4320edo: Difference between revisions

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==Theory==

== Theory ==

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]].

Line 15:

[[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]] 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.

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]].

Line 26:

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

===Prime harmonics===

=== Prime harmonics ===

Line 32:

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 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]].

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|Comma List]]~~

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

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

~~! colspan="2" |Tuning error~~

|-

![[~~TE error~~|~~Absolute~~]] ~~(¢)~~

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

![[~~TE simple badness|Relative~~]] (%)

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

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

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

! colspan="2" | Tuning error

|-

|2.3

| 2.3

|{{monzo|-6847 4320}}

| {{monzo|-6847 4320}}

|[{{val| 4320 6847}}]

| [{{val| 4320 6847}}]

|0.003

| +0.003

|0.003

| 0.003

|1.20

| 1.20

|-

|2.3.5

| 2.3.5

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

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

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

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

|~~<nowiki>-0~~.009~~</nowiki>~~

| −0.009

|0.017

| 0.017

|6.12

| 6.12

|-

|2.3.5.7

| 2.3.5.7

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

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

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

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

|~~<nowiki>-0~~.012~~</nowiki>~~

| −0.012

|0.016

| 0.016

|5.74

| 5.74

|-

|2.3.5.7.11

| 2.3.5.7.11

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

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

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

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

|~~<nowiki>-0~~.014~~</nowiki>~~

| −0.014

|0.015

| 0.015

|5.28

| 5.28

|-

|2.3.5.7.11.13

| 2.3.5.7.11.13

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

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

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

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

|~~<nowiki>-0~~.013~~</nowiki>~~

| −0.013

|0.014

| 0.014

|4.89

| 4.89

|-

|2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

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

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

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

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

|~~<nowiki>-0~~.012~~</nowiki>~~

| −0.012

|0.013

| 0.013

|4.53

| 4.53

|}

=== Rank-2 temperaments ===

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

~~!Periods per 8ve~~

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

~~!Generator (reduced)~~

~~!Cents (reduced)~~

~~!Associated ratio~~

~~!Temperaments~~

|-

~~|12~~

! Periods per 8ve

~~|2527\4320~~ ~~(7\2460)~~

! Generator*

~~|498.056~~<br~~>(1.944)~~

! Cents*

|4/~~3<br~~>~~(32805/32768)~~

! Associated ratio*

~~|[[Atomic]]~~

! Temperaments

|-

|60

| 12

|2527\4320 (7\2460)

| 2527\4320 (7\2460)

|498.056 (1.944)

| 498.056 (1.944)

|4/3 (32805/32768)

| 4/3 (32805/32768)

|[[~~Minutes~~]]

| [[Atomic]]

|-

|80

| 60

|1337\4320 (41\4320)

| 2527\4320 (7\2460)

|371.389 (11.389)

| 498.056 (1.944)

|2275/1836 (?)

| 4/3 (32805/32768)

|[[Mercury]]

| [[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 ==

Line 119:

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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==
+== Theory ==
 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]].
@@ Line 15: / Line 15: @@
 [[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]] 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.
+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]].
@@ Line 26: / Line 26: @@
 Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.
-===Prime harmonics===
+=== Prime harmonics ===
 {{harmonics in equal|4320}}
@@ Line 32: / Line 32: @@
 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 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]].
+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|Comma List]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal<br>8ve stretch (¢)
-! colspan="2" |Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! rowspan="2" | [[Subgroup]]
-![[TE simple badness|Relative]] (%)
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
 |-
-|2.3
+| 2.3
-|{{monzo|-6847 4320}}
+| {{monzo|-6847 4320}}
-|[{{val| 4320 6847}}]
+| [{{val| 4320 6847}}]
-|0.003
+| +0.003
-|0.003
+| 0.003
-|1.20
+| 1.20
 |-
-|2.3.5
+| 2.3.5
-|{{monzo|60  31 -47}}, {{monzo|161 -84 -12}}
+| {{monzo|60  31 -47}}, {{monzo|161 -84 -12}}
-|[{{val| 4320 6847 10031}}]
+| [{{val| 4320 6847 10031}}]
-|<nowiki>-0.009</nowiki>
+| −0.009
-|0.017
+| 0.017
-|6.12
+| 6.12
 |-
-|2.3.5.7
+| 2.3.5.7
-|250047/250000, {{monzo|-55 30 2 1}}, {{monzo|33 19 -3 -20}}
+| 250047/250000, {{monzo|-55 30 2 1}}, {{monzo|33 19 -3 -20}}
-|[{{val| 4320 6847 10031 12128}}]
+| [{{val| 4320 6847 10031 12128}}]
-|<nowiki>-0.012</nowiki>
+| −0.012
-|0.016
+| 0.016
-|5.74
+| 5.74
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|9801/9800, 250047/250000, {{monzo|24 -10 -5 0 1}}, {{monzo|17 19 4 -9 -9}}
+| 9801/9800, 250047/250000, {{monzo|24 -10 -5 0 1}}, {{monzo|17 19 4 -9 -9}}
-|[{{val| 4320 6847 10031 12128 14945}}]
+| [{{val| 4320 6847 10031 12128 14945}}]
-|<nowiki>-0.014</nowiki>
+| −0.014
-|0.015
+| 0.015
-|5.28
+| 5.28
 |-
-|2.3.5.7.11.13
+| 2.3.5.7.11.13
-|9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250
+| 9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250
-|[{{val| 4320 6847 10031 12128 14945<br>15986}}]
+| [{{val| 4320 6847 10031 12128 14945<br>15986}}]
-|<nowiki>-0.013</nowiki>
+| −0.013
-|0.014
+| 0.014
-|4.89
+| 4.89
 |-
-|2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-|9801/9800, 12376/12375, 194481/194480, 11275335/11275264, 63922176/63903125, 152649728/152628125
+| 9801/9800, 12376/12375, 194481/194480, 11275335/11275264, 63922176/63903125, 152649728/152628125
-|[{{val| 4320 6847 10031 12128 14945<br>15986 17658}}]
+| [{{val| 4320 6847 10031 12128 14945<br>15986 17658}}]
-|<nowiki>-0.012</nowiki>
+| −0.012
-|0.013
+| 0.013
-|4.53
+| 4.53
 |}
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-!Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-!Generator<br>(reduced)
-!Cents<br>(reduced)
-!Associated<br>ratio
-!Temperaments
 |-
-|12
+! Periods<br />per 8ve
-|2527\4320<br>(7\2460)
+! Generator*
-|498.056<br>(1.944)
+! Cents*
-|4/3<br>(32805/32768)
+! Associated<br />ratio*
-|[[Atomic]]
+! Temperaments
 |-
-|60
+| 12
-|2527\4320<br>(7\2460)
+| 2527\4320<br>(7\2460)
-|498.056<br>(1.944)
+| 498.056<br>(1.944)
-|4/3<br>(32805/32768)
+| 4/3<br>(32805/32768)
-|[[Minutes]]
+| [[Atomic]]
 |-
-|80
+| 60
-|1337\4320<br>(41\4320)
+| 2527\4320<br>(7\2460)
-|371.389<br>(11.389)
+| 498.056<br>(1.944)
-|2275/1836<br>(?)
+| 4/3<br>(32805/32768)
-|[[Mercury]]
+| [[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 ==
@@ Line 119: / Line 120: @@
 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]]