4320edo: Difference between revisions

Line 2:

==Theory==

4320edo is distinctly consistent in the [[23-odd-limit]]. While ~~this~~ 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]].

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 ===

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.

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]].

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.

Other notable divisors 4320edo has are [[12edo]], the dominant tuning system in the world today, [[15edo]], known for use by [[Easley Blackwood Jr]]., [[72edo]], which has found usage in Byzantine chanting and various other applications, [[96edo]] notable for its use by Julian Carrillo, [[270edo]], notable for its excellent closed representation of the 13-limit relative to its size, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure Dröbisch angle.

Line 22:

In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]]. It is a tuning for the [[hemiennealimmal]] temperament and the rank-3 temperament [[odin]].

Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.

===Prime harmonics===

@@ Line 2: / Line 2: @@
 {{EDO intro|4320}}
 ==Theory==
-edo is distinctly consistent in the [[23-odd-limit]]. While this 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]].
+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 ===
 '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.
-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]].
+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.
 Other notable divisors 4320edo has are [[12edo]], the dominant tuning system in the world today, [[15edo]], known for use by [[Easley Blackwood Jr]]., [[72edo]], which has found usage in Byzantine chanting and various other applications, [[96edo]] notable for its use by Julian Carrillo, [[270edo]], notable for its excellent closed representation of the 13-limit relative to its size, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure Dröbisch angle.
@@ Line 22: / Line 22: @@
 In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]]. It is a tuning for the [[hemiennealimmal]] temperament and the rank-3 temperament [[odin]].
+Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.
 ===Prime harmonics===