The wedgie: Difference between revisions
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The ''[[Wedgies_and_Multivals|wedgie]]'' is a way of defining and working with an [[Abstract_regular_temperament|abstract regular temperament]]. If one takes r independent [[Vals|vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [[Wedgies_and_Multivals|wedge product]] of the vals, 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, it 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; that is, regular temperaments where no tuning has been decided on. | The ''[[Wedgies_and_Multivals|wedgie]]'' is a way of defining and working with an [[Abstract_regular_temperament|abstract regular temperament]]. If one takes r independent [[Vals|vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [[Wedgies_and_Multivals|wedge product]] of the vals, 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, it 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; that is, regular temperaments where no tuning has been decided on. | ||
===How the period and generator falls out of a rank-2 wedgie=== | ===How the period and generator falls out of a rank-2 wedgie=== | ||
The following is both a procedure for finding a period and a generator for a rank-2 regular temperament given the temperament's wedgie, and a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed. | The following is both a procedure for finding a period and a generator for a rank-2 regular temperament on the 2.q_1.(...).q_n [[Subgroup temperaments|JI subgroup]]. given the temperament's wedgie, and a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the [[interval of equivalence|equave]] is the octave, but non-octave equaves can be substituted for the octave if needed. | ||
The following assumes that: | The following assumes that: | ||
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'''The procedure:''' | '''The procedure:''' | ||
Consider the entries of the wedgie W. The | Consider the entries of the wedgie W. The entries of W are W(2, q_1), ..., W(2, q_n), and W(q_i, q_j) for i < j. | ||
To find the '''period''': let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d. | To find the '''period''': let d = gcd(W(2, q_1), ..., W(2, q_n)). Then your period is 1\d. | ||