Wedgie/Archived version: Difference between revisions

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To find (a JI interpretation of) the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a JI ratio g = q_1^a_1 ... q_n^a_n (equivalently, 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.

Now choosing the optimal tuning for the temperament~~'s JI reading~~ is just a matter of choosing a way to measure error and minimizing it with linear algebra: for example, the TOP (Tenney-OPtimal) and POTE (Pure-Octave Tenney-Euclidean) are based on minimizing [[Tenney-Euclidean_temperament_measures#TE_error|TE error]].

Now choosing the optimal tuning for the temperament is just a matter of choosing a way to measure error from JI and minimizing it with linear algebra: for example, the TOP (Tenney-OPtimal) and POTE (Pure-Octave Tenney-Euclidean) are based on minimizing [[Tenney-Euclidean_temperament_measures#TE_error|TE error]].

[In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. This is the determinant of the tempered versions of u and v. The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.]

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 To find (a JI interpretation of) the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a JI ratio g = q_1^a_1 ... q_n^a_n (equivalently, 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.
-Now choosing the optimal tuning for the temperament's JI reading is just a matter of choosing a way to measure error and minimizing it with linear algebra: for example, the TOP (Tenney-OPtimal) and POTE (Pure-Octave Tenney-Euclidean) are based on minimizing [[Tenney-Euclidean_temperament_measures#TE_error|TE error]].
+Now choosing the optimal tuning for the temperament is just a matter of choosing a way to measure error from JI and minimizing it with linear algebra: for example, the TOP (Tenney-OPtimal) and POTE (Pure-Octave Tenney-Euclidean) are based on minimizing [[Tenney-Euclidean_temperament_measures#TE_error|TE error]].
 [In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. This is the determinant of the tempered versions of u and v. The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.]