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624edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|624}}
{{ED intro}}


== Theory ==
== Theory ==
624edo is [[consistent]] to the [[27-odd-limit]], tempering out 6115295232/6103515625 ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[250047/250000]], 2460375/2458624, and 134217728/133984375 in the 7-limit; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the 11-limit; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the 13-limit; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the 17-limit; [[1521/1520]], 2376/2375, 2432/2431, and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, and 3888/3887 in the 23-limit.
624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].
 
It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|624|columns=11}}
{{Harmonics in equal|624|columns=11}}
{{Harmonics in equal|624|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 624edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 624 = 2<sup>4</sup> × 3 × 13, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.
Since 624 factors into primes as {{nowrap| 2<sup>4</sup> × 3 × 13 }}, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 24: Line 28:
| 2.3
| 2.3
| {{monzo| -989 624 }}
| {{monzo| -989 624 }}
| [{{val| 624 989 }}]
| {{mapping| 624 989 }}
| +0.0101
| +0.0101
| 0.0101
| 0.0101
Line 31: Line 35:
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| [{{val| 624 989 1449 }}]
| {{mapping| 624 989 1449 }}
| -0.0256
| −0.0256
| 0.0510
| 0.0510
| 2.65
| 2.65
Line 38: Line 42:
| 2.3.5.7
| 2.3.5.7
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| [{{val| 624 989 1449 1752 }}]
| {{mapping| 624 989 1449 1752 }}
| -0.0552
| −0.0552
| 0.0678
| 0.0678
| 3.52
| 3.52
Line 45: Line 49:
| 2.3.5.7.11
| 2.3.5.7.11
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| [{{val| 624 989 1449 1752 2159 }}]
| {{mapping| 624 989 1449 1752 2159 }}
| -0.0792
| −0.0792
| 0.0772
| 0.0772
| 4.02
| 4.02
Line 52: Line 56:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
| [{{val| 624 989 1449 1752 2159 2309 }}]
| {{mapping| 624 989 1449 1752 2159 2309 }}
| -0.0595
| −0.0595
| 0.0831
| 0.0831
| 4.32
| 4.32
Line 59: Line 63:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
| [{{val| 624 989 1449 1752 2159 2309 2551 }}]
| {{mapping| 624 989 1449 1752 2159 2309 2551 }}
| -0.0795
| −0.0795
| 0.0911
| 0.0911
| 4.74
| 4.74
Line 66: Line 70:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
| [{{val| 624 989 1449 1752 2159 2309 2551 2651 }}]
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}
| -0.0861
| −0.0861
| 0.0870
| 0.0870
| 4.53
| 4.53
|-
| 2.3.5.7.11.13.17.19.23
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 2823 }}
| −0.0906
| 0.0830
| 4.32
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 87: Line 99:
| [[Counterschismic]]
| [[Counterschismic]]
|-
|-
|1
| 1
|311\624
| 311\624
|598.08
| 598.08
|847/600
| 847/600
|[[Vydubychi]]
| [[Vydubychi]]
|-
|-
| 2
| 2
Line 104: Line 116:
| 243/224
| 243/224
| [[Septichrome]]
| [[Septichrome]]
|-
| 6
| 177\624<br />(31\624)
| 340.38<br />(59.62)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
|-
|-
| 12
| 12
| 259\624<br>(1\624)
| 259\624<br />(1\624)
| 498.08<br>(1.92)
| 498.08<br />(1.92)
| 4/3<br>(32805/32768)
| 4/3<br />(32805/32768)
| [[Atomic]]
| [[Atomic]]
|-
|-
| 13
| 13
| 259\624<br>(19\624)
| 259\624<br />(19\624)
| 498.08<br>(36.54)
| 498.08<br />(36.54)
| 4/3<br>(?)
| 4/3<br />(?)
| [[Aluminium]]
| [[Aluminium]] (5-limit)
|-
| 16
| 259\624<br />(14\624)
| 498.08<br />(48.077)
| 4/3<br />(?)
| [[Sulfur]]
|-
|-
| 24
| 24
| 303\624<br>(17\624)
| 303\624<br />(17\624)
| 582.692<br>(32.692)
| 582.692<br />(32.692)
| 7/5<br>(?)
| 7/5<br />(?)
| [[Chromium]]
| [[Chromium]]
|-
|-
| 26
| 26
| 259\624<br>(19\624)
| 259\624<br />(19\624)
| 498.08<br>(36.54)
| 498.08<br />(36.54)
| 4/3<br>(?)
| 4/3<br />(?)
| [[Iron]]
| [[Iron]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=vEDajIHqRUw&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Ainismic]]
[[Category:Listen]]
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