391edo: Difference between revisions
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391edo has a sharp tendency, with [[prime harmonic]]s 3 to 13 all tuned sharp | 391edo has a sharp tendency, with [[prime harmonic]]s 3 to 13 all tuned sharp. | ||
The 391bcde [[val]] provides a tuning for 11-limit miracle very close to the POTE tuning. | As an equal temperament, it [[tempering out|tempers out]] [[5120/5103]], [[420175/419904]], and [[29360128/29296875]] in the [[7-limit]], and provides the [[optimal patent val]] for the [[aberschismic]] temperament, and [[septiquarter]], the {{nowrap| 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 === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 391 factors into {{ | Since 391 factors into primes as {{nowrap| 17 × 23}}, 391edo contains [[17edo]] and [[23edo]] as subsets. | ||
[[Category: | [[Category:Aberschismic]] | ||
[[Category:Septiquarter]] | [[Category:Septiquarter]] | ||
[[Category:Alphaquarter]] | [[Category:Alphaquarter]] | ||
Latest revision as of 06:50, 7 June 2026
| ← 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.
As an equal temperament, it tempers out 5120/5103, 420175/419904, and 29360128/29296875 in the 7-limit, and provides the optimal patent val for the aberschismic 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 primes as 17 × 23, 391edo contains 17edo and 23edo as subsets.