# 291edo

 ← 290edo 291edo 292edo →
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.