A Non-Pythagorean Musical Scale Based on Logarithms

TL;DR

This paper introduces a novel musical scale based on natural logarithms, which differs from traditional Pythagorean and equal temperament scales, offering unique harmonic possibilities through irrational pitch ratios.

Contribution

The author proposes a non-Pythagorean scale derived from logarithms, expanding musical tuning systems beyond traditional rational ratios and exploring new harmonic textures.

Findings

Logarithmic pitches do not correspond to traditional scales.

Beat frequencies reveal extended harmonic potential.

The scale introduces irrational pitch ratios for novel sounds.

Abstract

A new musical scale devised by the author, based on natural logarithms, is described. Most of the logarithmic pitches bear no correspondence to the twelve tones of the ancient tuning system attributed to Pythagoras, based on ratios of whole numbers, nor to the chromatic tones of scales in equal temperament used widely in the modern era. Logarithms obey a special arithmetic compared to whole and rational numbers, which can be heard in beat frequencies between tones of the scale, suggesting extended harmonic possibilities by incorporating the beat frequencies into compositions. The author uses the broad term "non-Pythagorean" to describe the logarithmic musical scale, as the ratios of pitches are usually irrational numbers.