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4320edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|4320}}
{{ED intro}}


==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]].
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]].


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


===Prime harmonics===
=== Prime harmonics ===
{{harmonics in equal|4320}}
{{harmonics in equal|4320}}


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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.
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 × 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| 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 ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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 ==
== Miscellany ==
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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.
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:Equal divisions of the octave|####]]
[[Category:Atomic]]
[[Category:Atomic]]
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