323edo
| ← 322edo | 323edo | 324edo → |
The 323 equal divisions of the octave (323edo), or the 323(-tone) equal temperament (323tet, 323et) when viewed from a regular temperament perspective, is the equal division of the octave into 323 parts of about 3.72 cents each.
Theory
323et 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 the 2.3.5.11 subgroup temperament tempering out 14641/14580, the semicanousma.
323 = 17 × 19, and shares the excellent approximations of 25/24 in 17edo and of the 28/27 and the 6/5 in 19edo.
Prime harmonics
Script error: No such module "primes_in_edo".
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 |
| 2.3.5.7.11 | 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 | 1375/1372, 4375/4374, 5632/5625, 14641/14580 | [⟨323 512 750 907 1117]] (323) | -0.0066 | 0.2399 | 6.46 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 26\323 | 96.59 | 200/189 | Hemiluna (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 |