231edo: Difference between revisions
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{{Infobox ET}} | |||
[[Category: | {{ED intro}} | ||
== Theory == | |||
In the 5-limit, 231et [[tempering out|tempers out]] the [[kleisma]], 15625/15552, and in the 7-limit [[1029/1024]], so that it [[support]]s the [[tritikleismic]] temperament, and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out [[385/384]], [[441/440]] and [[4000/3993]], leading to 11-limit tritikleismic for which it also gives the optimal patent val. | |||
231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a {{nowrap|41 & 149}} temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful edo harmonically, and it preserves the simple commas mentioned above. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|231}} | |||
=== Subsets and supersets === | |||
231 = 3 × 7 × 11, with subset edos {{EDOs| 3, 7, 11, 21, 33, and 77 }}. Since it contains [[77edo]], it can be used for playing such a tuning of the [[Carlos Alpha]] scale. | |||
== Regular temperament properties == | |||
{| 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 | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5 | |||
| 15625/15552, {{monzo| -64 36 3 }} | |||
| {{mapping| 231 366 536 }} | |||
| +0.410 | |||
| 0.334 | |||
| 6.43 | |||
|- | |||
| 2.3.5.7 | |||
| 1029/1024, 15625/15552, 823543/820125 | |||
| {{mapping| 231 366 536 648 }} | |||
| +0.539 | |||
| 0.365 | |||
| 7.01 | |||
|- | |||
| 2.3.5.7.11 | |||
| 385/384, 441/440, 4000/3993, 823543/820125 | |||
| {{mapping| 231 366 536 648 799 }} | |||
| +0.469 | |||
| 0.354 | |||
| 6.81 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 26\231 | |||
| 135.06 | |||
| 27/25 | |||
| [[Superlimmal]] | |||
|- | |||
| 1 | |||
| 27\231 | |||
| 140.26 | |||
| 243/224 | |||
| [[Septichrome]] | |||
|- | |||
| 1 | |||
| 45\231 | |||
| 233.77 | |||
| 8/7 | |||
| [[Slendric]] | |||
|- | |||
| 1 | |||
| 61\231 | |||
| 316.88 | |||
| 6/5 | |||
| [[Hanson]] | |||
|- | |||
| 1 | |||
| 62\231 | |||
| 322.08 | |||
| 135/112 | |||
| [[Dee leap week]] | |||
|- | |||
| 1 | |||
| 73\231 | |||
| 379.22 | |||
| 56/45 | |||
| [[Marthirds]] | |||
|- | |||
| 3 | |||
| 61\231<br />(16\231) | |||
| 316.88<br />(83.12) | |||
| 6/5<br />(21/20) | |||
| [[Tritikleismic]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | |||
; [[Mercury Amalgam]] | |||
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022) | |||
[[Category:Listen]] | |||
[[Category:Tritikleismic]] | |||