317edo

 ← 316edo 317edo 318edo →
Prime factorization 317 (prime)
Step size 3.78549¢
Fifth 185\317 (700.315¢)
Semitones (A1:m2) 27:26 (102.2¢ : 98.42¢)
Dual sharp fifth 186\317 (704.101¢)
Dual flat fifth 185\317 (700.315¢)
Dual major 2nd 54\317 (204.416¢)
Consistency limit 5
Distinct consistency limit 5

317 equal divisions of the octave (abbreviated 317edo or 317ed2), also called 317-tone equal temperament (317tet) or 317 equal temperament (317et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 317 equal parts of about 3.79 ¢ each. Each step represents a frequency ratio of 21/317, or the 317th root of 2.

Theory

317edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway its steps. To start with, it can be considered in the 2.9.5.7 subgroup, in which it is strong, tempering out 420175/419904, 703125/702464, and 33554432/33480783.

Using the patent val nonetheless, the equal temperament tempers out 78732/78125 (sensipent comma in the 5-limit; 16875/16807, 65625/65536, 589824/588245, and 49009212/48828125 in the 7-limit. In the 11-limit, the 317e val tempers out 540/539, 1375/1372, and 3025/3024, whereas the patent val tempers out 441/440, 4000/3993, and 14700/14641.

Odd harmonics

Approximation of odd harmonics in 317edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.64 -0.19 +0.26 +0.51 +1.36 -0.15 -1.83 +1.04 +1.54 -1.38 +0.12
Relative (%) -43.3 -5.1 +6.8 +13.4 +36.0 -3.9 -48.4 +27.4 +40.7 -36.5 +3.1
Steps
(reduced)
502
(185)
736
(102)
890
(256)
1005
(54)
1097
(146)
1173
(222)
1238
(287)
1296
(28)
1347
(79)
1392
(124)
1434
(166)

Subsets and supersets

317edo is the 66th prime edo. 634edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [1005 -317 [317 1005]] -0.0799 0.0799 2.11
2.9.5 [-53 5 16, [33 -17 9 [317 1005 736]] -0.0254 0.1009 2.67
2.9.5.7 420175/419904, 703125/702464, 33554432/33480783 [317 1005 736 890]] -0.0422 0.0921 2.43
2.9.5.7.11 6250/6237, 12005/11979, 46656/46585, 151263/151250 [317 1005 736 890 1097]] -0.1126 0.1631 4.31
2.9.5.7.11.13 1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728 [317 1005 736 890 1097 1173]] -0.0871 0.1594 4.21
2.9.5.7.11.13.17 936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957 [317 1005 736 890 1097 1173 1296]] -0.1109 0.1587 4.19

Francium