Harmony as an underlying ingredient in the numerical search for suitable equal temperaments

Abstract

This work investigates whether there are harmonic sets which are universally prioritized when fitting the chromatic just scale with equal-tempered tunings. We design a method to find which temperaments properly and non-ambiguously fit a set of pure intervals and show that: (a) Within range N ∈ [9, 54], temperaments with 12, 19, 22, 24, 31, 34, 41 and 53 divisions are suitable for adjusting the twelve-tone chromatic just scale – in particular, 53 divisions provide the most accurate fit, (b) For M-element subsets of the chromatic universe with M ∈ [3, 7], these solutions are always more faithfully reproduced when the target set is a major scale rather than a minor one, and (c) Sequences of notes uniformly distributed over the octave are disadvantageous references for finding suitable temperaments numerically. The latter observation suggests a mathematical background for understanding the preference for non-uniform scales in world music revealed by recent studies in ethnomusicology.

Keywords

• Music theory
• equal-temperament
• harmony
• pure intervals