624edo: Difference between revisions
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+infobox; +RTT table and rank-2 temperaments |
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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 --> | ||