Wedgie/Archived version: Difference between revisions

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The key fact about the determinant we use here is that two integer vectors v1, v2 form a basis for the rank-2 integer lattice '''Z'''2 iff det(v1, v2) = ±1. So in order to find a period and generator for our temperament, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.

Let d = gcd(W(2/1, q1), ..., W(2/1, qn)). This tells you that for any JI ratio v in your JI subgroup, W(2/1, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2/1 with any JI ratio u that is equated to 2/1. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.

Let d = gcd(W(2/1, q1), ..., W(2/1, q''n'')). This tells you that for any JI ratio v in your JI subgroup, W(2/1, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2/1 with any JI ratio u that is equated to 2/1. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.

Choose a basis e1, e2 for the temperament group and write (the image of) 2/1 as 2/1 = k1 e1 + k2 e2. Then:

*W(2/1, e1) = W(k2 e2, e1) = -k2 W(e1, e2) = -k2

*W(2/1, e2) = W(k1 e1, e2) = k1 W(e1, e2) = k1.

Divisibility by d and the fact that e1 and e2 represent JI ratios in the 2.q1.[...].qn subgroup imply that k1 and k2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q1), ..., W(2, qn)) = d, we can always find a linear combination g = c1 q1 + ... + cn qn such that W(2, g) = c1 W(2, q1) + ... cn W(2,qn) = d using the extended Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da!

Divisibility by d and the fact that e1 and e2 represent JI ratios in the 2.q1.[...].q''n'' subgroup imply that k1 and k2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q1), ..., W(2, q''n'')) = d, we can always find a linear combination g = c1 q1 + ... + c''n'' q''n'' such that W(2, g) = c1 W(2, q1) + ... c''n'' W(2,q''n'') = d using the extended Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da!

== Technical introduction ==

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 The key fact about the determinant we use here is that two integer vectors v<sub>1</sub>, v<sub>2</sub> form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v<sub>1</sub>, v<sub>2</sub>) = ±1. So in order to find a period and generator for our temperament, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.
-Let d = gcd(W(2/1, q<sub>1</sub>), ..., W(2/1, q<sub>n</sub>)). This tells you that for any JI ratio v in your JI subgroup, W(2/1, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2/1 with any JI ratio u that is equated to 2/1. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.
+Let d = gcd(W(2/1, q<sub>1</sub>), ..., W(2/1, q<sub>''n''</sub>)). This tells you that for any JI ratio v in your JI subgroup, W(2/1, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2/1 with any JI ratio u that is equated to 2/1. This tells us that for W(p, g) = 1, we (up to some choices) need p to be an interval such that d*p is equated to 2/1, i.e. p represents 1/d of the octave.
 Choose a basis e<sub>1</sub>, e<sub>2</sub> for the temperament group and write (the image of) 2/1 as 2/1 = k<sub>1</sub> e<sub>1</sub> + k<sub>2</sub> e<sub>2</sub>. Then:
 *W(2/1, e<sub>1</sub>) = W(k<sub>2</sub> e<sub>2</sub>, e<sub>1</sub>) = -k<sub>2</sub> W(e<sub>1</sub>, e<sub>2</sub>) = -k<sub>2</sub>
 *W(2/1, e<sub>2</sub>) = W(k<sub>1</sub> e<sub>1</sub>, e<sub>2</sub>) = k<sub>1</sub> W(e<sub>1</sub>, e<sub>2</sub>) = k<sub>1</sub>.
-Divisibility by d and the fact that e<sub>1</sub> and e<sub>2</sub> represent JI ratios in the 2.q<sub>1</sub>.[...].q<sub>n</sub> subgroup imply that k<sub>1</sub> and k<sub>2</sub> are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q<sub>1</sub>), ..., W(2, q<sub>n</sub>)) = d, we can always find a linear combination g = c<sub>1</sub> q<sub>1</sub> + ... + c<sub>n</sub> q<sub>n</sub> such that W(2, g) = c<sub>1</sub> W(2, q<sub>1</sub>) + ... c<sub>n</sub> W(2,q<sub>n</sub>) = d using the extended Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da!
+Divisibility by d and the fact that e<sub>1</sub> and e<sub>2</sub> represent JI ratios in the 2.q<sub>1</sub>.[...].q<sub>''n''</sub> subgroup imply that k<sub>1</sub> and k<sub>2</sub> are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q<sub>1</sub>), ..., W(2, q<sub>''n''</sub>)) = d, we can always find a linear combination g = c<sub>1</sub> q<sub>1</sub> + ... + c<sub>''n''</sub> q<sub>''n''</sub> such that W(2, g) = c<sub>1</sub> W(2, q<sub>1</sub>) + ... c<sub>''n''</sub> W(2,q<sub>''n''</sub>) = d using the extended Euclidean algorithm. Then since W(2, g) = W(d*p, g) = d*W(p, g) = d, we have W(p,g) = 1. Ta-da!
 == Technical introduction ==