624edo: Difference between revisions

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{{Infobox ET
| Prime factorization = 2<sup>4</sup> × 3 × 13
| Step size = 1.92308¢
| Fifth = 365\624 (701.92¢)
| Semitones = 59:47 (113.46¢ : 90.38¢)
| Consistency = 27
}}
{{EDO intro|624}}
{{EDO intro|624}}


== 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]], 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.


=== Prime harmonics ===
{{Harmonics in equal|624|columns=11}}
{{Harmonics in equal|624|columns=11}}
== Regular temperament properties ==
{| 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]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -989 624 }}
| [{{val| 624 989 }}]
| +0.0101
| 0.0101
| 0.52
|-
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| [{{val| 624 989 1449 }}]
| -0.0256
| 0.0510
| 2.65
|-
| 2.3.5.7
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
| [{{val| 624 989 1449 1752 }}]
| -0.0552
| 0.0678
| 3.52
|-
| 2.3.5.7.11
| 9801/9800, 46656/46585, 131072/130977, 151263/151250
| [{{val| 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
| [{{val| 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
| [{{val| 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
| [{{val| 624 989 1449 1752 2159 2309 2551 2651 }}]
| -0.0861
| 0.0870
| 4.53
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>Ratio
! Temperaments
|-
| 1
| 73\624
| 140.38
| 243/224
| [[Septichrome]]
|-
| 1
| 259\624
| 498.08
| 4/3
| [[Counterschismic]]
|-
| 2
| 37\624
| 71.15
| 25/24
| [[Vishnu]] (5-limit)
|-
| 12
| 259\624<br>(1\624)
| 498.08<br>(1.92)
| 4/3<br>(32805/32768)
| [[Atomic]]
|}


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

Revision as of 23:26, 11 September 2022

← 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, 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.

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)

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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 73\624 140.38 243/224 Septichrome
1 259\624 498.08 4/3 Counterschismic
2 37\624 71.15 25/24 Vishnu (5-limit)
12 259\624
(1\624) 		498.08
(1.92) 		4/3
(32805/32768) 		Atomic
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