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Partition Entropy as a Measure of Regularity of Music Scales
Version 1
: Received: 17 April 2024 / Approved: 18 April 2024 / Online: 18 April 2024 (08:48:08 CEST)
A peer-reviewed article of this Preprint also exists.
Cubarsi, R. Partition Entropy as a Measure of Regularity of Music Scales. Mathematics 2024, 12, 1658. Cubarsi, R. Partition Entropy as a Measure of Regularity of Music Scales. Mathematics 2024, 12, 1658.
Abstract
The entropy of the partition generated by an n-tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition, and its complementary, here referred to as the bias, allow to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales having limited bias because their tones are distributed along interval fractions of a regular partition. The most typical case in music concerns partitions associated with well-formed scales generated by a single tone h. These scales are maximal even sets that combine two elementary intervals. Then, the normalized entropy depends on the number of tones as well as the relative size of both elementary intervals. When well-formed scales are refined, several nested families stand out with increasing regularity. It is proven that a scale of minimal bias, i.e. with less bias than those with fewer tones, is always a best rational approximation of log2h.
Keywords
metric entropy; convex sets; convex functions; continued fractions; best rational approximation; maximal even sets; well-formed scales; pythagorean tuning
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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