Wedgie/Archived version: Difference between revisions

Line 1:

__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). 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 '''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.

A wedgie is defined by its values on the basis of the temperament's JI subgroup. Let a and b be non-[[contorted]] vals on a [[JI subgroup]] q1.[...].qn (where the qi need not be prime). Then the entries of the wedgie W corresponding to the temperament a&b on the JI subgroup q1.[...].qn are:

A wedgie is defined by its values on the basis of the temperament's JI subgroup. Let a and b be non-[[contorted]] vals on a [[JI subgroup]] q1.[...].qn (where the qi need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b on the JI subgroup q1.[...].qn are:

Line 15:

<math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>

More generally, 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.

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

@@ Line 1: / Line 1: @@
 __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). 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 '''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.
-A wedgie is defined by its values on the basis of the temperament's JI subgroup. Let a and b be non-[[contorted]] vals on a [[JI subgroup]] q<sub>1</sub>.[...].q<sub>n</sub> (where the q<sub>i</sub> need not be prime). Then the entries of the wedgie W corresponding to the temperament a&b on the JI subgroup q<sub>1</sub>.[...].q<sub>n</sub> are:
+A wedgie is defined by its values on the basis of the temperament's JI subgroup. Let a and b be non-[[contorted]] vals on a [[JI subgroup]] q<sub>1</sub>.[...].q<sub>n</sub> (where the q<sub>i</sub> need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b on the JI subgroup q<sub>1</sub>.[...].q<sub>n</sub> are:
 <math>W(q_i, q_j) = a(q_i)b(q_j) - a(q_j)b(q_i) \text{ for } i < j.</math>
@@ Line 15: / Line 15: @@
 <math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math>
+More generally, 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.
 ==How the period and generator falls out of a rank-2 wedgie==