232edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
232edo [[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 [[ | 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 [[minthmic chords]], and because it tempers out 364/363 it allows [[ | 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 === | ||
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Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}. | Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}. | ||
==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" | [[Comma list]] | |||
| | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
| | ! colspan="2" | Tuning error | ||
| | |||
|- | |- | ||
|2.3.5 | ! [[TE error|Absolute]] (¢) | ||
| | ! [[TE simple badness|Relative]] (%) | ||
|{{ | |- | ||
| | | 2.3.5 | ||
| 393216/390625, {{monzo| 46 -29 0 }} | |||
| {{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 | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|75\232 | | 61\232 | ||
|387.93 | | 315.52 | ||
|5/4 | | 6/5 | ||
|[[Würschmidt]] | | [[Acrokleismic]] (7-limit, 232d) | ||
|- | |||
| 1 | |||
| 75\232 | |||
| 387.93 | |||
| 5/4 | |||
| [[Würschmidt]] (5-limit) | |||
|- | |||
| 8 | |||
| 113\232<br>(3\232) | |||
| 584.48<br>(15.52) | |||
| 7/5<br>(100/99) | |||
| [[Octoid]] (232d) | |||
|- | |||
| 29 | |||
| 96\232<br>(3\232) | |||
| 496.55<br>(15.52) | |||
| 4/3<br>(105/104) | |||
| [[Mystery]] (232) | |||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
== Scales == | |||
* [[Mystery58]] | |||
[[Category:Mystery]] | [[Category:Mystery]] | ||
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[[Category:Trimyna]] | [[Category:Trimyna]] | ||
[[Category:Mynucumic]] | [[Category:Mynucumic]] | ||