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.
The equal temperament 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 = 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
Template:Comma basis begin |- | 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 |- style="border-top: double;" | 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 |- style="border-top: double;" | 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 Template:Comma basis end
- 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past 289 and followed by 388.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf