323edo
Theory
323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the 31-limit.
It tempers out the vulture comma, [24 -21 4⟩ and the luna comma, [38 -2 -15⟩, in the 5-limit; 4375/4374, 589824/588245, and 703125/702464 in the 7-limit, supporting 7-limit vulture, lunatic, enneadecal, and gamera.
In the 11-limit, the 323e val and the patent val are comparable in errors. 1375/1372, 5632/5625, 14641/14580, and 19712/19683 are tempered out in the patent val; 540/539, 6250/6237, 12005/11979, and 16384/16335 are tempered out in the 323e val. It provides the optimal patent val for the rank-5 temperament tempering out 1573/1568, the lambeth comma, as well as 13-limit stockhausenic, and deuteromere, the 2.3.5.11 subgroup temperament tempering out 14641/14580.
Since 323 factors into 17 × 19, 323edo shares the excellent approximations of 25/24 in 17edo and of 28/27 and the 6/5 in 19edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.21 | +0.06 | +0.83 | -1.47 | -0.90 | -0.93 | -0.30 | -0.41 | -0.48 | -0.76 |
| Relative (%) | +0.0 | +5.7 | +1.7 | +22.4 | -39.6 | -24.2 | -25.0 | -8.1 | -11.1 | -12.8 | -20.5 | |
| Steps (reduced) |
323 (0) |
512 (189) |
750 (104) |
907 (261) |
1117 (148) |
1195 (226) |
1320 (28) |
1372 (80) |
1461 (169) |
1569 (277) |
1600 (308) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [512 -323⟩ | [⟨323 512]] | −0.0669 | 0.0669 | 1.80 |
| 2.3.5 | [24 -21 4⟩, [38 -2 -15⟩ | [⟨323 512 750]] | −0.0538 | 0.0577 | 1.55 |
| 2.3.5.7 | 4375/4374, 589824/588245, 703125/702464 | [⟨323 512 750 907]] | −0.1146 | 0.1165 | 3.14 |
| 2.3.5.7.13 | 676/675, 4096/4095, 4375/4374, 16848/16807 | [⟨323 512 750 907 1195]] | −0.0431 | 0.1770 | 4.76 |
| 2.3.5.7.13.17 | 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374 | [⟨323 512 750 907 1195 1320]] | +0.0020 | 0.1905 | 5.13 |
| 2.3.5.7.11 | 1375/1372, 4375/4374, 5632/5625, 14641/14580 | [⟨323 512 750 907 1117]] (323) | −0.0066 | 0.2399 | 6.46 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374 | [⟨323 512 750 907 1117 1195]] (323) | +0.0350 | 0.2380 | 6.40 |
| 2.3.5.7.11 | 540/539, 4375/4374, 12005/11979, 16384/16335 | [⟨323 512 750 907 1118]] (323e) | −0.2213 | 0.2375 | 6.39 |
| 2.3.5.7.11.13 | 364/363, 540/539, 676/675, 4096/4095, 4375/4374 | [⟨323 512 750 907 1118 1195]] (323e) | −0.1440 | 0.2773 | 7.47 |
- 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past 289 and followed by 388.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 26\323 | 96.59 | 200/189 | Hemiluna (323) |
| 1 | 30\323 | 111.46 | 16/15 | Stockhausenic (323) |
| 1 | 31\323 | 115.17 | 77/72 | Semigamera (323) |
| 1 | 52\323 | 193.19 | 352/315 | Luna / lunatic (323e) |
| 1 | 62\323 | 230.34 | 8/7 | Gamera |
| 1 | 128\323 | 475.54 | 320/243 | Vulture |
| 17 | 134\323 (9\323) |
248.92 (33.44) |
[-23 5 9 -2⟩ (100352/98415) |
Chlorine |
| 19 | 134\323 (2\323) |
497.83 (7.43) |
4/3 (225/224) |
Enneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct