Radical interval: Difference between revisions
Wikispaces>xenwolf **Imported revision 180708243 - Original comment: I hope this destination was intended** |
Wikispaces>genewardsmith **Imported revision 197268384 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-30 19:50:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>197268384</tt>.<br> | ||
: The revision comment was: <tt> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
| Line 18: | Line 18: | ||
If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1. | If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1. | ||
There are various [[http://en.wikipedia.org/wiki/Matrix_norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[ | There are various [[http://en.wikipedia.org/wiki/Matrix_norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted rms tuning which can be found using the [[pseudoinverse]]. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the [[http://en.wikipedia.org/wiki/Singular_value_decomposition|singular value decomposition]]. | ||
The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting. | The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting. | ||
| Line 45: | Line 45: | ||
If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br /> | If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br /> | ||
<br /> | <br /> | ||
There are various <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Matrix_norm" rel="nofollow">matrix norms</a>, and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the <a class="wiki_link" href="/ | There are various <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Matrix_norm" rel="nofollow">matrix norms</a>, and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">Frobenius tuning</a>, which is the same as the unweighted rms tuning which can be found using the <a class="wiki_link" href="/pseudoinverse">pseudoinverse</a>. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Singular_value_decomposition" rel="nofollow">singular value decomposition</a>.<br /> | ||
<br /> | <br /> | ||
The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.<br /> | The projection matrix of the Frobenius tuning is symmetrical. Because of this, r vals spanning the subspace of the temperament can be taken to be r eigenmonzos instead. The same considerations apply to the projection matrix of RMS-TOP tuning in weighted coordinates. This projection matrix can be transformed to a matrix in unweighted coordinates by monzo weighting the columns and val weighting the rows, but the resulting matrix is no longer symmetrical. However, a val (in unweighted coordinates) can be converted to a left eigenvalue with eigenvalue 1 by doubly val weighing it; that is, val weighting it, and then val weighting it again. By finding fractional monzos which approximate r such left eigenvalues, a projection map with fractional monzos as rows which approximates TOP-RMS tuning may be obtained. An easy way to do this is to use an equal temperament val to approximate val weighting.<br /> | ||