181edo
| ← 180edo | 181edo | 182edo → |
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
| 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 | +12 | -27 | -13 | -16 | +22 | +17 | +13 | +24 | -29 | +29 | |
| 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
| 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