161edo: Difference between revisions
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The '''161 equal | {{Infobox ET | ||
| Prime factorization = 7 × 23 | |||
| Step size = 7.45324¢ | |||
| Fifth = 94\161 (700.62¢) | |||
| Semitones = 14:13 (104.35¢ : 96.89) | |||
| Consistency = 7 | |||
}} | |||
The '''161 equal divisions of the octave''' ('''161edo'''), or the '''161(-tone) equal temperament''' ('''161tet''', '''161et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 161 [[equal]] parts of about 7.45 [[cent]]s each. | |||
== Prime harmonics == | == Theory == | ||
161edo 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 [[1188/1183]], [[351/350]], [[847/845]], [[1575/1573]], [[1001/1000]] 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-limits. | |||
=== Prime harmonics === | |||
161edo is notable as being low in [[29-limit]] relative error in the 100 to 200 range. | 161edo is notable as being low in [[29-limit]] relative error in the 100 to 200 range. | ||
{{Harmonics in equal|161}} | {{Harmonics in equal|161}} | ||
Revision as of 14:30, 6 March 2022
| ← 160edo | 161edo | 162edo → |
The 161 equal divisions of the octave (161edo), or the 161(-tone) equal temperament (161tet, 161et) when viewed from a regular temperament perspective, divides the octave into 161 equal parts of about 7.45 cents each.
Theory
161edo 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 1188/1183, 351/350, 847/845, 1575/1573, 1001/1000 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-limits.
Prime harmonics
161edo is notable as being low in 29-limit relative error in the 100 to 200 range.
| 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) | |
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 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 15\161 | 111.80 | 16/15 | Vavoom |
| 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 |