperament systems

Reinterprets classical music theory concepts mathematically

Creates systematic sequences for musical facts

Abstract

Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal-temperament system. We reexamine the mathematical understandings and interpretations of ideas in classical music theory, such as the circle of fifths, enharmonic equivalent, degrees such as the dominant and the subdominant, and the leading tone, and endow them with meaning outside of the 12-TET system. In the process of deriving scales, we create various kinds of sequences to describe facts in music theory, and we name these sequences systematically and unambiguously with the aim to facilitate future research.