Wedgie/Archived version: Difference between revisions

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By the First Isomorphism Theorem it follows that M' is the group of intervals in the rank-2 temperament in question.

The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±1. So in order to find a period and generator for our ~~tempearment~~, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.

The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±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_1), ..., 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.

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*W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2

*W(2/1, e_2) = W(a_1 e_1, e_2) = a_1 W(e_1, e_2) = a_1.

Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup ~~impliy~~ that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_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!

Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup imply that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_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 ==

@@ Line 64: / Line 64: @@
 By the First Isomorphism Theorem it follows that M' is the group of intervals in the rank-2 temperament in question.
-The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±1. So in order to find a period and generator for our tempearment, we need a pair of vectors {p, g} such that W(p, g) = 1 and p is 1\d for some integer d.
+The key fact about the determinant we use here is that two integer vectors v_1, v_2 form a basis for the rank-2 integer lattice '''Z'''<sup>2</sup> iff det(v_1, v_2) = ±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_1), ..., 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.
@@ Line 71: / Line 71: @@
 *W(2/1, e_1) = W(a_2 e_2, e_1) = -a_2 W(e_1, e_2) = -a_2
 *W(2/1, e_2) = W(a_1 e_1, e_2) = a_1 W(e_1, e_2) = a_1.
-Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup impliy that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_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!
+Divisibility (by d = gcd(W(2, q_1), ..., W(2, q_n))) and the fact that e1 and e2 represent JI ratios in the 2.q_1. ... q_n subgroup imply that a_1 and a_2 are both divisible by d, and hence 2/1 is a dth power in M' (the temperament space). Since gcd(W(2, q_1), ..., W(2, q_n)) = d, we can always find a linear combination g = a_1 q_1 + ... + a_n q_n such that W(2, g) = a_1 W(2, q_1) + ... a_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 ==