3178edo
| ← 3177edo | 3178edo | 3179edo → |
3178 equal divisions of the octave (abbreviated 3178edo or 3178ed2), also called 3178-tone equal temperament (3178tet) or 3178 equal temperament (3178et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3178 equal parts of about 0.378 ¢ each. Each step represents a frequency ratio of 21/3178, or the 3178th root of 2.
3178edo is quite accurate in the 23-limit, consistent to the 27-odd-limit, and has an exceptional approximation of harmonic 13. However, like most edos of this size, it is rather impractical to use. It tempers out several of the smaller 23-limit superparticular commas, including 9801/9800 in the 11-limit; 10648/10647 and 123201/123200 in the 13-limit; 5832/5831, 14400/14399, and 28561/28560 in the 17-limit; 6175/6174, 10830/10829, 12636/12635, 14080/14079, 14365/14364, 23409/23408, 28900/28899, and 43681/43680 in the 19-limit; 8625/8624, 11271/11270, 12168/12167 and 43264/43263 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.004 | -0.033 | +0.085 | -0.028 | +0.001 | +0.016 | +0.033 | +0.045 | +0.127 | -0.165 |
| Relative (%) | +0.0 | -1.1 | -8.7 | +22.6 | -7.4 | +0.3 | +4.3 | +8.6 | +12.0 | +33.6 | -43.6 | |
| Steps (reduced) |
3178 (0) |
5037 (1859) |
7379 (1023) |
8922 (2566) |
10994 (1460) |
11760 (2226) |
12990 (278) |
13500 (788) |
14376 (1664) |
15439 (2727) |
15744 (3032) | |
Subsets and supersets
Since 3178 factors into 2 × 7 × 227, 3178edo has subset edos 2, 7, 14, 227, 454, and 1589.