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== Gene Ward Smith's introduction == | == Gene Ward Smith's introduction == | ||
An alternating [[Wikipedia: Multilinear map|multilinear map]] which is a multilinear function taking a certain number n of [[monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. 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. | An alternating [[Wikipedia: Multilinear map|multilinear map]] which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value we may call an '''''n''-map'''. 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 [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: Linear map|linear]]: | 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 | 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 (n+m)-map, the [[Wikipedia: Exterior algebra|wedge product]] of f and g, written | 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: | ||
<math>\displaystyle f\wedge g = \sum_s sgn(s)f(x_s(1),x_s(2),...,x_s(n))g(x_s(n+1),...,x_s(n+m))</math> | <math>\displaystyle f\wedge g = \sum_s sgn(s)f\left(x_s(1),x_s(2),...,x_s\left(n\right)\right)g\left(x_s\left(n+1\right),...,x_s\left(n+m\right)\right)</math> | ||
where the sum is taken over S(n,m), the set of all [[Wikipedia:Permutation|permutations]] of the first n+m integers which are an [[Wikipedia:(p,q)_shuffle|(n,m) | 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: | 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</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 | ||
<math>( | <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> | ||
We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields & | We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\wedgie{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone. | ||
This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: Greatest common divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s. | This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: Greatest common divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s. | ||
These reduced n-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, | These reduced ''n''-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, <math>E_{24} = \val{24 38 56}</math> is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments. | ||
===== Computing the previous example in Maple ===== | ===== Computing the previous example in Maple ===== | ||
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, | 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 | 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 | [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]. | ||
A form is said to be decomposable if it can be written as an exterior product of 1 forms. So given above [math] \alpha [/math] how do we know if it is decomposable or not? Let us introduce a linear map [math] L (b)=b\wedge \alpha [/math]. This is a map from 1 forms to 3 forms. Now a kernel or nullspace of this map are all 1 forms such that [math] L(b)=0 [/math]. A basis for this nullspace in the present case is | A form is said to be decomposable if it can be written as an exterior product of 1 forms. So given above [math] \alpha [/math] how do we know if it is decomposable or not? Let us introduce a linear map [math] L (b)=b\wedge \alpha [/math]. This is a map from 1 forms to 3 forms. Now a kernel or nullspace of this map are all 1 forms such that [math] L(b)=0 [/math]. A basis for this nullspace in the present case is | ||
[math] \displaystyle | [math] \displaystyle b_1=dx_1-4dx_3-13dx_4 \quad, \quad b_2=dx_2+4dx_3+10dx_4 [/math]. | ||
Now one can check that [math] \alpha=b_1\wedge b_2 [/math]. All these computations can be done easily in Maple when the things are properly set up. But is this useful to anyone? | Now one can check that [math] \alpha=b_1\wedge b_2 [/math]. All these computations can be done easily in Maple when the things are properly set up. But is this useful to anyone? | ||
| Line 123: | Line 123: | ||
The original [math] (e_{19}, e_{31}) [/math] is a different basis of the nullspace. In matrix terms the connection between them is as follows. If | The original [math] (e_{19}, e_{31}) [/math] is a different basis of the nullspace. In matrix terms the connection between them is as follows. If | ||
[math] \displaystyle | [math] \displaystyle A=\begin{pmatrix} 19 & 30 & 44 & 53 \\ 31 & 49 & 72 & 87 \end{pmatrix}[/math] | ||
then its Hermite (normal) form is | then its Hermite (normal) form is | ||
[math] \displaystyle | [math] \displaystyle H=\begin{pmatrix} 1 & 0 & -4 & -13 \\ 0 & 1 & 4 & 10 \end{pmatrix}[/math] | ||
Let us take another [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT|example]]. Suppose we have [math] \alpha_0=dx_1\wedge dx_2\wedge dx_3+2dx_1\wedge dx_2\wedge dx_4-2dx_1\wedge dx_3\wedge dx_4 -5dx_2\wedge dx_3\wedge dx_4[/math]. Now we have [math] L_0 (b)=b\wedge \alpha_0 [/math] and the basis for nullspace is | Let us take another [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT|example]]. Suppose we have [math] \alpha_0=dx_1\wedge dx_2\wedge dx_3+2dx_1\wedge dx_2\wedge dx_4-2dx_1\wedge dx_3\wedge dx_4 -5dx_2\wedge dx_3\wedge dx_4[/math]. Now we have [math] L_0 (b)=b\wedge \alpha_0 [/math] and the basis for nullspace is | ||
[math] \displaystyle | [math] \displaystyle b_1=dx_1+5dx_4 \quad, \quad b_2=dx_2+2dx_4 \quad, \quad b_3=dx_3+2dx_4 [/math]. | ||
and one can check that | and one can check that [math] \alpha=b_1\wedge b_2 \wedge b_3[/math]. Note by the way that {{nowrap|''n'' − 1}} forms are always decomposable (here n=4 and we computed the decomposition of 3 form). | ||
== Truncation of wedgies == | == Truncation of wedgies == | ||