317edo
| ← 316edo | 317edo | 318edo → |
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
| 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 |