1323edo: Difference between revisions
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Note: 7-limit temperaments supported by 441et are not included. | Note: 7-limit temperaments supported by 441et are not included. | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 27 | | 27 | ||
| Line 24: | Line 31: | ||
| 1375/1176<br />(?) | | 1375/1176<br />(?) | ||
| [[Trinealimmal]] | | [[Trinealimmal]] | ||
|} | |||
<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 13:23, 16 November 2024
| ← 1322edo | 1323edo | 1324edo → |
Theory
1323edo is the smallest edo distinctly consistent in the 29-odd-limit. It is enfactored in the 7-limit, sharing the same excellent 7-limit approximation with 441edo, but it makes for a great higher-limit system by splitting each step of 441edo into three.
It provides the optimal patent val for the 11-limit trinealimmal temperament, which has a period of 1\27 octave.
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
Since 1323 factors into 33 × 72, 1323edo has subset edos 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, of which 441edo is a member of the zeta edos.
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