Optimal patent val: Difference between revisions

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~~Given any collection of~~ ''p''~~-limit commas~~, there is a finite list of ~~''p''-limit~~ [[patent val]]s ~~tempering~~ out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the ~~unique patent val~~ which has the lowest [[TE error]]; this is the (TE) ~~'''~~optimal patent val~~'''~~ for the temperament ~~defined by the commas~~. Note that other definitions of error, ~~such~~ as ~~maximum ''p''-limit error~~, or ~~maximum ''q''-limit error where q~~ is the ~~largest odd number less than the prime above p, lead to different results~~.

The '''optimal patent val''' for a [[regular temperament]] is the unique [[patent val]] that [[support]]s the temperament with the lowest [[error]].

Given any temperament, which is characterized by the [[comma]]s it [[tempering out|tempers out]], there is a finite list of [[patent val]]s that temper out all the commas of the temperament in the same [[subgroup]]. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the one which has the lowest [[TE error]]; this is the (TE) optimal patent val for the temperament. Note that other definitions of error lead to different results.

On this wiki, the optimal patent val for each temperament is given as the last patent val in the [[optimal ET sequence]], or stated explicitly in case it is not a member of the sequence.

== Instructions ==

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{{todo

| improve layout

~~| add introduction~~

~~| introduce lemma~~

| increase applicability

| text = add structure by switching to tables

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-Given any collection of ''p''-limit commas, there is a finite list of ''p''-limit [[patent val]]s tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[TE error]]; this is the (TE) '''optimal patent val''' for the temperament defined by the commas. Note that other definitions of error, such as maximum ''p''-limit error, or maximum ''q''-limit error where q is the largest odd number less than the prime above p, lead to different results.
+The '''optimal patent val''' for a [[regular temperament]] is the unique [[patent val]] that [[support]]s the temperament with the lowest [[error]].
+Given any temperament, which is characterized by the [[comma]]s it [[tempering out|tempers out]], there is a finite list of [[patent val]]s that temper out all the commas of the temperament in the same [[subgroup]]. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the one which has the lowest [[TE error]]; this is the (TE) optimal patent val for the temperament. Note that other definitions of error lead to different results.
+On this wiki, the optimal patent val for each temperament is given as the last patent val in the [[optimal ET sequence]], or stated explicitly in case it is not a member of the sequence.
 == Instructions ==
@@ Line 2,023: / Line 2,027: @@
 {{todo
 | improve layout
-| add introduction
-| introduce lemma
 | increase applicability
 | text = add structure by switching to tables