Radical interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 163349667 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 163552187 - Original comment: ** |
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<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:genewardsmith|genewardsmith]] and made on <tt>2010-09-17 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-17 19:29:39 UTC</tt>.<br> | ||
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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 [[RMS tuning|Frobenius tuning]], which is the same as the unweighted rms tuning which can be found using the [[RMS tuning|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://Singular%20value%20decomposition|singular value decomposition]]. | 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 [[RMS tuning|Frobenius tuning]], which is the same as the unweighted rms tuning which can be found using the [[RMS tuning|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://Singular%20value%20decomposition|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. | |||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
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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="/RMS%20tuning">Frobenius tuning</a>, which is the same as the unweighted rms tuning which can be found using the <a class="wiki_link" href="/RMS%20tuning">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://Singular%20value%20decomposition" rel="nofollow">singular value decomposition</a>.<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="/RMS%20tuning">Frobenius tuning</a>, which is the same as the unweighted rms tuning which can be found using the <a class="wiki_link" href="/RMS%20tuning">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://Singular%20value%20decomposition" rel="nofollow">singular value decomposition</a>.<br /> | ||
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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.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Algebraic considerations</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Algebraic considerations</h3> | ||
For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension n) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html></pre></div> | For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension n) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html></pre></div> | ||