Wedgie/Archived version: Difference between revisions

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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 [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: 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>\val{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{{frac|1|4}} (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{{frac|1|4}} 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.

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>\val{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{{frac|1|4}} (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{{frac|1|4}} 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.

Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows:

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where the sum is taken over {{nowrap|S(''n'', ''m'')}}, the set of all [[Wikipedia:Permutation|permutations]] of the first {{nowrap|''n'' + ''m''}} integers which are an [[Wikipedia:(p,q)_shuffle|{{nowrap|(''n'', ''m'')}} shuffle]], and sgn(''t'') is the [[Wikipedia: Parity of a permutation|parity of the permutation]] ''t'', which is +1 if ''t'' is even meaning an even number of transpositions of two numbers will get to ''t'', and −1 if ''t'' is odd.

If ''f'' and ''g'' are both vals (1-maps) then this becomes especially easy: {{nowrap|(''f'' ∧ ''g'')(''u'', ''v'') {{=}} ''f''(''u'')''g''(''v'') − ''f''(''v'')''g''(''u'')}}. Let's consider a specific example. Suppose ~~{{nowrap|E~~<~~sub~~>19~~ {{~~=}} {~~{val|~~ 19 30 44 53 }~~}}}~~ is the equal temperament val for septimal 19et, and ~~{{nowrap|E~~<~~sub~~>31~~ {{~~=}} {~~{val|~~ 31 49 72 87 }~~}}}~~ is the val for septimal 31et. Then writing intervals multiplicatively, we have

If ''f'' and ''g'' are both vals (1-maps) then this becomes especially easy: {{nowrap|(''f'' ∧ ''g'')(''u'', ''v'') {{=}} ''f''(''u'')''g''(''v'') − ''f''(''v'')''g''(''u'')}}. Let's consider a specific example. Suppose <math>E_{19} = \val{19 & 30 & 44 & 53}</math> is the equal temperament val for septimal 19et, and <math>E_{31} = \val{31 & 49 & 72 & 87}</math> is the val for septimal 31et. Then writing intervals multiplicatively, we have

<math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>

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In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms, [math] dx_i \wedge dx_j [/math] 2 forms etc.

In this way let's write <math>E_{19} = \val{19 30 44 53}</math> and <math>E_{31} = \val{31 49 72 87}</math> as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product

In this way let's write <math>E_{19} = \val{19 30 44 53}</math> and <math>E_{31} = \val{31 & 49 & 72 & 87}</math> as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product

[math] \displaystyle \alpha =e_{19} \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].

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 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 [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: 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'') {{=}} &minus;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>\val{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.
+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>\val{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.
 Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows:
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 where the sum is taken over {{nowrap|S(''n'', ''m'')}}, the set of all [[Wikipedia:Permutation|permutations]] of the first {{nowrap|''n'' + ''m''}} integers which are an [[Wikipedia:(p,q)_shuffle|{{nowrap|(''n'', ''m'')}} shuffle]], and sgn(''t'') is the [[Wikipedia: Parity of a permutation|parity of the permutation]] ''t'', which is +1 if ''t'' is even meaning an even number of transpositions of two numbers will get to ''t'', and &minus;1 if ''t'' is odd.
-If ''f'' and ''g'' are both vals (1-maps) then this becomes especially easy: {{nowrap|(''f'' ∧ ''g'')(''u'', ''v'') {{=}} ''f''(''u'')''g''(''v'') &minus; ''f''(''v'')''g''(''u'')}}. Let's consider a specific example. Suppose {{nowrap|E<sub>19</sub> {{=}} {{val| 19 30 44 53 }}}} is the equal temperament val for septimal 19et, and {{nowrap|E<sub>31</sub> {{=}} {{val| 31 49 72 87 }}}} is the val for septimal 31et. Then writing intervals multiplicatively, we have
+If ''f'' and ''g'' are both vals (1-maps) then this becomes especially easy: {{nowrap|(''f'' ∧ ''g'')(''u'', ''v'') {{=}} ''f''(''u'')''g''(''v'') &minus; ''f''(''v'')''g''(''u'')}}. Let's consider a specific example. Suppose <math>E_{19} = \val{19 & 30 & 44 & 53}</math> is the equal temperament val for septimal 19et, and <math>E_{31} = \val{31 & 49 & 72 & 87}</math> is the val for septimal 31et. Then writing intervals multiplicatively, we have
 <math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>
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 In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms, [math] dx_i \wedge dx_j [/math] 2 forms etc.
-In this way let's write <math>E_{19} = \val{19 30 44 53}</math> and <math>E_{31} = \val{31 49 72 87}</math>  as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and  [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product
+In this way let's write <math>E_{19} = \val{19 30 44 53}</math> and <math>E_{31} = \val{31 & 49 & 72 & 87}</math>  as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and  [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product
 [math] \displaystyle \alpha =e_{19} \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].