836edo: Difference between revisions

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{{EDO intro|836}}
{{EDO intro|836}}


836edo is a strong 11-limit system, having the lowest absolute error beating [[612edo]].  
== Theory ==
836edo is a strong 11-limit system, having the lowest absolute error, beating [[612edo]].
 
The equal temperament 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.


836edo is a tuning for the [[enneadecal]] in the 7-limit as well as the [[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.
=== Prime harmonics ===
=== Prime harmonics ===
{{harmonics in equal|836}}
{{Harmonics in equal|836}}


=== Subsets and supersets ===
=== Subsets and supersets ===
836edo has subset edos {{EDOs|1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418}}.
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 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1325 836 }}
| {{mapping| 836 1325 }}
| +0.0130
| 0.0130
| 0.90
|-
| 2.3.5
| {{monzo| -14 -19 19 }}, {{monzo| -69 45 -1 }}
| {{mapping| 836 1325 1941 }}
| +0.0358
| 0.0340
| 2.37
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| 41 -4 2 -14 }}
| {{mapping| 836 1325 1941 2347 }}
| +0.0203
| 0.0399
| 2.78
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| 22 -4 2 -6 -1 }}
| {{mapping| 836 1325 1941 2347 2892 }}
| +0.0233
| 0.0362
| 2.52
|}
* 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 ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>Ratio
! Temperaments
|-
| 1
| 347\836
| 498.09
| 4/3
| [[Counterschismic]]
|-
| 2
| 161\836
| 231.10
| 8/7
| [[Orga]] (836f)
|-
| 2
| 265\836<br>(56\836)
| 380.38<br>(80.38)
| 81/65<br>(22/21)
| [[Quasithird]] (836)
|-
| 19
| 347\836<br>(5\836)
| 498.09<br>(7.18)
| 4/3<br>(225/224)
| [[Enneadecal]]
|-
| 38
| 347\836<br>(5\836)
| 498.09<br>(7.18)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
|-
| 44
| 347\836<br>(5\836)
| 498.09<br>(7.18)
| 4/3<br>(18375/18304)
| [[Ruthenium]]
|}
 
[[Category:Quasithird]]

Revision as of 13:49, 8 September 2023

← 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

Template:EDO intro

Theory

836edo is a strong 11-limit system, having the lowest absolute error, beating 612edo.

The equal temperament 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.

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.

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
  • 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
(Reduced) 		Cents
(Reduced) 		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
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
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