232edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|232}} | {{EDO intro|232}} | ||
== 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. | 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. | ||
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 [[3125/3087]] and [[245/243]], supporting [[bohpier]]. 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]], [[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 [[3125/3087]] and [[245/243]], supporting [[bohpier]]. 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]], [[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 [[gentle 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 [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. | ||
=== 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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
| | | 393216/390625, {{monzo| 46 -29 0 }} | ||
|{{ | | {{mapping| 232 368 539 }} | ||
| -0.5461 | | -0.5461 | ||
| 0.3989 | | 0.3989 | ||
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|+Table of rank-2 temperaments by generator | |+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 | ||
|- | |- | ||
|1 | | 1 | ||
|75\232 | | 75\232 | ||
|387.93 | | 387.93 | ||
|5/4 | | 5/4 | ||
|[[Würschmidt]] | | [[Würschmidt]] | ||
|} | |} | ||