# 181edo

 ← 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

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