181edo

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181edo

182edo →

Prime factorization	181 (prime)
Step size	6.62983¢
Fifth	106\181 (702.762¢)
Semitones (A1:m2)	18:13 (119.3¢ : 86.19¢)
Consistency limit	7
Distinct consistency limit	7

181 equal divisions of the octave (abbreviated 181edo or 181ed2), also called 181-tone equal temperament (181tet) or 181 equal temperament (181et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 181 equal parts of about 6.63 ¢ each. Each step represents a frequency ratio of 2^1/181, or the 181st root of 2.

Theory

181et tempers out 2109375/2097152 (semicomma) and [14 -22 9⟩ 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, 2200/2187, and 4000/3993 in the 11-limit; 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit.

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)

Subsets and supersets

181edo is the 42nd prime edo.

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 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	18\181	119.34	15/14	Septidiasemi
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)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Francium

• "Today Or Tomorrow?" from Questions (2024) – Spotify | Bandcamp | YouTube – slurpee in 181edo tuning

See also

• 181edo and stretched hemififths