Wedgie/Archived version: Difference between revisions

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'''This page goes over supplementary information on "wedgies", such as Gene Smith's introduction, the "wedgie method", and other such things. The main page may be found at [[Plücker coordinates]].'''

== Gene Smith's introduction ==

An '''n-map''' is an alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

The simplest kind of ''n''-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.

The simplest kind of ''n''-map is the 1-map, which is a [[val]] taken as a vector. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.

One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have <math>\tval{31 & 49 & 72 & 87 & 107}</math>. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map {{nowrap|"meantone(''u'', ''v'')"}} which tells us, roughly speaking, how many generator steps it takes to get to ''v'' assuming ''u'' is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have {{nowrap|meantone(2, 3) {{=}} 1}}, {{nowrap|meantone(2, 5) {{=}} 4}}, {{nowrap|meantone(2, 7) {{=}} 10}}. With 3 as a period and 3/2 as a generator, we get {{nowrap|meantone(3, 5) {{=}} 4}} and {{nowrap|meantone(3, 7) {{=}} 13}}. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5<sup>{{frac|1|4}}</sup> (or equivalently, {{frac|3|2}}) is the basic period. Using {{frac|3|2}} as a period and {{frac|9|8}} as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5<sup>{{frac|1|4}}</sup> gives us {{nowrap|meantone(5, 7) {{=}} 12}}. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.

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== Using wedgies to find temperaments ==

'''The wedgie method''' is a placeholder for an unknown method of finding temperaments<ref>This page was created as the outcome of discussion here: [[Talk:Catalog of seven-limit rank two temperaments #The "wedgie method"]]</ref>. The original drafts of several temperament-cataloging wiki pages<ref>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]]</ref> included links to the page called "[[The wedgie]]", claiming that this page explained the method by which the cataloged temperaments were obtained. However, the method and its explanation are missing.

The links to the page were not anchored to a subsection of the page; they were simply directed to the page. However, the page is not itself a dedicated explanation of such a temperament-finding method, and does not clearly appear to include an explanation of such a method anywhere within it. In 2012, when this page and the several pages linking to it for an explanation of this temperament-finding method were first created, still no such explanation existed, and it does not appear to exist in an easy-to-find place in page edit histories. Furthermore, the original author of these pages, [[Gene Ward Smith]], has unfortunately passed away, so he cannot be consulted on the matter. And so for now this page here is intended to serve as a placeholder for this missing information, in case anyone is eventually able to fill it in.

For now, one possible explanation of this method could be: iterate through all temperaments by complexity using wedgies, and filter out those which are non-temperaments and/or are of high badness.

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].}}

== 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 ==

@@ Line 1: / Line 1: @@
 {{texmap}}
 {{inacc}}
+'''This page goes over supplementary information on "wedgies", such as Gene Smith's introduction, the "wedgie method", and other such things. The main page may be found at [[Plücker coordinates]].'''
+== Gene Smith's introduction ==
 An '''n-map''' is an alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
-The simplest kind of ''n''-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.
+The simplest kind of ''n''-map is the 1-map, which is a [[val]] taken as a vector. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.
 One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have <math>\tval{31 & 49 & 72 & 87 & 107}</math>. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map {{nowrap|"meantone(''u'', ''v'')"}} which tells us, roughly speaking, how many generator steps it takes to get to ''v'' assuming ''u'' is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have {{nowrap|meantone(2, 3) {{=}} 1}}, {{nowrap|meantone(2, 5) {{=}} 4}}, {{nowrap|meantone(2, 7) {{=}} 10}}. With 3 as a period and 3/2 as a generator, we get {{nowrap|meantone(3, 5) {{=}} 4}} and {{nowrap|meantone(3, 7) {{=}} 13}}. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5<sup>{{frac|1|4}}</sup> (or equivalently, {{frac|3|2}}) is the basic period. Using {{frac|3|2}} as a period and {{frac|9|8}} as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5<sup>{{frac|1|4}}</sup> gives us {{nowrap|meantone(5, 7) {{=}} 12}}. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.
@@ Line 129: / Line 134: @@
 == Using wedgies to find temperaments ==
+'''The wedgie method''' is a placeholder for an unknown method of finding temperaments<ref>This page was created as the outcome of discussion here: [[Talk:Catalog of seven-limit rank two temperaments #The "wedgie method"]]</ref>. The original drafts of several temperament-cataloging wiki pages<ref>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]]</ref> included links to the page called "[[The wedgie]]", claiming that this page explained the method by which the cataloged temperaments were obtained. However, the method and its explanation are missing.
 The links to the page were not anchored to a subsection of the page; they were simply directed to the page. However, the page is not itself a dedicated explanation of such a temperament-finding method, and does not clearly appear to include an explanation of such a method anywhere within it. In 2012, when this page and the several pages linking to it for an explanation of this temperament-finding method were first created, still no such explanation existed, and it does not appear to exist in an easy-to-find place in page edit histories. Furthermore, the original author of these pages, [[Gene Ward Smith]], has unfortunately passed away, so he cannot be consulted on the matter. And so for now this page here is intended to serve as a placeholder for this missing information, in case anyone is eventually able to fill it in.
 For now, one possible explanation of this method could be: iterate through all temperaments by complexity using wedgies, and filter out those which are non-temperaments and/or are of high badness.
+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].}}
+== 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 ==