4320edo: Difference between revisions

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==Theory==

4320edo is distinctly consistent in the [[23-odd-limit]]. While this fact is not remarkable on 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]]. 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]].

===~~Harmonics~~===

=== Divisors ===

4320's divisors are 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 3.5 = exactly 7/2.

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]].

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, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure Drobisch angle.

=== Regular temperament theory ===

4320edo supports the period-80 temperament [[mercury]].

=== Possible usage in Georgian music ===

4320edo maps the 3/2 interval to 2527 steps, which factors as 7 x 19^2, and thus 4/3 to 1793 steps, factoring as 11 x 163. Since Georgian traditional music is based on dividing 3/2 and 4/3 into an arbitrary number of steps, it is able to support a variety of [[Kartvelian scales]] on the patent val, for example a combination of [[7edf]] and [[11ed4/3]].

===Prime harmonics===

== Miscellaneous properties ==

4320edo is the 69th highly abundant EDO. Nice.

[[Category:Equal divisions of the octave|####]]

@@ Line 2: / Line 2: @@
 {{EDO intro|4320}}
 ==Theory==
-edo is distinctly consistent in the [[23-odd-limit]]. While this fact is not remarkable on 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]]. 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]].
-===Harmonics===
+=== Divisors ===
+'s divisors are 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 3.5 = exactly 7/2.
+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]].
+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, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure Drobisch angle.
+=== Regular temperament theory ===
+edo supports the period-80 temperament [[mercury]].
+=== Possible usage in Georgian music ===
+edo maps the 3/2 interval to 2527 steps, which factors as 7 x 19^2, and thus 4/3 to 1793 steps, factoring as 11 x 163. Since Georgian traditional music is based on dividing 3/2 and 4/3 into an arbitrary number of steps, it is able to support a variety of [[Kartvelian scales]] on the patent val, for example a combination of [[7edf]] and [[11ed4/3]].
+===Prime harmonics===
 {{harmonics in equal|4320}}
+== Miscellaneous properties ==
+edo is the 69th highly abundant EDO. Nice.
 [[Category:Equal divisions of the octave|####]]