940edo
| ← 939edo | 940edo | 941edo → |
940 equal divisions of the octave (abbreviated 940edo or 940ed2), also called 940-tone equal temperament (940tet) or 940 equal temperament (940et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 940 equal parts of about 1.28 ¢ each. Each step represents a frequency ratio of 21/940, or the 940th root of 2.
940edo is distinctly consistent through the 11-odd-limit. The equal temperament tempers out 2401/2400 in the 7-limit and 5632/5625 and 9801/9800 in the 11-limit, which means it supports decoid and in fact gives an excellent tuning for it. In the 13-limit, it tempers out 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095 and 4225/4224, so that it supports and gives the optimal patent val for 13-limit decoid. It also gives the optimal patent val for the greenland and baffin temperaments, and for the rank-5 temperament tempering out 676/675.
The non-patent val ⟨940 1491 2184 2638 3254 3481] gives a tuning almost identical to the POTE tuning for the 13-limit pele temperament, tempering out 196/195, 352/351 and 364/363.
In higher limits, it is a satisfactory no-13's 23-limit tuning. Another thing to note is that its 15th harmonic is just barely off of the sum of mappings for 3rd and 5th harmonics, which adds a peculiarity to it where it has good 3/2 and 5/4, but 15/8 is one step off from the closest location.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.173 | +0.495 | +0.110 | +0.345 | +0.171 | -0.528 | -0.609 | -0.275 | -0.066 | +0.283 | -0.189 |
| Relative (%) | +13.5 | +38.8 | +8.6 | +27.0 | +13.4 | -41.3 | -47.7 | -21.5 | -5.2 | +22.2 | -14.8 | |
| Steps (reduced) |
1490 (550) |
2183 (303) |
2639 (759) |
2980 (160) |
3252 (432) |
3478 (658) |
3672 (852) |
3842 (82) |
3993 (233) |
4129 (369) |
4252 (492) | |
Subsets and supersets
Since 940 factors into 22 × 5 × 47, 940edo has subset edos 2, 4, 5, 10, 20, 47, 94, 188, 235, 470, of which 94edo is notable.
1880edo, which doubles 940edo, provides good correction for harmonics 5 and 13, though the error of 3 has accumulated to the point of inconsistency in the 9-odd-limit.