232edo
| ← 231edo | 232edo | 233edo → |
232 equal divisions of the octave (abbreviated 232edo or 232ed2), also called 232-tone equal temperament (232tet) or 232 equal temperament (232et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 232 equal parts of about 5.17 ¢ each. Each step represents a frequency ratio of 21/232, or the 232nd root of 2.
Theory
232 = 8 × 29, and 232edo shares its fifth with 29edo. The equal temperament supports and provides the optimal patent val for the 13-limit mystery temperament, the rank-3 pele temperament and the rank-3 trimyna temperament and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank-5 temperament.
Aside from its patent val, the 232d val ⟨232 368 539 652 803 859] is worth considering. Both temper out the würschmidt 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 4375/4374 and 16875/16807, supporting octoid. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 676/675, and 847/845, 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 a variant 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 major minthmic chords, and because it tempers out 364/363 it allows minor minthmic chords, and because it tempers out 847/845 it allows the cuthbert chords, making it a very flexible harmonic system.
Odd harmonics
| 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 (%) | +29 | +31 | -31 | -42 | +41 | +50 | -40 | -29 | +48 | -2 | -47 | |
| Steps (reduced) |
368 (136) |
539 (75) |
651 (187) |
735 (39) |
803 (107) |
859 (163) |
906 (210) |
948 (20) |
986 (58) |
1019 (91) |
1049 (121) | |
Subsets and supersets
Since 232 factors into 23 × 29, 232edo has subset edos 2, 4, 8, 29, 58, and 116.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 393216/390625, [46 -29 0⟩ | [⟨232 368 539]] | -0.5461 | 0.3989 | 7.71 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 61\232 | 315.52 | 6/5 | Acrokleismic (7-limit, 232d) |
| 1 | 75\232 | 387.93 | 5/4 | Würschmidt (5-limit) |
| 8 | 113\232 (3\232) |
584.48 (15.52) |
7/5 (100/99) |
Octoid (232d) |
| 29 | 96\232 (3\232) |
496.55 (15.52) |
4/3 (105/104) |
Mystery (232) |