1323edo: 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 == | ||
1323edo is the smallest | 1323edo is the smallest edo [[consistency|distinctly consistent]] 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 | It provides the [[optimal patent val]] for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave. In addition, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|1323}} | {{Harmonics in equal|1323}} | ||
[[ | === Subsets and supersets === | ||
1323's divisors are {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 441edo is a member of the [[zeta edo]]s. 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. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | |||
Note: 7-limit temperaments supported by 441et are not included. | |||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods<br>per 8ve | ! Periods<br>per 8ve | ||
!Generator | ! Generator* | ||
!Cents | ! Cents* | ||
!Associated<br> | ! Associated<br>Ratio | ||
!Temperaments | ! Temperaments | ||
<!-- | |||
|- | |- | ||
|3 | | 3 | ||
|177\1323 | | 177\1323 | ||
|160.544 | | 160.544 | ||
|154478651796875/140737488355328 | | 154478651796875/140737488355328 | ||
|[[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]] | | [[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]] | ||
--> | |||
|- | |- | ||
|27 | | 27 | ||
|299\1323<br>(5\1323) | | 299\1323<br>(5\1323) | ||
|271.201<br>(4.535) | | 271.201<br>(4.535) | ||
|1375/1176<br>(?) | | 1375/1176<br>(?) | ||
|[[Trinealimmal]] | | [[Trinealimmal]] | ||
|}< | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 08:13, 17 October 2023
| ← 1322edo | 1323edo | 1324edo → |
Theory
1323edo is the smallest edo distinctly consistent 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 addition, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.
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) | |
Subsets and supersets
1323's divisors are 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.
Regular temperament properties
Rank-2 temperaments
Note: 7-limit temperaments supported by 441et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 27 | 299\1323 (5\1323) |
271.201 (4.535) |
1375/1176 (?) |
Trinealimmal |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct