323edo: Difference between revisions

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== Theory ==

The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]] and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].

~~323edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the [[31-limit]].~~

=== Prime harmonics ===

== Regular temperament properties ==

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* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo|289]] and followed by [[441edo|441]].

=== Rank-2 temperaments ===

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 == Theory ==
-The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.
+edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].
+The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]] and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].
+In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.
 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].
-edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the [[31-limit]].
 === Prime harmonics ===
-{{Harmonics in equal|323|columns=11}}
+{{Harmonics in equal|323}}
 == Regular temperament properties ==
@@ Line 86: / Line 88: @@
 | 6.40
 |}
+* 323et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[289edo|289]] and followed by [[441edo|441]].
 === Rank-2 temperaments ===