836edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{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]]. | ||
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 | 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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| 0.0337 | | 0.0337 | ||
| 2.35 | | 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 | | 2.3.5.7.11.13.17 | ||
| 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) | ||
| | | −0.0014 | ||
| 0.0747 | | 0.0747 | ||
| 5.21 | | 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 | | 2.3.5.7.11.13.17 | ||
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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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 113: | Line 115: | ||
|- | |- | ||
| 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 | | 22 | ||
| 347\836<br>(5\836) | | 347\836<br />(5\836) | ||
| 498.09<br>(7.18) | | 498.09<br />(7.18) | ||
| 4/3<br>({{monzo| 16 -13 2 }}) | | 4/3<br />({{monzo| 16 -13 2 }}) | ||
| [[Major arcana]] | | [[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>* | <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]] | ||