191edo
| ← 190edo | 191edo | 192edo → |
191 equal divisions of the octave (abbreviated 191edo or 191ed2), also called 191-tone equal temperament (191tet) or 191 equal temperament (191et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 191 equal parts of about 6.28 ¢ each. Each step represents a frequency ratio of 21/191, or the 191st root of 2.
191edo is inconsistent to the 5-odd-limit and higher limits, with two mappings possible for the 5-limit: ⟨191 303 443] (patent val) and ⟨191 303 444] (191c). Using the patent val, it tempers out the tetracot comma, 20000/19683 and [-52 5 19⟩ in the 5-limit; 245/243, 2401/2400, and 68359375/67108864 in the 7-limit; 385/384, 896/891, 1375/1372, and 118125/117128 in the 11-limit; 352/351, 364/363, 1625/1617, 1875/1859, and 2197/2187 in the 13-limit. Using the 191c val, it tempers out the amity comma, 1600000/1594323 and [39 -7 -12⟩ in the 5-limit; 4375/4374, 5120/5103, and 823543/810000 in the 7-limit; 441/440, 896/891, 6912/6875, and 14641/14580 in the 11-limit; 196/195, 352/351, 364/363, 2197/2187, and 3146/3125 in the 13-limit. Using the alternative 191cd val, it tempers out 1728/1715, 3136/3125, and 1605632/1594323 in the 7-limit; 176/175, 540/539, 1331/1323, and 655360/649539 in the 11-limit; 351/350, 352/351, 640/637, 1573/1568, and 2197/2187 in the 13-limit, supporting the semisept temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.71 | -3.07 | -1.29 | -2.86 | +1.56 | +1.36 | -1.36 | +1.85 | -2.23 | +0.42 | -0.00 |
| Relative (%) | +27.2 | -48.8 | -20.5 | -45.6 | +24.9 | +21.6 | -21.6 | +29.5 | -35.4 | +6.7 | -0.0 | |
| Steps (reduced) |
303 (112) |
443 (61) |
536 (154) |
605 (32) |
661 (88) |
707 (134) |
746 (173) |
781 (17) |
811 (47) |
839 (75) |
864 (100) | |
Subsets and supersets
191edo is the 43rd prime edo.