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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 [[Optimal_patent_val|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.
{{Infobox ET}}
[[Category:tritikleismic]]
{{ED intro}}
 
== Theory ==
231et [[tempering out|tempers out]] the kleisma, [[15625/15552]] in the [[5-limit]], and [[1029/1024]] in the [[7-limit]], 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 ===
Since 231 factors into primes as {{nowrap| 3 × 7 × 11 }}, 231edo has 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. [[1848edo]], which divides its step into eight, provides a near-just representation of the 11-limit.
 
== 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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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]]

Latest revision as of 13:14, 12 July 2026

← 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

231et tempers out the kleisma, 15625/15552 in the 5-limit, and 1029/1024 in the 7-limit, 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

Since 231 factors into primes as 3 × 7 × 11, 231edo has 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. 1848edo, which divides its step into eight, provides a near-just representation of the 11-limit.

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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