 Partition Entropy as a Measure of Regularity of Music ScalesRafael Cubarsi ()
Mathematics, 2024, vol. 12, issue 11, 1-23 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 us to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales with limited bias because their tones are distributed along specific 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 each number of intervals as well as their relative size. 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 l o g 2 h . Keywords: metric entropy; convex sets; convex functions; continued fractions; best rational approximation; maximal even sets; well-formed scales; Pythagorean tuning (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:11:p:1658-:d:1401823 Access Statistics for this article Mathematics is currently edited by Ms. Emma He More articles in Mathematics from MDPI |
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