Generator preimage: Difference between revisions
todo cleanup, see also |
m →Example: Spelling (update link) |
||
| (7 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
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= | =Technical Definition= | ||
| Line 22: | Line 24: | ||
=Finding the generator preimage transversal= | =Finding the generator preimage transversal= | ||
{{todo|cleanup}} | {{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]]. | 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]]. | ||
| Line 121: | Line 123: | ||
=== Example === | === Example === | ||
Our mapping is 5-limit meantone, in [[equave-reduced generator form]]: | Our mapping is 5-limit meantone, in [[equave-reduced-generator form]]: | ||
| Line 412: | Line 414: | ||
getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *) | getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *) | ||
</syntaxhighlight>}} | </syntaxhighlight>}} | ||
[[Category:generator]] | [[Category:generator]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:todo:reduce_mathslang]] | [[Category:todo:reduce_mathslang]] | ||