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]] | ||
Revision as of 09:12, 16 March 2022
The 231 equal temperament divides the octave into 231 equal parts of 5.195 cents each.
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.66 | -1.90 | -2.59 | -1.31 | -0.67 | +1.03 | -2.55 | -1.06 | -1.41 | +1.95 | +0.30 |
| Relative (%) | -12.6 | -36.5 | -49.9 | -25.3 | -12.9 | +19.8 | -49.2 | -20.4 | -27.1 | +37.5 | +5.7 | |
| Steps (reduced) |
366 (135) |
536 (74) |
648 (186) |
732 (39) |
799 (106) |
855 (162) |
902 (209) |
944 (20) |
981 (57) |
1015 (91) |
1045 (121) | |
In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports 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 - see here.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, [-64, 36, 3⟩ | [⟨231 366 536]] | 0.410 | 0.334 | 6.43 |
| 2.3.5.7 | 1029/1024, 15625/15552, 823543/820125 | [⟨231 366 536 648]] | 0.539 | 0.365 | 7.01 |
| 2.3.5.7.11 | 385/384, 441/440, 14700/14641, 2460375/2458624 | [⟨231 366 536 648 799]] | 0.469 | 0.354 | 6.81 |