323edo

The 323 equal divisions of the octave (323edo) is the equal division of the octave into 323 parts of 3.7152 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.

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

Table of rank-2 temperaments by generator

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