Wedgie/Archived version: Difference between revisions

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These reduced ''n''-vals, and particularly reduced bivals, are called '''wedgies''' (or [[Plücker coordinates]]), 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} = \tval{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, [math] dx_i \wedge dx_j [/math] 2 forms etc.

@@ Line 24: / Line 24: @@
 These reduced ''n''-vals, and particularly reduced bivals, are called '''wedgies''' (or [[Plücker coordinates]]), 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} = \tval{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, [math] dx_i \wedge dx_j [/math] 2 forms etc.