The wedgie: Difference between revisions
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Let d = gcd(W(2, q_1), ..., W(2, q_n)). This tells you that for any JI ratio v in your JI subgroup, W(2, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2 with any JI ratio u that is equated to 2. 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, q_1), ..., W(2, q_n)). This tells you that for any JI ratio v in your JI subgroup, W(2, v) = 2n(v) for some number n(v) [that depends linearly on v]. This equation is also true when we replace 2 with any JI ratio u that is equated to 2. 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 2/1 = a1 e1 + a2 e2. Then: | Choose a basis e1, e2 for the temperament group and write (the image of) 2/1 as 2/1 = a1 e1 + a2 e2. Then: | ||
*W(2/1, e1) = W(a2 e2, e1) = -a2 W(e1, e2) = -a2 | *W(2/1, e1) = W(a2 e2, e1) = -a2 W(e1, e2) = -a2 | ||
*W(2/1,e2) = W(a1e1, e2) = a1 W(e1, e2) = a1. | *W(2/1,e2) = W(a1e1, e2) = a1 W(e1, e2) = a1. | ||