Wedgie/Archived version: Difference between revisions

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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, <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.

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 =====

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].

@@ Line 106: / Line 106: @@
 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, <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.
+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 =====
 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].