181edo

181edo is the equal division of the octave into 181 parts of 6.6298 cents each.

Theory

181et tempers out 2109375/2097152 (semicomma) and 32000000000/31381059609 in the 5-limit; 2401/2400, 5120/5103, and 390625/387072 in the 7-limit (supporting the hemififths and the cotritone). Using the patent val, it tempers out 385/384, 1375/1372, 2200/2197, and 4000/3993 in the 11-limit; 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit.

181edo is the 42nd prime EDO.

Prime harmonics

Approximation of prime harmonics in 181edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.81	-1.78	-0.87	-1.04	+1.46	+1.12	+0.83	+1.56	-1.95	+1.93
	Relative (%)	+0.0	+12.2	-26.9	-13.1	-15.7	+22.0	+16.9	+12.5	+23.5	-29.5	+29.0
Steps (reduced)		181 (0)	287 (106)	420 (58)	508 (146)	626 (83)	670 (127)	740 (16)	769 (45)	819 (95)	879 (155)	897 (173)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[287 -181⟩	[⟨181 287]]	-0.255	0.255	3.84
2.3.5	2109375/2097152, [14 -22 9⟩	[⟨181 287 420]]	+0.086	0.525	7.92
2.3.5.7	2401/2400, 5120/5103, 390625/387072	[⟨181 287 420 508]]	+0.142	0.465	7.01
2.3.5.7.11	385/384, 1375/1372, 2200/2187, 4000/3993	[⟨181 287 420 508 626]]	+0.174	0.421	6.35
2.3.5.7.11.13	325/324, 352/351, 385/384, 1375/1372, 1575/1573	[⟨181 287 420 508 626 670]]	+0.079	0.439	6.62
2.3.5.7.11.13.17	325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274	[⟨181 287 420 508 626 670 740]]	+0.028	0.425	6.40
2.3.5.7.11.13.17.19	325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274	[⟨181 287 420 508 626 670 740 769]]	+0.000	0.404	6.09

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	35\181	232.04	8/7	Quadrawell
1	39\181	258.56	[-32 13 5⟩	Lafa
1	41\181	271.82	75/64	Orson
1	53\181	351.38	49/40	Hemififths (7-limit)
1	88\181	583.43	7/5	Cotritone (181f)