One categorization of microtonal scales

TL;DR

This paper analyzes rational approximations of the musical constant log2(3/2), evaluating their accuracy and introducing a new way to categorize microtonal scales based on these approximations.

Contribution

It provides a detailed analysis of convergents and secondary convergents of the musical constant, including non-convergents, to categorize microtonal scales.

Findings

Identified the quality of secondary convergents for the musical constant

Positioned microtonal scales using non-convergents approximations

Compared approximation quality with best Huygens approximations

Abstract

This study considers rational approximations of musical constant $\beta=\log_2(3/2)$, which defines perfect fifth. This constant has been the subject of the numerous studies, and this paper determines quality of rational approximations in regards to absolute error. We analysed convergents and secondary convergents (some of these are the best Huygens approximations). Especially, we determined quality of the secondary convergents which are not the best Huygens approximations - in this paper we called them non-convergents approximations. Some of the microtonal scales have been positioned and determined by using non-convergents approximation of music constant which defines perfect fifth.