231edo: Difference between revisions
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The ''231 equal temperament'' divides the octave into 231 equal parts of 5.195 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it [[support]]s [[Kleismic_family#Tritikleismic|tritikleismic temperament]], and in fact provides the [[ | The ''231 equal temperament'' divides the octave into 231 equal parts of 5.195 cents each. | ||
== Theory == | |||
{{Harmonics in equal|231}} | |||
In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it [[support]]s [[Kleismic_family#Tritikleismic|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 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 - [http://x31eq.com/cgi-bin/rt.cgi?ets=41%26231&limit=11 see here.] | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |Subgroup | |||
! rowspan="2" |[[Comma list]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal | |||
8ve stretch (¢) | |||
! colspan="2" |Tuning error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3.5 | |||
|15625/15552, [-64, 36, 3⟩ | |||
|[{{val|231 366 536}}] | |||
|0.410 | |||
|0.334 | |||
|6.43 | |||
|- | |||
|2.3.5.7 | |||
|1029/1024, 15625/15552, 823543/820125 | |||
|[{{val|231 366 536 648}}] | |||
|0.539 | |||
|0.365 | |||
|7.01 | |||
|- | |||
|2.3.5.7.11 | |||
|385/384, 441/440, 14700/14641, 2460375/2458624 | |||
|[{{val|231 366 536 648 799}}] | |||
|0.469 | |||
|0.354 | |||
|6.81 | |||
|} | |||
== References == | |||
https://individual.utoronto.ca/kalendis/leap/index.htm | |||
[[Category:tritikleismic]] | [[Category:tritikleismic]] | ||