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.
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