836edo: Difference between revisions

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== 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]].

It [[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.

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=== Subsets and supersets ===

Since 836 factors into ~~{{factorization|836}}~~, 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]].

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 ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|836}}
+{{ED intro}}
 == Theory ==
-edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].
+edo is a strong 11-limit system, having the record of lowest absolute error and beating [[612edo]].
-It [[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.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 836 factors into {{factorization|836}}, 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]].
+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 ==