The wedgie: Difference between revisions
m →How the period and generator falls out of a rank-2 wedgie: Should be 5-limit meantone |
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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== | ||
The following is 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. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1. | The following is 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. We also give a (hopefully convincing and enlightening) proof of why the procedure always works. We'll assume that the equave (interval of equivalence) is the octave, but non-octave JI equaves can be substituted for the octave if needed, by substituting the appropriate JI ratio for 2/1. | ||
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* you know what the words "basis", "linear map", and "determinant" mean. | * you know what the words "basis", "linear map", and "determinant" mean. | ||
===The procedure=== | |||
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. | 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. | ||
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To find the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a linear combination g = a_1 q_1 + ... + a_n q_n (i.e. a JI ratio g = q_1^a_1 ... q_n^a_n) such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d. | To find the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a linear combination g = a_1 q_1 + ... + a_n q_n (i.e. a JI ratio g = q_1^a_1 ... q_n^a_n) such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d. | ||
===Example=== | |||
Consider the wedgie W = <<1 4 4|| for 2.3.5 meantone. We have W(2,3) = 1 and W(2,5) = 4, so d = 1, and our period is 1\1. We already have W(2,3) = 1, so we can use 3/1 as our generator. Alternatively, W(2, 3/2) = W(2,3) - W(2, 2) = W(2, 3) = 1, so 3/2 is a valid generator for meantone as well. | Consider the wedgie W = <<1 4 4|| for 2.3.5 meantone. We have W(2,3) = 1 and W(2,5) = 4, so d = 1, and our period is 1\1. We already have W(2,3) = 1, so we can use 3/1 as our generator. Alternatively, W(2, 3/2) = W(2,3) - W(2, 2) = W(2, 3) = 1, so 3/2 is a valid generator for meantone as well. | ||
===Proof=== | |||
The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy. | The period ''p'' (fraction of octave) and generator ''g'' form a basis for all the intervals of a rank-2 temperament. For example, p = 2/1 and g = 3/2 form a basis for meantone. But from a linear algebra perspective, there's nothing special about the basis {p, g}; I could have chosen the basis p' = 3/1 and g' = 2/1. What makes the wedgie a unique identifier for a temperament is that rather than specify a basis directly, the wedgie acts more like a set of constraints that any basis for the temperament must satisfy. | ||