Wedgie/Archived version: Difference between revisions

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__FORCETOC__

The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). If one takes r independent [[vals]] V1, ..., Vr in a p-limit group of n primes, then the wedgie for the rank-r temperament V1& ...&Vr is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. ~~Wedgies~~ are in a ~~one~~-~~to-one relationship with abstract regular temperaments~~.

The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). Wedgies are in a one-to-one relationship with abstract regular temperaments. If one takes r independent [[vals]] V1, ..., Vr in a p-limit group of n primes, then the wedgie for the rank-r temperament V1& ...&Vr is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie.

The entries of a wedgie W for the p_n-limit temperament a&b are W(p_i, p_j) = a(p_i)b(p_j) - a(p_j)b(p_i) for i < j. They are listed in the order {{val|{{val|W(2, 3) ... W(2, p_n) W(3, 5) ... W(3, p_n) ... W(p_(n-1), p_n)}}}}. For example, a 7-limit wedgie is of the form {{val|{{val|W(2, 3), W(2,5) W(2, 7) W(3, 5) W(3, 7), W(5, 7)}}}}.

==How the period and generator falls out of a rank-2 wedgie==

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* you know what a [[val]] is and how to work with one.

===The procedure===

Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p). ~~(This is how the wedge product of two 1-forms a and b works.)~~

Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p).

To find the '''period''': Let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d.

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 __FORCETOC__
-The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). If one takes r independent [[vals]] V1, ..., Vr in a p-limit group of n primes, then the wedgie for the rank-r temperament V1& ...&Vr is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. Wedgies are in a one-to-one relationship with abstract regular temperaments.
+The '''wedgie''' is a way of defining and working with an [[abstract regular temperament]] (that is, a regular temperament where no tuning has been decided on). Wedgies are in a one-to-one relationship with abstract regular temperaments. If one takes r independent [[vals]] V1, ..., Vr in a p-limit group of n primes, then the wedgie for the rank-r temperament V1& ...&Vr is defined by taking the [https://en.wikipedia.org/wiki/Wedge_product wedge product] of the vals (called a '''multival'''), and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, the multival is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie.
+The entries of a  wedgie W for the p_n-limit temperament a&b are W(p_i, p_j) = a(p_i)b(p_j) - a(p_j)b(p_i) for i < j. They are listed in the order {{val|{{val|W(2, 3) ... W(2, p_n) W(3, 5) ... W(3, p_n) ... W(p_(n-1), p_n)}}}}. For example, a 7-limit wedgie is of the form {{val|{{val|W(2, 3), W(2,5) W(2, 7) W(3, 5) W(3, 7), W(5, 7)}}}}.
 ==How the period and generator falls out of a rank-2 wedgie==
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 * you know what a [[val]] is and how to work with one.
 ===The procedure===
-Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p). (This is how the wedge product of two 1-forms a and b works.)
+Consider the rank-2 temperament a&b, where a and b are two [[val]]s. Then the entries of the wedgie W corresponding to a&b are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j, and the entry W(p,q) is given by a(p)b(q) - a(q)b(p).
 To find the '''period''': Let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d.