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)}}


=== Miscellaneous properties ===
=== 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 Octave
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 73\624
| 140.38
| 243/224
| [[Septichrome]]
|-
|-
| 1
| 1
Line 92: Line 98:
| 4/3
| 4/3
| [[Counterschismic]]
| [[Counterschismic]]
|-
| 1
| 311\624
| 598.08
| 847/600
| [[Vydubychi]]
|-
|-
| 2
| 2
Line 98: Line 110:
| 25/24
| 25/24
| [[Vishnu]] (5-limit)
| [[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
| 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
| 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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Ainismic]]
[[Category:Listen]]

Latest revision as of 16:18, 5 June 2026

← 623edo 624edo 625edo →
Prime factorization 24 × 3 × 13
Step size 1.92308 ¢ 
Fifth 365\624 (701.923 ¢)
Semitones (A1:m2) 59:47 (113.5 ¢ : 90.38 ¢)
Consistency limit 27
Distinct consistency limit 27

624 equal divisions of the octave (abbreviated 624edo or 624ed2), also called 624-tone equal temperament (624tet) or 624 equal temperament (624et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 624 equal parts of about 1.92 ¢ each. Each step represents a frequency ratio of 21/624, or the 624th root of 2.

Theory

624edo is consistent to the 27-odd-limit. As an equal temperament, it tempers out [23 6 -14⟩ (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, 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

Approximation of prime harmonics in 624edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.032 +0.225 +0.405 +0.605 -0.143 +0.814 +0.564 +0.572 -0.731 -0.805
Relative (%) +0.0 -1.7 +11.7 +21.1 +31.5 -7.4 +42.3 +29.3 +29.7 -38.0 -41.8
Steps
(reduced) 		624
(0) 		989
(365) 		1449
(201) 		1752
(504) 		2159
(287) 		2309
(437) 		2551
(55) 		2651
(155) 		2823
(327) 		3031
(535) 		3091
(595)
Approximation of prime harmonics in 624edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.579 -0.216 +0.021 -0.122 -0.428 +0.444 +0.423 -0.461 -0.850 -0.866 +0.848
Relative (%) +30.1 -11.2 +1.1 -6.3 -22.2 +23.1 +22.0 -24.0 -44.2 -45.1 +44.1
Steps
(reduced) 		3251
(131) 		3343
(223) 		3386
(266) 		3466
(346) 		3574
(454) 		3671
(551) 		3701
(581) 		3785
(41) 		3837
(93) 		3862
(118) 		3934
(190)

Subsets and supersets

Since 624 factors into primes as 24 × 3 × 13, 624edo has subset edos 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-989 624⟩ [⟨624 989]] +0.0101 0.0101 0.52
2.3.5 [23 6 -14⟩, [-69 45 -1⟩ [⟨624 989 1449]] −0.0256 0.0510 2.65
2.3.5.7 250047/250000, 2460375/2458624, [27 0 -8 -3⟩ [⟨624 989 1449 1752]] −0.0552 0.0678 3.52
2.3.5.7.11 9801/9800, 46656/46585, 131072/130977, 151263/151250 [⟨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 [⟨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 [⟨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 [⟨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 [⟨624 989 1449 1752 2159 2309 2551 2651 2823]] −0.0906 0.0830 4.32

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(31\624) 		340.38
(59.62) 		162/133
(88/85) 		Semiseptichrome
12 259\624
(1\624) 		498.08
(1.92) 		4/3
(32805/32768) 		Atomic
13 259\624
(19\624) 		498.08
(36.54) 		4/3
(?) 		Aluminium (5-limit)
16 259\624
(14\624) 		498.08
(48.077) 		4/3
(?) 		Sulfur
24 303\624
(17\624) 		582.692
(32.692) 		7/5
(?) 		Chromium
26 259\624
(19\624) 		498.08
(36.54) 		4/3
(?) 		Iron

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
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