The Two-Step Property and the Mathematics of Musical Scale Size

TL;DR

This paper mathematically characterizes Pythagorean scales with the 2-step property, including well-known musical scales like pentatonic, diatonic, and chromatic, highlighting their even spacing and mathematical structure.

Contribution

It provides a mathematician's account of the characterization of Pythagorean scales with the 2-step property, including classical musical scales, for audiences without musical background.

Findings

Includes 5-note, 7-note, and 12-note scales as 2-step Pythagorean scales.

Shows these scales are the most 'evenly-spaced' within their class.

Connects mathematical properties to well-known musical scales.

Abstract

A Pythagorean scale is a mathematical encoding of a musical scale as a finite list of numbers of the form 3^b/2^a. Previous work of the first author discussed the 2-step property as a way to measure which Pythagorean scales are the most "evenly-spaced." In this paper, we give a mathematician's account of the characterization of the Pythagorean scales that have the 2-step property; compellingly, the list includes the 5-note, 7-note, and 12-note Pythagorean scales, which are well-known as the pentatonic, diatonic, and chromatic scales of music theory. (After this preprint was initially posted, it was brought to our attention that these results are not new: they have previously appeared in the work of Carey and Clampitt, and related work has been done by several other members of the music theory community, now cited below. We leave this preprint available because it is written for mathematicians with no musical background, and as such may provide helpful exposition for some audiences, but we no longer make any claim to originality of the content.)