181edo

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← 180edo181edo182edo →
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

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 octave
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

See also