391edo
| ← 390edo | 391edo | 392edo → |
391 equal divisions of the octave (abbreviated 391edo or 391ed2), also called 391-tone equal temperament (391tet) or 391 equal temperament (391et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 391 equal parts of about 3.07 ¢ each. Each step represents a frequency ratio of 21/391, or the 391st root of 2.
391edo has a sharp tendency, with prime harmonics 3 to 13 all tuned sharp. The equal temperament tempers out 5120/5103, 420175/419904, and 29360128/29296875 in the 7-limit, and provides the optimal patent val for the hemifamity temperament, and septiquarter, the 99 & 292 temperament. It tempers out 3025/3024, 4000/3993, 5632/5625, and 6250/6237 in the 11-limit; and 676/675, 1716/1715 and 4225/4224 in the 13-limit, and provides further optimal patent vals for temperaments tempering out 5120/5103 such as alphaquarter.
The 391bcde val provides a tuning for 11-limit miracle very close to the POTE tuning.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.86 | +0.39 | +1.00 | -1.35 | +1.11 | +0.39 | +1.25 | -0.61 | +0.19 | -1.22 | +0.88 |
| Relative (%) | +28.0 | +12.6 | +32.4 | -44.1 | +36.2 | +12.8 | +40.6 | -19.8 | +6.0 | -39.6 | +28.7 | |
| Steps (reduced) |
620 (229) |
908 (126) |
1098 (316) |
1239 (66) |
1353 (180) |
1447 (274) |
1528 (355) |
1598 (34) |
1661 (97) |
1717 (153) |
1769 (205) | |
Subsets and supersets
Since 391 factors into 17 × 23, 391edo contains 17edo and 23edo as subsets.