4320edo: Difference between revisions

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4320edo is distinctly consistent in the [[23-odd-limit]] and it is an excellent no-29s 37-limit tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].

=== ~~Divisors~~ ===

=== Subsets ===

4320's divisors are {{EDOs|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093|highly abundant]] (although not superabundant)~~. It's~~ abundancy index is 2.5 = exactly 5/2.

4320's divisors are {{EDOs|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093|highly abundant]] (although not superabundant), with an abundancy index of 2.5 = exactly 5/2, as well as [[highly composite equal division#highly factorable numbers|highly factorable EDO]], with a total of 382 ways of being split into subset EDOs.

Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.

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 edo is distinctly consistent in the [[23-odd-limit]] and it is an excellent no-29s 37-limit tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].
-=== Divisors ===
+=== Subsets ===
-'s divisors are {{EDOs|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093|highly abundant]] (although not superabundant). It's abundancy index is 2.5 = exactly 5/2.
+'s divisors are {{EDOs|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093|highly abundant]] (although not superabundant), with an abundancy index of 2.5 = exactly 5/2, as well as [[highly composite equal division#highly factorable numbers|highly factorable EDO]], with a total of 382 ways of being split into subset EDOs.
 Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.