232edo: Difference between revisions
→Regular temperament properties: +acrokleismic |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| 393216/390625, {{monzo| 46 -29 0 }} | | 393216/390625, {{monzo| 46 -29 0 }} | ||
| {{mapping| 232 368 539 }} | | {{mapping| 232 368 539 }} | ||
| | | −0.5461 | ||
| 0.3989 | | 0.3989 | ||
| 7.71 | | 7.71 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Mystery]] (232) | | [[Mystery]] (232) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Mystery]] | [[Category:Mystery]] | ||
Revision as of 14:21, 20 February 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.
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