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836edo: Difference between revisions

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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, beating [[612edo]].  
836edo is a strong 11-limit system, having the record of lowest absolute error and 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.
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.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
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 }}.
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 ==
{| 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 0.0362
| 0.0362
| 2.52
| 2.52
|-
| 2.3.5.7.11.17
| 2500/2499, 3025/3024, 4375/4374, 57375/57344, 108086/108045
| {{mapping| 836 1325 1941 2347 2892 3417 }}
| +0.0264
| 0.0337
| 2.35
|- style="border-top: double;"
| 2.3.5.7.11.13
| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
| {{mapping| 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
| {{mapping| 836 1325 1941 2347 2892 3094 3417 }} (836)
| −0.0014
| 0.0747
| 5.21
|- style="border-top: double;"
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256
| {{mapping| 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
| {{mapping| 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 [[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]].
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 265\836<br>(56\836)
| 265\836<br />(56\836)
| 380.38<br>(80.38)
| 380.38<br />(80.38)
| 81/65<br>(22/21)
| 81/65<br />(22/21)
| [[Quasithird]] (836)
| [[Quasithird]] (836)
|-
|-
| 19
| 19
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|-
| 22
| 347\836<br />(5\836)
| 498.09<br />(7.18)
| 4/3<br />({{monzo| 16 -13 2 }})
| [[Major arcana]]
|-
|-
| 38
| 38
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|-
|-
| 44
| 44
| 347\836<br>(5\836)
| 347\836<br />(5\836)
| 498.09<br>(7.18)
| 498.09<br />(7.18)
| 4/3<br>(18375/18304)
| 4/3<br />(18375/18304)
| [[Ruthenium]]
| [[Ruthenium]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


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