Generator preimage: Difference between revisions

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Also known as a generator preimage transversal or 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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=Finding the generator preimage transversal=

Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].

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and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a prime-count vector represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.

and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.

Essentially we need a way to do:

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[[Category:generator]]

~~[[Category:theory]]~~

[[Category:Regular temperament theory]]

[[Category:todo:reduce_mathslang]]

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-Also known as a generator preimage transversal or 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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 =Finding the generator preimage transversal=
+{{todo|cleanup|inline=1|text=Add simpler algorithm}}
 Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].
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-and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a prime-count vector represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.
+and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.
 Essentially we need a way to do:
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 [[Category:generator]]
-[[Category:theory]]
 [[Category:Regular temperament theory]]
 [[Category:todo:reduce_mathslang]]