624edo: Difference between revisions

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== Theory ==
== 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]].
624edo is [[consistent]] to the [[27-odd-limit]]. It [[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.  
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.  
Line 19: Line 19:
! rowspan="2" | [[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 35: Line 35:
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{mapping| 624 989 1449 }}
| {{mapping| 624 989 1449 }}
| &minus;0.0256
| −0.0256
| 0.0510
| 0.0510
| 2.65
| 2.65
Line 42: Line 42:
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| {{mapping| 624 989 1449 1752 }}
| {{mapping| 624 989 1449 1752 }}
| &minus;0.0552
| −0.0552
| 0.0678
| 0.0678
| 3.52
| 3.52
Line 49: Line 49:
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| {{mapping| 624 989 1449 1752 2159 }}
| {{mapping| 624 989 1449 1752 2159 }}
| &minus;0.0792
| −0.0792
| 0.0772
| 0.0772
| 4.02
| 4.02
Line 56: Line 56:
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
| {{mapping| 624 989 1449 1752 2159 2309 }}
| {{mapping| 624 989 1449 1752 2159 2309 }}
| &minus;0.0595
| −0.0595
| 0.0831
| 0.0831
| 4.32
| 4.32
Line 63: Line 63:
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
| {{mapping| 624 989 1449 1752 2159 2309 2551 }}
| {{mapping| 624 989 1449 1752 2159 2309 2551 }}
| &minus;0.0795
| −0.0795
| 0.0911
| 0.0911
| 4.74
| 4.74
Line 70: Line 70:
| 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
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}
| &minus;0.0861
| −0.0861
| 0.0870
| 0.0870
| 4.53
| 4.53
Line 77: Line 77:
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645
| 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 }}
| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 2823 }}
| &minus;0.0906
| −0.0906
| 0.0830
| 0.0830
| 4.32
| 4.32
Line 86: Line 86:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 6
| 6
| 177\624<br>(31\624)
| 177\624<br />(31\624)
| 340.38<br>(59.62)
| 340.38<br />(59.62)
| 162/133<br>(88/85)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
| [[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]] (5-limit)
| [[Aluminium]] (5-limit)
|-
|-
| 16
| 16
| 259\624<br>(14\624)
| 259\624<br />(14\624)
| 498.08<br>(48.077)
| 498.08<br />(48.077)
| 4/3<br>(?)
| 4/3<br />(?)
| [[Sulfur]]
| [[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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; [[Eliora]]
; [[Eliora]]
* [https://www.youtube.com/watch?v=vEDajIHqRUw&t=3s&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)
* [https://www.youtube.com/watch?v=vEDajIHqRUw&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)


[[Category:Ainismic]]
[[Category:Ainismic]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 15:01, 16 January 2025

← 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

Template:EDO intro

Theory

624edo is consistent to the 27-odd-limit. 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)

Subsets and supersets

Since 624 factors into 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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