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Partition Entropy as a Measure of Regularity of Music Scales.
- Source :
- Mathematics (2227-7390); Jun2024, Vol. 12 Issue 11, p1658, 23p
- Publication Year :
- 2024
-
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 . [ABSTRACT FROM AUTHOR]
- Subjects :
- ENTROPY
CONTINUED fractions
CONVEX sets
CONVEX functions
Subjects
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 12
- Issue :
- 11
- Database :
- Complementary Index
- Journal :
- Mathematics (2227-7390)
- Publication Type :
- Academic Journal
- Accession number :
- 177856812
- Full Text :
- https://doi.org/10.3390/math12111658