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]].

== Theory ==

===~~Harmonics~~===

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]].

=== 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), 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.

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''.

==== Proposal for an interval size measure ====

[[Eliora]] proposes that 1 step of 4320edo be called a '''click''' as an [[interval size measure]]. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.

A [[semitone (interval size measure)|semitone]] therefore is 360 clicks, a [[quartertone]] is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.

Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the [[patent val]].

=== Regular temperament theory ===

4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the [[atomic]] temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].

In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them. In the 13-limit, it tempers out [[6656/6655]], 67392/67375, 151263/151250. In the 17-limit, it tempers out [[12376/12375]], 14400/14399, 28561/28560, and also commas associated with 80edo, such as [[80-17-comma]] and 80-11/10-comma, that is {{monzo|-91 0 -80 0 80}}.

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

=== Prime harmonics ===

=== Other scales and techniques ===

Due to being consistent in the 23-limit, 4320edo is capable of consistently supporting the "[[factor 9 grid]]". It's quite coincidental that the number 4320 is divisible by 432, the number of Hertz in absolute pitch to which the alleged mystical properties of the scale are ascribed, except this time it is the cardinality of an EDO supporting the scale.

4320edo has a possible usage in [[Georgian]] folk music. 4320edo maps the 3/2 interval to 2527 steps, which factors as {{nowrap|7 × 192}}, and thus 4/3 to 1793 steps, factoring as {{nowrap|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]].

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

| 2.3

| {{monzo|-6847 4320}}

| [{{val| 4320 6847}}]

| +0.003

| 0.003

| 1.20

|-

| 2.3.5

| {{monzo|60 31 -47}}, {{monzo|161 -84 -12}}

| [{{val| 4320 6847 10031}}]

| −0.009

| 0.017

| 6.12

|-

| 2.3.5.7

| 250047/250000, {{monzo|-55 30 2 1}}, {{monzo|33 19 -3 -20}}

| [{{val| 4320 6847 10031 12128}}]

| −0.012

| 0.016

| 5.74

|-

| 2.3.5.7.11

| 9801/9800, 250047/250000, {{monzo|24 -10 -5 0 1}}, {{monzo|17 19 4 -9 -9}}

| [{{val| 4320 6847 10031 12128 14945}}]

| −0.014

| 0.015

| 5.28

|-

| 2.3.5.7.11.13

| 9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250

| [{{val| 4320 6847 10031 12128 14945 15986}}]

| −0.013

| 0.014

| 4.89

|-

| 2.3.5.7.11.13.17

| 9801/9800, 12376/12375, 194481/194480, 11275335/11275264, 63922176/63903125, 152649728/152628125

| [{{val| 4320 6847 10031 12128 14945 15986 17658}}]

| −0.012

| 0.013

| 4.53

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 12

| 2527\4320 (7\2460)

| 498.056 (1.944)

| 4/3 (32805/32768)

| [[Atomic]]

|-

| 60

| 2527\4320 (7\2460)

| 498.056 (1.944)

| 4/3 (32805/32768)

| [[Minutes]]

|-

| 80

| 1337\4320 (41\4320)

| 371.389 (11.389)

| 2275/1836 (?)

| [[Mercury]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Miscellany ==

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

When it comes to interval size measures, a curious observation is also that 4320 km/h is close enough to whole integer to equal to 2684 mph, and [[2684edo]] is a [[zeta]] peak EDO.

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

[[Category:Atomic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|4320}}
+{{ED intro}}
-==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]].
+== Theory ==
-===Harmonics===
+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]].
+=== 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), 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.
+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''.
+==== Proposal for an interval size measure ====
+[[Eliora]] proposes that 1 step of 4320edo be called a '''click''' as an [[interval size measure]]. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.
+A [[semitone (interval size measure)|semitone]] therefore is 360 clicks, a [[quartertone]] is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.
+Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the [[patent val]].
+=== Regular temperament theory ===
+edo tempers out the [[Kirnberger's atom]], and aside from tuning the [[atomic]] temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].
+In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them. In the 13-limit, it tempers out [[6656/6655]], 67392/67375, 151263/151250. In the 17-limit, it tempers out [[12376/12375]], 14400/14399, 28561/28560, and also commas associated with 80edo, such as [[80-17-comma]] and 80-11/10-comma, that is {{monzo|-91 0 -80 0 80}}.
+Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.
+=== Prime harmonics ===
 {{harmonics in equal|4320}}
+=== Other scales and techniques ===
+Due to being consistent in the 23-limit, 4320edo is capable of consistently supporting the "[[factor 9 grid]]". It's quite coincidental that the number 4320 is divisible by 432, the number of Hertz in absolute pitch to which the alleged mystical properties of the scale are ascribed, except this time it is the cardinality of an EDO supporting the scale.
+edo has a possible usage in [[Georgian]] folk music. 4320edo maps the 3/2 interval to 2527 steps, which factors as {{nowrap|7 × 19<sup>2</sup>}}, and thus 4/3 to 1793 steps, factoring as {{nowrap|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]].
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+| 2.3
+| {{monzo|-6847 4320}}
+| [{{val| 4320 6847}}]
+| +0.003
+| 0.003
+| 1.20
+|-
+| 2.3.5
+| {{monzo|60  31 -47}}, {{monzo|161 -84 -12}}
+| [{{val| 4320 6847 10031}}]
+| −0.009
+| 0.017
+| 6.12
+|-
+| 2.3.5.7
+| 250047/250000, {{monzo|-55 30 2 1}}, {{monzo|33 19 -3 -20}}
+| [{{val| 4320 6847 10031 12128}}]
+| −0.012
+| 0.016
+| 5.74
+|-
+| 2.3.5.7.11
+| 9801/9800, 250047/250000, {{monzo|24 -10 -5 0 1}}, {{monzo|17 19 4 -9 -9}}
+| [{{val| 4320 6847 10031 12128 14945}}]
+| −0.014
+| 0.015
+| 5.28
+|-
+| 2.3.5.7.11.13
+| 9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250
+| [{{val| 4320 6847 10031 12128 14945<br>15986}}]
+| −0.013
+| 0.014
+| 4.89
+|-
+| 2.3.5.7.11.13.17
+| 9801/9800, 12376/12375, 194481/194480, 11275335/11275264, 63922176/63903125, 152649728/152628125
+| [{{val| 4320 6847 10031 12128 14945<br>15986 17658}}]
+| −0.012
+| 0.013
+| 4.53
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 12
+| 2527\4320<br>(7\2460)
+| 498.056<br>(1.944)
+| 4/3<br>(32805/32768)
+| [[Atomic]]
+|-
+| 60
+| 2527\4320<br>(7\2460)
+| 498.056<br>(1.944)
+| 4/3<br>(32805/32768)
+| [[Minutes]]
+|-
+| 80
+| 1337\4320<br>(41\4320)
+| 371.389<br>(11.389)
+| 2275/1836<br>(?)
+| [[Mercury]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Miscellany ==
+edo is the 69th highly abundant EDO. Nice.
+When it comes to interval size measures, a curious observation is also that 4320 km/h is close enough to whole integer to equal to 2684 mph, and [[2684edo]] is a [[zeta]] peak EDO.
 [[Category:Equal divisions of the octave|####]]
+[[Category:Atomic]]