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The ''231 equal temperament'' divides the octave into 231 equal parts of 5.195 cents each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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}}
{{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.]
=== 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 ==
{| class="wikitable center-4 center-5 center-6"
{| 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]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! 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
| 2.3.5
|15625/15552, [-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
|6.43
| 6.43
|-
|-
|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
|7.01
| 7.01
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|385/384, 441/440, 14700/14641, 2460375/2458624
| 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
|6.81
| 6.81
|}
|}


== Rank two temperaments by generator ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per octave
|-
!Generator
! Periods<br />per 8ve
!Cents
! Generator*
!Associated
! Cents*
ratio
! Associated<br />ratio*
!Temperaments
! Temperaments
|-
|-
|1
| 1
|62\231
| 26\231
|322.08
| 135.06
|
| 27/25
|Dee / Iranian Leap Week
| [[Superlimmal]]
|-
|-
|3
| 1
|61\231
| 27\231
(16\231)
| 140.26
|316.88
| 243/224
(83.12)
| [[Septichrome]]
|6/5
|-
|Tritrikleismic
| 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]]
|}
|}
== References ==
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
https://individual.utoronto.ca/kalendis/leap/index.htm
 
[[Category:tritikleismic]]
== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)
 
[[Category:Listen]]
[[Category:Tritikleismic]]

Latest revision as of 14:20, 20 February 2025

← 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

231 equal divisions of the octave (abbreviated 231edo or 231ed2), also called 231-tone equal temperament (231tet) or 231 equal temperament (231et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 231 equal parts of about 5.19 ¢ each. Each step represents a frequency ratio of 21/231, or the 231st root of 2.

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

Table of rank-2 temperaments by generator
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 distinct

Music

Mercury Amalgam
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