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'''624EDO''' is the [[EDO|equal division of the octave]] into 624 parts of 1.92308 [[cent]]s each. It is consistent to the [[27-odd-limit|27-limit]], tempering out 6115295232/6103515625 (vishnuzma) and |-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.
{{Infobox ET}}
{{ED intro}}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Theory ==
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 ===
{{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 ===
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 ==
{| 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
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -989 624 }}
| {{mapping| 624 989 }}
| +0.0101
| 0.0101
| 0.52
|-
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{mapping| 624 989 1449 }}
| −0.0256
| 0.0510
| 2.65
|-
| 2.3.5.7
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| {{mapping| 624 989 1449 1752 }}
| −0.0552
| 0.0678
| 3.52
|-
| 2.3.5.7.11
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| {{mapping| 624 989 1449 1752 2159 }}
| −0.0792
| 0.0772
| 4.02
|-
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
| {{mapping| 624 989 1449 1752 2159 2309 }}
| −0.0595
| 0.0831
| 4.32
|-
| 2.3.5.7.11.13.17
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
| {{mapping| 624 989 1449 1752 2159 2309 2551 }}
| −0.0795
| 0.0911
| 4.74
|-
| 2.3.5.7.11.13.17.19
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}
| −0.0861
| 0.0870
| 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 ===
{| 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
|-
| 1
| 259\624
| 498.08
| 4/3
| [[Counterschismic]]
|-
| 1
| 311\624
| 598.08
| 847/600
| [[Vydubychi]]
|-
| 2
| 37\624
| 71.15
| 25/24
| [[Vishnu]] (5-limit)
|-
| 3
| 73\624
| 140.38
| 243/224
| [[Septichrome]]
|-
| 6
| 177\624<br />(31\624)
| 340.38<br />(59.62)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
|-
| 12
| 259\624<br />(1\624)
| 498.08<br />(1.92)
| 4/3<br />(32805/32768)
| [[Atomic]]
|-
| 13
| 259\624<br />(19\624)
| 498.08<br />(36.54)
| 4/3<br />(?)
| [[Aluminium]] (5-limit)
|-
| 16
| 259\624<br />(14\624)
| 498.08<br />(48.077)
| 4/3<br />(?)
| [[Sulfur]]
|-
| 24
| 303\624<br />(17\624)
| 582.692<br />(32.692)
| 7/5<br />(?)
| [[Chromium]]
|-
| 26
| 259\624<br />(19\624)
| 498.08<br />(36.54)
| 4/3<br />(?)
| [[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:Ainismic]]
[[Category:Listen]]
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