624edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
Line 34: Line 26:
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
| {{mapping| 624 989 1449 }}
| {{mapping| 624 989 1449 }}
| -0.0256
| &minus;0.0256
| 0.0510
| 0.0510
| 2.65
| 2.65
Line 41: Line 33:
| 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 }}
| -0.0552
| &minus;0.0552
| 0.0678
| 0.0678
| 3.52
| 3.52
Line 48: Line 40:
| 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 }}
| -0.0792
| &minus;0.0792
| 0.0772
| 0.0772
| 4.02
| 4.02
Line 55: Line 47:
| 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 }}
| -0.0595
| &minus;0.0595
| 0.0831
| 0.0831
| 4.32
| 4.32
Line 62: Line 54:
| 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 }}
| -0.0795
| &minus;0.0795
| 0.0911
| 0.0911
| 4.74
| 4.74
Line 69: Line 61:
| 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 }}
| -0.0861
| &minus;0.0861
| 0.0870
| 0.0870
| 4.53
| 4.53
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 108: Line 94:
|-
|-
| 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]]
| [[Aluminium]]
|-
|-
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Music ==
== Music ==

Revision as of 05:06, 16 November 2024

← 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.

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 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 |- | 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 Template:Rank-2 end Template:Orf

Music

Eliora
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