232edo: Difference between revisions
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== Theory == | == Theory == | ||
232 = 8 × 29, and 232edo shares its [[3/2|fifth]] with [[29edo]]. The equal temperament [[support]]s 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. | 232 = 8 × 29, and 232edo shares its [[3/2|fifth]] with [[29edo]]. The equal temperament [[support]]s 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 {{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. | Aside from its [[patent val]], the 232d val {{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. | 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. 232edo is also the first edo that approximates [[6/5]] more accurately than [[19edo]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
Revision as of 23:14, 19 December 2025
| ← 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. 232edo is also the first edo that approximates 6/5 more accurately than 19edo.
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 (%) | +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) | |
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) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct