1323edo
| ← 1322edo | 1323edo | 1324edo → |
1323 equal divisions of the octave (abbreviated 1323edo or 1323ed2), also called 1323-tone equal temperament (1323tet) or 1323 equal temperament (1323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1323 equal parts of about 0.907 ¢ each. Each step represents a frequency ratio of 21/1323, or the 1323rd root of 2.
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 | +9 | +9 | -13 | +17 | +32 | +29 | -1 | +33 | -11 | -40 | |
| 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