624edo: Difference between revisions

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Cleanup; clarify the title row of the rank-2 temp table; -redundant categories
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== 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]]. The equal temperament [[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, and 3888/3887 in the 23-limit.


=== Prime harmonics ===
=== Prime harmonics ===
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! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| -989 624 }}
| {{monzo| -989 624 }}
| [{{val| 624 989 }}]
| {{mapping| 624 989 }}
| +0.0101
| +0.0101
| 0.0101
| 0.0101
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| 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
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| 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
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| 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
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| 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
Line 59: Line 59:
| 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
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| 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
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Iron]]
| [[Iron]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 07:51, 25 October 2023

← 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. The equal temperament 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, 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)

Subsets and supersets

Since 624 = 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

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
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
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 it is distinct

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