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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|836}}
{{ED intro}}


== Theory ==
== Theory ==
836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].  
836edo is a strong 11-limit system, having the record of lowest absolute error and beating [[612edo]]. Its perfect fifth is notable as the midpoint between that of [[19edo]] and [[22edo]], which are nearly the same distance away from a pure 3/2 but in opposite directions, thus being quite an accurate approximation.


The equal temperament [[tempering out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.  
As an equal temperament, it [[tempering out|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.  


Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.  
Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.  
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 836 factors into {{factorization|836}}, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. [[1672edo]], which doubles it, provides a good correction for [[harmonic]] [[13/1|13]].  
Since 836 factors into 2<sup>2</sup> × 11 × 19, 836edo has subset edos {{EDOs| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. [[1672edo]], which doubles it, provides a good correction for [[harmonic]] [[13/1|13]].


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| 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
| 2.3
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| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
| {{mapping| 836 1325 1941 2347 2892 3094 }} (836)
| {{mapping| 836 1325 1941 2347 2892 3094 }} (836)
| &minus;0.0085
| −0.0085
| 0.0785
| 0.0785
| 5.47
| 5.47
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| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
| &minus;0.0014
| −0.0014
| 0.0747
| 0.0747
| 5.21
| 5.21
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| 0.0747
| 0.0747
| 5.20
| 5.20
{{comma basis end}}
|}
* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo|612]] and before [[1084edo|1084]].
* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo|612]] and before [[1084edo|1084]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| 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
| 1
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| 4/3<br />(18375/18304)
| 4/3<br />(18375/18304)
| [[Ruthenium]]
| [[Ruthenium]]
{{rank-2 end}}
|}
{{orf}}
<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


[[Category:Quasithird]]
[[Category:Quasithird]]

Latest revision as of 04:27, 11 April 2026

← 835edo 836edo 837edo →
Prime factorization 22 × 11 × 19
Step size 1.43541 ¢ 
Fifth 489\836 (701.914 ¢)
Semitones (A1:m2) 79:63 (113.4 ¢ : 90.43 ¢)
Consistency limit 11
Distinct consistency limit 11

836 equal divisions of the octave (abbreviated 836edo or 836ed2), also called 836-tone equal temperament (836tet) or 836 equal temperament (836et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 836 equal parts of about 1.44 ¢ each. Each step represents a frequency ratio of 21/836, or the 836th root of 2.

Theory

836edo is a strong 11-limit system, having the record of lowest absolute error and beating 612edo. Its perfect fifth is notable as the midpoint between that of 19edo and 22edo, which are nearly the same distance away from a pure 3/2 but in opposite directions, thus being quite an accurate approximation.

As an equal temperament, it tempers out the counterschisma and the enneadeca in the 5-limit; 4375/4374, 703125/702464 in the 7-limit; 3025/3024 and 9801/9800 in the 11-limit. It supports enneadecal in the 7-limit as well as hemienneadecal in the 11-limit. It also tunes orga and quasithird. In addition, it is divisible by 44 and in light of that it tunes ruthenium in the 7-limit and also 11-limit.

Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the patent val, it tempers out 2200/2197, 4096/4095, 31250/31213 in the 13-limit; and 1275/1274, 2500/2499, 2601/2600 in the 17-limit. It provides the optimal patent val for 13-limit quasithird. Using the 836f val, it tempers out 1716/1715, 2080/2079, 15379/15360 in the 13-limit; and 2431/2430, 2500/2499, 4914/4913, 5832/5831, 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.

Prime harmonics

Approximation of prime harmonics in 836edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.041 -0.189 +0.074 -0.122 +0.621 -0.171 -0.384 +0.434 -0.391 +0.419
Relative (%) +0.0 -2.9 -13.2 +5.1 -8.5 +43.2 -11.9 -26.7 +30.2 -27.2 +29.2
Steps
(reduced) 		836
(0) 		1325
(489) 		1941
(269) 		2347
(675) 		2892
(384) 		3094
(586) 		3417
(73) 		3551
(207) 		3782
(438) 		4061
(717) 		4142
(798)

Subsets and supersets

Since 836 factors into 22 × 11 × 19, 836edo has subset edos 2, 4, 11, 19, 22, 38, 44, 76, 209, 418. 1672edo, which doubles it, provides a good correction for harmonic 13.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1325 836⟩ [⟨836 1325]] +0.0130 0.0130 0.90
2.3.5 [-14 -19 19⟩, [-69 45 -1⟩ [⟨836 1325 1941]] +0.0358 0.0340 2.37
2.3.5.7 4375/4374, 703125/702464, [41 -4 2 -14⟩ [⟨836 1325 1941 2347]] +0.0203 0.0399 2.78
2.3.5.7.11 3025/3024, 4375/4374, 234375/234256, [22 -4 2 -6 -1⟩ [⟨836 1325 1941 2347 2892]] +0.0233 0.0362 2.52
2.3.5.7.11.17 2500/2499, 3025/3024, 4375/4374, 57375/57344, 108086/108045 [⟨836 1325 1941 2347 2892 3417]] +0.0264 0.0337 2.35
2.3.5.7.11.13 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213 [⟨836 1325 1941 2347 2892 3094]] (836) −0.0085 0.0785 5.47
2.3.5.7.11.13.17 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374 [⟨836 1325 1941 2347 2892 3094 3417]] (836) −0.0014 0.0747 5.21
2.3.5.7.11.13 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256 [⟨836 1325 1941 2347 2892 3093]] (836f) +0.0561 0.0805 5.60
2.3.5.7.11.13.17 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4914/4913, 11271/11264 [⟨836 1325 1941 2347 2892 3093 3417]] (836f) +0.0541 0.0747 5.20
  • 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past 612 and before 1084.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 347\836 498.09 4/3 Counterschismic
2 161\836 231.10 8/7 Orga (836f)
2 265\836
(56\836) 		380.38
(80.38) 		81/65
(22/21) 		Quasithird (836)
19 347\836
(5\836) 		498.09
(7.18) 		4/3
(225/224) 		Enneadecal
22 347\836
(5\836) 		498.09
(7.18) 		4/3
([16 -13 2⟩) 		Major arcana
38 347\836
(5\836) 		498.09
(7.18) 		4/3
(225/224) 		Hemienneadecal
44 347\836
(5\836) 		498.09
(7.18) 		4/3
(18375/18304) 		Ruthenium

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

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