Wedgie/Archived version: Difference between revisions

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The pages are: Catalog of eleven-limit rank three temperaments, Catalog of eleven-limit rank two temperaments, Catalog of thirteen-limit rank two temperaments, Catalog of seven-limit rank two temperaments, and User:TallKite/Catalog of eleven-limit rank three temperaments with Color names

~~{{Todo|add definition|update|inline=1|comment=Document this method via [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17912.html Temperament seaches using wedgies only].}}~~

See also: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17912.html

== Converting to wedgies from reduced row echelon form ==

''Main article: [[Mathematical theory of regular temperaments]]''

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank ''r'' in ''n'' dimensions (where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes in the ''p''-limit) take a wedge product of basis vectors involving {{nowrap|''r'' − 1}} basis elements (i.e., the wedge product of {{nowrap|''r'' − 1}} elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an ''r''-fold wedge product with sign ±1. Take the corresponding element of the wedgie times the ±1 sign (which is computed from the parity of the permutation of the ''r'' elements.) This gives a val; do this for every combination of {{nowrap|''r'' − 1}} basis elements to obtain ''n'' choose {{nowrap|''r'' − 1}} vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.

An alternative explanation of this process is provided here: [[Intro to exterior algebra for RTT#Converting varianced multivectors to matrices]]

== See also ==

* [[Intro to exterior algebra for RTT]]: for detailed background about and further explanations for how to work with multivectors and multicovectors such as wedgies

~~[[Category:Math]]~~

[[Category:Exterior algebra]]

~~{{todo| improve readability }}~~

== Notes ==

@@ Line 141: / Line 141: @@
 The pages are: Catalog of eleven-limit rank three temperaments, Catalog of eleven-limit rank two temperaments, Catalog of thirteen-limit rank two temperaments, Catalog of seven-limit rank two temperaments, and User:TallKite/Catalog of eleven-limit rank three temperaments with Color names
-{{Todo|add definition|update|inline=1|comment=Document this method via [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17912.html Temperament seaches using wedgies only].}}
+See also: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17912.html
+== Converting to wedgies from reduced row echelon form ==
+''Main article: [[Mathematical theory of regular temperaments]]''
+To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank ''r'' in ''n'' dimensions (where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes in the ''p''-limit) take a wedge product of basis vectors involving {{nowrap|''r'' − 1}} basis elements (i.e., the wedge product of {{nowrap|''r'' − 1}} elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an ''r''-fold wedge product with sign ±1. Take the corresponding element of the wedgie times the ±1 sign (which is computed from the parity of the permutation of the ''r'' elements.) This gives a val; do this for every combination of {{nowrap|''r'' − 1}} basis elements to obtain ''n'' choose {{nowrap|''r'' − 1}} vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.
+An alternative explanation of this process is provided here: [[Intro to exterior algebra for RTT#Converting varianced multivectors to matrices]]
 == See also ==
 * [[Intro to exterior algebra for RTT]]: for detailed background about and further explanations for how to work with multivectors and multicovectors such as wedgies
-[[Category:Math]]
 [[Category:Exterior algebra]]
-{{todo| improve readability }}
 == Notes ==
 <references />