Wedgie/Archived version: Difference between revisions

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To find the '''period''': Let ''d'' = gcd(W(2, ''q''1), …, W(2, ''q''''n'')). Then your period is 1\''d''.

To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''a''1 W(2, ''q''1) + … ''a''''n'' W(2, q''n'') = ''d'' for the coefficients ''a''1, ..., ''a''''n'' (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is ''g'' = (the tempered version of) ''q''1''a''1 … ''q''''n''''a''''n'' (written additively, a linear combination g = ''a''1''q''1 + … + ''a''''n''''q''''n'').

To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''c''1 W(2, ''q''1) + … ''c''''n'' W(2, q''n'') = ''d'' for the coefficients ''c''1, ..., ''c''''n'' (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is ''g'' = (the tempered version of) ''q''1''a''1 … ''q''''n''''c''''n'' (written additively, a linear combination g = ''c''1''q''1 + … + ''c''''n''''q''''n'').

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. For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using the x31eq temperament finder.

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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.

Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = ~~a_1~~ e_1 + ~~a_2~~ e_2. Then:

Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = r_1 e_1 + r_2 e_2. Then:

*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_1) = W(r_2 e_2, e_1) = -r_2 W(e_1, e_2) = -r_2

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

*W(2/1, e_2) = W(r_1 e_1, e_2) = r_1 W(e_1, e_2) = r_1.

Divisibility by d and the fact that e_1 and e_2 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!

Divisibility by d and the fact that e_1 and e_2 represent JI ratios in the 2.q_1.[...].q_n subgroup imply that r_1 and r_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 = c_1 q_1 + ... + c_n q_n such that W(2, g) = c_1 W(2, q_1) + ... 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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 To find the '''period''': Let ''d'' = gcd(W(2, ''q''<sub>1</sub>), …, W(2, ''q''<sub>''n''</sub>)). Then your period is 1\''d''.
-To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''a''<sub>1</sub> W(2, ''q''<sub>1</sub>) + … ''a''<sub>''n''</sub> W(2, q<sub>''n''</sub>) = ''d'' for the coefficients ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is ''g'' = (the tempered version of) ''q''<sub>1</sub><sup>''a''<sub>1</sub></sup> … ''q''<sub>''n''</sub><sup>''a''<sub>''n''</sub></sup> (written additively, a linear combination g = ''a''<sub>1</sub>''q''<sub>1</sub> + … + ''a''<sub>''n''</sub>''q''<sub>''n''</sub>).
+To find (a JI interpretation of) the '''generator''': Solve the equation W(2, ''g'') = ''c''<sub>1</sub> W(2, ''q''<sub>1</sub>) + … ''c''<sub>''n''</sub> W(2, q<sub>''n''</sub>) = ''d'' for the coefficients ''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub> (using some algorithm such as the [[Wikipedia: Extended Euclidean algorithm|extended Euclidean algorithm]]). Then one valid generator for the temperament is ''g'' = (the tempered version of) ''q''<sub>1</sub><sup>''a''<sub>1</sub></sup> … ''q''<sub>''n''</sub><sup>''c''<sub>''n''</sub></sup> (written additively, a linear combination g = ''c''<sub>1</sub>''q''<sub>1</sub> + … + ''c''<sub>''n''</sub>''q''<sub>''n''</sub>).
 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. For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]], and those tunings can be found using the x31eq temperament finder.
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 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.
-Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = a_1 e_1 + a_2 e_2. Then:
+Choose a basis e_1, e_2 for the temperament group and write (the image of) 2/1 as 2/1 = r_1 e_1 + r_2 e_2. Then:
-*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_1) = W(r_2 e_2, e_1) = -r_2 W(e_1, e_2) = -r_2
-*W(2/1, e_2) = W(a_1 e_1, e_2) = a_1 W(e_1, e_2) = a_1.
+*W(2/1, e_2) = W(r_1 e_1, e_2) = r_1 W(e_1, e_2) = r_1.
-Divisibility by d and the fact that e_1 and e_2 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!
+Divisibility by d and the fact that e_1 and e_2 represent JI ratios in the 2.q_1.[...].q_n subgroup imply that r_1 and r_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 = c_1 q_1 + ... + c_n q_n such that W(2, g) = c_1 W(2, q_1) + ... 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 ==