181edo

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← 180edo 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 21/181, or the 181st root of 2.

Theory

181edo is only consistent to the 7-odd-limit, though besides 10/9 and 9/5 all intervals are consistent in the 21-odd-limit, and it has low errors on primes all the way up to 43. It even does well beyond that, with less than 30% error on all primes up to 137, besides 47, 61, 73, 97, and 113. 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)
Approximation of prime harmonics in 181edo
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.59 +1.88 -1.02 -2.52 +1.63 +1.60 -3.07 +0.25 -0.69 -2.38 +0.10
Relative (%) +8.9 +28.3 -15.4 -38.1 +24.6 +24.2 -46.3 +3.8 -10.4 -35.8 +1.6
Steps
(reduced) 		943
(38) 		970
(65) 		982
(77) 		1005
(100) 		1037
(132) 		1065
(160) 		1073
(168) 		1098
(12) 		1113
(27) 		1120
(34) 		1141
(55)
Approximation of prime harmonics in 181edo
Harmonic 83 89 97 101 103 107 109 113 127 131 137
Error Absolute (¢) +0.78 -0.71 +2.76 -0.90 -1.70 -1.36 -0.27 -3.00 +0.32 -0.33 +1.70
Relative (%) +11.8 -10.8 +41.6 -13.6 -25.7 -20.6 -4.1 -45.2 +4.8 -5.0 +25.6
Steps
(reduced) 		1154
(68) 		1172
(86) 		1195
(109) 		1205
(119) 		1210
(124) 		1220
(134) 		1225
(139) 		1234
(148) 		1265
(179) 		1273
(6) 		1285
(18)

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 distinct

Music

Francium

See also

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