323edo: Difference between revisions

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== Theory ==

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]].

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]].

As an equal temperament, it [[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.

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.

~~Since 323 factors into~~ {{~~factorisation~~|323}}~~, 323edo shares the excellent approximations of [[25/24]]~~ in ~~[[17edo]] and~~ of ~~[[28/27]] and the [[6/5]]~~ in ~~[[19edo]].~~

=== Prime harmonics ===

{{Harmonics in equal|323|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 323edo (continued)}}

=== ~~Prime harmonics~~ ===

=== Subsets and supersets ===

Since 323 factors into primes as {{nowrap| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].

== Regular temperament properties ==

@@ Line 3: / Line 3: @@
 == Theory ==
-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]].
+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]].
 As an equal temperament, it [[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.
+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.
-Since 323 factors into {{factorisation|323}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].
+=== Prime harmonics ===
+{{Harmonics in equal|323|columns=11}}
+{{Harmonics in equal|323|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 323edo (continued)}}
-=== Prime harmonics ===
+=== Subsets and supersets ===
-{{Harmonics in equal|323}}
+Since 323 factors into primes as {{nowrap| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].
 == Regular temperament properties ==