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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 7 × 11
{{ED intro}}
| Step size = 5.19481¢
| Fifth = 135\231 (701.30¢) (→ [[77edo|45\77]])
| Semitones = 21:18 (109.09¢ : 93.51¢)
| Consistency = 11
}}
The '''231 equal divisions of the octave''' ('''231edo'''), or the '''231(-tone) equal temperament''' ('''231tet''', '''231et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 231 [[equal]] parts of about 5.19 [[cent]]s each.


== 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 {{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.  
 
Since the patent val mapping of fifth in 231edo is divisible by 9, it can be used for playing 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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal <br> 8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 31: Line 27:
| 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
| 6.43
| 6.43
Line 38: Line 34:
| 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
Line 45: Line 41:
| 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
| 6.81
| 6.81
Line 53: Line 49:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods <br> per octave
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator
|-
! Cents
! Periods<br />per 8ve
! Associated <br> ratio
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 26\231
| 135.06
| 27/25
| [[Superlimmal]]
|-
|-
| 1
| 1
Line 90: Line 94:
|-
|-
| 3
| 3
| 61\231<br>(16\231)
| 61\231<br />(16\231)
| 316.88<br>(83.12)
| 316.88<br />(83.12)
| 6/5<br>(21/20)
| 6/5<br />(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)


[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Tritikleismic]]
[[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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