Mathematical theory of regular temperaments: Difference between revisions
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Cmloegcmluin (talk | contribs) →Translation between methods of specifying temperaments: add link to Maple code to the top so it more clearly applies to all subsections and is easier found; also add link to alternative explanation inspired by it |
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The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality. | The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality. | ||
Maple code for the parts of these which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]]. | |||
=== Wedgies === | === Wedgies === | ||
To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (i.e., the wedge product of r-1 elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers. | To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (i.e., the wedge product of r-1 elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers. | ||
An alternative explanation of this process is provided here: [[Varianced Exterior Algebra#converting vectorals to matrices]] | |||
=== Frobenius projection maps === | === Frobenius projection maps === | ||
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To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one. | To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one. | ||
== The Geometry of Regular Temperaments == | == The Geometry of Regular Temperaments == | ||