Difference Tones in "Non-Pythagorean" Scales Based on Logarithms

TL;DR

This paper investigates logarithmic musical scales inspired by Schneider's non-Pythagorean approach, aiming to refine them so difference tones align with the scale, and shows such scales must have integer ratios if all difference tones are included.

Contribution

It demonstrates that scales containing all their difference tones as octave equivalents must be composed of integer ratios and proposes methods for creating logarithmic scales with many difference tones.

Findings

Scales with all difference tones as octave equivalents are necessarily integer ratio scales.

Methods are proposed for constructing logarithmic scales with many, but not all, difference tones.

The study bridges logarithmic scale design with traditional integer ratio constraints.

Abstract

In order to explore tonality outside of the `Pythagorean' paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate.