232edo
The 232 equal division divides the octave into 232 equal parts of 5.172 cents each. It provides the optimal patent val for 13-limit mystery temperament, the rank three pele temperament and the rank three trimyna temperament and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank five temperament.
| ← 231edo | 232edo | 233edo → |
Aside from its patent val, the 232d val is worth considering. Both temper out the wuerschmidt comma, 393216/390625, in the 5-limit.
In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 3125/3097 and 245/243, supporting bohpier temperament. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 441/440 and 676/675, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports variants of octoid.
Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows minthmic chords, and because it tempers out 364/363 it allows gentle chords, and because it tempers out 847/845 it allows the cuthbert triad, making it a very flexible harmonic system.
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.49 | +1.62 | -1.58 | -2.19 | +2.13 | +2.58 | -2.06 | -1.51 | +2.49 | -0.09 | -2.41 |
| Relative (%) | +28.9 | +31.3 | -30.6 | -42.3 | +41.2 | +49.8 | -39.9 | -29.1 | +48.1 | -1.8 | -46.6 | |
| Steps (reduced) |
368 (136) |
539 (75) |
651 (187) |
735 (39) |
803 (107) |
859 (163) |
906 (210) |
948 (20) |
986 (58) |
1019 (91) |
1049 (121) | |