291edo

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← 290edo291edo292edo →
Prime factorization 3 × 97
Step size 4.12371¢ 
Fifth 170\291 (701.031¢)
Semitones (A1:m2) 26:23 (107.2¢ : 94.85¢)
Consistency limit 3
Distinct consistency limit 3

291 equal divisions of the octave (abbreviated 291edo or 291ed2), also called 291-tone equal temperament (291tet) or 291 equal temperament (291et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 291 equal parts of about 4.12 ¢ each. Each step represents a frequency ratio of 21/291, or the 291st root of 2.

291edo is inconsistent to the 5-odd-limit and higher limits, with three mappings possible for the 5-limit: 291 461 676] (patent val), 291 462 676] (291b), and 291 461 675] (291c).

Using the patent val, it tempers out 393216/390625 and [-47 37 -5 in the 5-limit; 2401/2400, 3136/3125, and 1162261467/1146880000 in the 7-limit; 243/242, 441/440, 5632/5625, and 58720256/58461513 in the 11-limit; 351/350, 1001/1000, 1575/1573, 3584/3575, and 43940/43923 in the 13-limit, so that it provides the optimal patent val for the 13-limit hemiwürschmidt temperament.

Using the 291b val, it tempers out 15625/15552 and [80 -46 -3 in the 5-limit.

Using the 291c val, it tempers out 390625000/387420489 and 1121008359375/1099511627776 in the 5-limit.

Prime harmonics

Approximation of prime harmonics in 291edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.92 +1.32 +0.25 +1.26 +0.71 -1.86 -0.61 -1.47 +1.35 +1.36
Relative (%) +0.0 -22.4 +31.9 +6.0 +30.5 +17.2 -45.2 -14.7 -35.7 +32.8 +32.9
Steps
(reduced)
291
(0)
461
(170)
676
(94)
817
(235)
1007
(134)
1077
(204)
1189
(25)
1236
(72)
1316
(152)
1414
(250)
1442
(278)

Subsets and supersets

Since 291 factors into 3 × 97, 291edo contains 3edo and 97edo as its subsets.