Wedgie/Archived version: Difference between revisions

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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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 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.