1342edo
| ← 1341edo | 1342edo | 1343edo → |
1342 equal divisions of the octave (abbreviated 1342edo or 1342ed2), also called 1342-tone equal temperament (1342tet) or 1342 equal temperament (1342et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1342 equal parts of about 0.894 ¢ each. Each step represents a frequency ratio of 21/1342, or the 1342nd root of 2.
1342edo is consistent to the 9-odd-limit, but there is a large relative delta for harmonics 7 and 11. Its notability lies in the utility as every other step of the full 13-limit monster – 2684edo, so it probably makes more sense as a 2.3.5.13 subgroup temperament. In the 5-limit it tempers out kwazy, [-53 10 16⟩, senior [-17 62 -35⟩, and egads, [-36 52 51⟩; in the 2.3.5.13 subgroup it tempers out 140625/140608.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.018 | -0.025 | -0.421 | +0.396 | +0.009 | -0.335 | +0.252 | +0.340 | -0.367 | +0.419 |
| relative (%) | +0 | -2 | -3 | -47 | +44 | +1 | -38 | +28 | +38 | -41 | +47 | |
| Steps (reduced) |
1342 (0) |
2127 (785) |
3116 (432) |
3767 (1083) |
4643 (617) |
4966 (940) |
5485 (117) |
5701 (333) |
6071 (703) |
6519 (1151) |
6649 (1281) | |
Subsets and supersets
Since 1342 factors as 2 × 11 × 61, 1342edo has subset edos 2, 11, 22, 61, 122, and 671. 2684edo, which doubles it, corrects the harmonics 7 and 11 to near-just qualities.