161edo
| ← 160edo | 161edo | 162edo → |
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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf