231edo: Difference between revisions

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== Theory ==
== Theory ==
In the 5-limit, 231et 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.
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 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.  
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.  
231 = 3 × 7 × 11, with subset 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.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|231}}
{{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 ==
== Regular temperament properties ==
Line 25: Line 26:
| 2.3.5
| 2.3.5
| 15625/15552, {{monzo| -64 36 3 }}
| 15625/15552, {{monzo| -64 36 3 }}
| [{{val| 231 366 536 }}]
| {{mapping| 231 366 536 }}
| 0.410
| 0.410
| 0.334
| 0.334
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| 2.3.5.7
| 2.3.5.7
| 1029/1024, 15625/15552, 823543/820125
| 1029/1024, 15625/15552, 823543/820125
| [{{val| 231 366 536 648 }}]
| {{mapping| 231 366 536 648 }}
| 0.539
| 0.539
| 0.365
| 0.365
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| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 441/440, 4000/3993, 823543/820125
| 385/384, 441/440, 4000/3993, 823543/820125
| [{{val| 231 366 536 648 799 }}]
| {{mapping| 231 366 536 648 799 }}
| 0.469
| 0.469
| 0.354
| 0.354
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Tritikleismic]]
| [[Tritikleismic]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM Sins of Stoicism (Demo Version, March 2022)] by [[Mercury Amalgam]]
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Tritikleismic]]
[[Category:Tritikleismic]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 10:02, 26 March 2024

← 230edo 231edo 232edo →
Prime factorization 3 × 7 × 11
Step size 5.19481 ¢ 
Fifth 135\231 (701.299 ¢) (→ 45\77)
Semitones (A1:m2) 21:18 (109.1 ¢ : 93.51 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

In the 5-limit, 231et tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports 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 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

Approximation of odd harmonics in 231edo
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)

Subsets and supersets

231 = 3 × 7 × 11, with subset 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

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, 4000/3993, 823543/820125 [⟨231 366 536 648 799]] 0.469 0.354 6.81

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
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
(16\231) 		316.88
(83.12) 		6/5
(21/20) 		Tritikleismic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Mercury Amalgam
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