# 323edo

 ← 322edo 323edo 324edo →
Prime factorization 17 × 19
Step size 3.71517¢
Fifth 189\323 (702.167¢)
Semitones (A1:m2) 31:24 (115.2¢ : 89.16¢)
Consistency limit 9
Distinct consistency limit 9

323 equal divisions of the octave (abbreviated 323edo or 323ed2), also called 323-tone equal temperament (323tet) or 323 equal temperament (323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 323 equal parts of about 3.72 ¢ each. Each step represents a frequency ratio of 21/323, or the 323rd root of 2.

## 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.

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

Approximation of prime harmonics in 323edo
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
2.3.5.7.11 540/539, 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
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
• 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past 289 and followed by 388.

### Rank-2 temperaments

Table of rank-2 temperaments by generator
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)