Wedgie/Archived version: Difference between revisions
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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. 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 | The entries of a wedgie W for the temperament a&b on a JI subgroup p_1.[...].p_n are | ||
<math>W(p_i, p_j) = a(p_i)b(p_j) - a(p_j)b(p_i) \text{ for } i < j.</math> | |||
For the p_n-limit, those entries are listed in the order | |||
<math>\langle\langle W(2, 3) \ ... \ W(2, p_n) \ W(3, 5) \ ... \ W(3, p_n) \ ... W(p_{n-2}, p_{n-1}) \ W(p_{n-2}, p_n)\ W(p_{n-1}, p_n)]].</math> | |||
For example, a 7-limit wedgie is of the form | |||
<math>\langle \langle W(2, 3) \ W(2,5) \ W(2, 7) \ W(3, 5) \ W(3, 7) \ W(5, 7)]].</math> | |||
==How the period and generator falls out of a rank-2 wedgie== | ==How the period and generator falls out of a rank-2 wedgie== | ||