Revising the Musical Equal Temperament

TL;DR

This paper quantitatively confirms that slightly increasing the semitone frequency ratio improves harmonicity, proposing a revised temperament as a better standard than the traditional equal temperament in Western music.

Contribution

It introduces a quantitative analysis showing that a modified temperament with a larger ratio than 2^{1/12} enhances harmonicity, challenging the traditional equal temperament standard.

Findings

Harmonicity peaks at ratios larger than 2^{1/12}.

Entropy effectively measures harmonicity.

Recommends adopting a harmonized temperament as the new standard.

Abstract

Western music is predominantly based on the equal temperament with a constant semitone frequency ratio of $2^{1/12}$. Although this temperament has been in use since the 19th century and in spite of its high degree of symmetry, various musicians have repeatedly expressed their discomfort with the harmonicity of certain intervals. Recently it was suggested that this problem can be overcome by introducing a modified temperament with a constant but slightly increased frequency ratio. In this paper we confirm this conjecture quantitatively. Using entropy as a measure for harmonicity, we show numerically that the harmonic optimum is in fact obtained for frequency ratios larger than $2^{1/12}$. This suggests that the equal temperament should be replaced by a harmonized temperament as a new standard.