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624edo: Difference between revisions

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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]]. 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.  
It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[retraction]] thereof.  


=== Prime harmonics ===
=== Prime harmonics ===
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! 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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -989 624 }}
| {{Monzo| -989 624 }}
| {{mapping| 624 989 }}
| {{Mapping| 624 989 }}
| +0.0101
| +0.0101
| 0.0101
| 0.0101
Line 34: Line 34:
|-
|-
| 2.3.5
| 2.3.5
| {{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
| −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, 134217728/133984375
| {{mapping| 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
| {{mapping| 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
| {{mapping| 624 989 1449 1752 2159 2309 }}
| {{Mapping| 624 989 1449 1752 2159 2309 }}
| −0.0595
| −0.0595
| 0.0831
| 0.0831
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| 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
| {{mapping| 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
| {{mapping| 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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| 2.3.5.7.11.13.17.19.23
| 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
| 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 }}
| −0.0906
| −0.0906
| 0.0830
| 0.0830
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|+ 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)
| 31\624
| 340.38<br />(59.62)
| 59.62
| 162/133<br />(88/85)
| 88/85
| [[Semiseptichrome]]
| [[Semiseptichrome]]
|-
| 6
| 37\624
| 71.15
| 25/24
| [[Trivish]]
|-
|-
| 12
| 12
| 259\624<br />(1\624)
| 1\624
| 498.08<br />(1.92)
| 1.92
| 4/3<br />(32805/32768)
| 32805/32768
| [[Atomic]]
| [[Atomic]]
|-
|-
| 13
| 13
| 259\624<br />(19\624)
| 19\624
| 498.08<br />(36.54)
| 36.54
| 4/3<br />(?)
| ?
| [[Aluminium]] (5-limit)
| [[Aluminium]] (5-limit)
|-
|-
| 16
| 16
| 259\624<br />(14\624)
| 14\624
| 498.08<br />(48.077)
| 48.08
| 4/3<br />(?)
| ?
| [[Sulfur]]
| [[Sulfur]]
|-
|-
| 24
| 24
| 303\624<br />(17\624)
| 9\624
| 582.692<br />(32.692)
| 17.31
| 7/5<br />(?)
| 100/99
| [[Chromium]]
| [[Chromium]]
|-
|-
| 26
| 26
| 259\624<br />(19\624)
| 19\624
| 498.08<br />(36.54)
| 36.54
| 4/3<br />(?)
| ?
| [[Iron]]
| [[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
<nowiki/>* In [[Normal forms #Minimal-generator form|minimal-generator form]]


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/624edo"