161edo
| ← 160edo | 161edo | 162edo → |
161 equal divisions of the octave (abbreviated 161edo or 161ed2), also called 161-tone equal temperament (161tet) or 161 equal temperament (161et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 161 equal parts of about 7.45 ¢ each. Each step represents a frequency ratio of 21/161, or the 161st root of 2.
Theory
161et tempers out the würschmidt comma, 393216/390625, in the 5-limit; 3136/3125, 6144/6125 and 2401/2400 in the 7-limit; 243/242, 441/440, 540/539 and 5632/5625 in the 11-limit; and 351/350, 847/845, 1001/1000, 1188/1183, 1575/1573 and 1716/1715 in the 13-limit. It serves as the optimal patent val for the mintone temperament in the 5-, 7-, 11- and 13-limit.
Prime harmonics
In the range of edos from 100 to 200, 161edo is notable as being low in 29-limit relative error.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.33 | +1.26 | +0.12 | +0.23 | +1.71 | -0.61 | +0.62 | -2.19 | -1.01 | +2.79 |
| Relative (%) | +0.0 | -17.9 | +17.0 | +1.6 | +3.2 | +22.9 | -8.2 | +8.4 | -29.3 | -13.5 | +37.4 | |
| Steps (reduced) |
161 (0) |
255 (94) |
374 (52) |
452 (130) |
557 (74) |
596 (113) |
658 (14) |
684 (40) |
728 (84) |
782 (138) |
798 (154) | |
Subsets and supersets
Since 161 factors into 7 × 23, 161edo contains 7edo and 23edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-255 161⟩ | [⟨161 255]] | +0.421 | 0.421 | 5.65 |
| 2.3.5 | 393216/390625, [-17 21 -7⟩ | [⟨161 255 374]] | +0.099 | 0.570 | 7.65 |
| 2.3.5.7 | 2401/2400, 3136/3125, 177147/175000 | [⟨161 255 374 452]] | +0.064 | 0.498 | 6.67 |
| 2.3.5.7.11 | 243/242, 441/440, 3136/3125, 35937/35840 | [⟨161 255 374 452 557]] | +0.037 | 0.448 | 6.01 |
| 2.3.5.7.11.13 | 243/242, 351/350, 441/440, 847/845, 3136/3125 | [⟨161 255 374 452 557 596]] | −0.046 | 0.449 | 6.03 |
| 2.3.5.7.11.13.17 | 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088 | [⟨161 255 374 452 557 596 658]] | −0.018 | 0.422 | 5.66 |
| 2.3.5.7.11.13.17.19 | 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512 | [⟨161 255 374 452 557 596 658 684]] | −0.034 | 0.397 | 5.32 |
- 161et has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by 183et.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 15\161 | 111.80 | 16/15 | Vavoom |
| 1 | 16\161 | 119.25 | 15/14 | Septidiasemi |
| 1 | 17\161 | 126.71 | 14/13 | Mowglic |
| 1 | 25\161 | 186.34 | 10/9 | Mintone |
| 1 | 26\161 | 193.79 | 28/25 | Hemiwürschmidt |
| 1 | 38\161 | 283.23 | 33/28 | Neominor (161f) |
| 1 | 52\161 | 387.58 | 5/4 | Würschmidt (5-limit) |
| 1 | 79\161 | 588.82 | 45/32 | Aufo |
| 7 | 67\161 (2\161) |
499.38 (14.91) |
4/3 (81/80) |
Absurdity |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct