Wedgie/Archived version: Difference between revisions

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Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[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.]

===Example===

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 Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[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.]
 ===Example===