Generator preimage: Difference between revisions

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Also known as a '''generator preimage transversal''' or a '''generator detempering'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generated by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].

=Technical Definition=

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getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)

</syntaxhighlight>}}

~~== See also ==~~

* [[Detempering]]

[[Category:generator]]

[[Category:Regular temperament theory]]

[[Category:todo:reduce_mathslang]]

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-Also known as a '''generator preimage transversal''' or a '''generator detempering'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.
+Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.
+Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generated by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].
 =Technical Definition=
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 getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
 </syntaxhighlight>}}
-== See also ==
-* [[Detempering]]
 [[Category:generator]]
 [[Category:Regular temperament theory]]
 [[Category:todo:reduce_mathslang]]