836edo: Difference between revisions
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17-limit notability; point to 1672edo; clarify the title row of the rank-2 temp table |
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== Theory == | == Theory == | ||
836edo is a strong 11-limit system, having the lowest absolute error | 836edo is a strong 11-limit system, having the 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. | 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. | ||
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. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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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 | |||
| style="border-top: double;" | 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213 | |||
| style="border-top: double;" | {{mapping| 836 1325 1941 2347 2892 3094 }} (836) | |||
| style="border-top: double;" | -0.0085 | |||
| style="border-top: double;" | 0.0785 | |||
| style="border-top: double;" | 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 | |||
| style="border-top: double;" | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256 | |||
| style="border-top: double;" | {{mapping| 836 1325 1941 2347 2892 3093 }} (836f) | |||
| style="border-top: double;" | +0.0561 | |||
| style="border-top: double;" | 0.0805 | |||
| style="border-top: double;" | 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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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
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| [[Ruthenium]] | | [[Ruthenium]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Quasithird]] | [[Category:Quasithird]] | ||