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

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 introduced the 2-step property as a way to measure which Pythagorean scales are the most “evenly-spaced.” In this paper, we give a complete 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.

💡 Research Summary

The paper investigates the “two‑step property” of Pythagorean scales—finite collections of frequencies of the form 3^b/2^a that lie in the interval