391edo

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← 390edo391edo392edo →
Prime factorization 17 × 23
Step size 3.06905¢ 
Fifth 229\391 (702.813¢)
Semitones (A1:m2) 39:28 (119.7¢ : 85.93¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 391edo
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.