1323edo: Difference between revisions
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Rank 2 temps, also notated 441 & 1308 despite not having a name, because it is listed on the augmented-cloudy equivalence continuum page. |
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1323edo is the smallest uniquely consistent EDO in the 29-odd-limit. | 1323edo is the smallest uniquely consistent EDO in the 29-odd-limit. | ||
It provides the optimal patent val for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave. | It provides the optimal patent val for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave. In additoin, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum. | ||
1323's divisors are {{EDOs|1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441}}, of which 441EDO is a member of the zeta edos. 1323edo shares the 7-limit mapping with 441edo. As such, it can be interpreted as an improvement for 441edo into the 29-limit by splitting each step of 441edo into three. | 1323's divisors are {{EDOs|1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441}}, of which 441EDO is a member of the zeta edos. 1323edo shares the 7-limit mapping with 441edo. As such, it can be interpreted as an improvement for 441edo into the 29-limit by splitting each step of 441edo into three. | ||
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{{Harmonics in equal|1323}} | {{Harmonics in equal|1323}} | ||
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number --> | [[Category:Equal divisions of the octave|####]] | ||
== Regular temperament properties == | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
!Periods | |||
per octave | |||
!Generator | |||
(reduced) | |||
!Cents | |||
(reduced) | |||
!Associated | |||
ratio | |||
!Temperaments | |||
|- | |||
|3 | |||
|177\1323 | |||
|160.544 | |||
|154478651796875/140737488355328 | |||
|441 & 1308 | |||
|- | |||
|27 | |||
|299\1323 | |||
(5\1323) | |||
|271.201 | |||
(4.535) | |||
|1375/1176 | |||
(?) | |||
|Trinealimmal | |||
|}<!-- 4-digit number --> | |||
Revision as of 15:13, 23 October 2022
| ← 1322edo | 1323edo | 1324edo → |
Theory
1323edo is the smallest uniquely consistent EDO in the 29-odd-limit.
It provides the optimal patent val for the 11-limit trinealimmal temperament, which has a period of 1\27 octave. In additoin, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.
1323's divisors are 1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, of which 441EDO is a member of the zeta edos. 1323edo shares the 7-limit mapping with 441edo. As such, it can be interpreted as an improvement for 441edo into the 29-limit by splitting each step of 441edo into three.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.086 | +0.081 | -0.118 | +0.156 | +0.289 | +0.260 | -0.007 | +0.297 | -0.099 | -0.364 |
| Relative (%) | +0.0 | +9.5 | +8.9 | -13.1 | +17.2 | +31.8 | +28.7 | -0.8 | +32.8 | -10.9 | -40.2 | |
| Steps (reduced) |
1323 (0) |
2097 (774) |
3072 (426) |
3714 (1068) |
4577 (608) |
4896 (927) |
5408 (116) |
5620 (328) |
5985 (693) |
6427 (1135) |
6554 (1262) | |
Regular temperament properties
Rank-2 temperaments
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 3 | 177\1323 | 160.544 | 154478651796875/140737488355328 | 441 & 1308 |
| 27 | 299\1323
(5\1323) |
271.201
(4.535) |
1375/1176
(?) |
Trinealimmal |