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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|836}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
836edo has subset edos {{EDOs| | 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]] | |||