Tenney–Euclidean metrics: Difference between revisions
Wikispaces>genewardsmith **Imported revision 174915615 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 174996747 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-10- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-30 16:12:43 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>174996747</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : 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> | ||
| Line 12: | Line 12: | ||
Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a [[http://en.wikipedia.org/wiki/Positive-definite_matrix|positive semidefinite matrix]], so it defines a [[http://en.wikipedia.org/wiki/Definite_bilinear_form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W^(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = **P**. From the semidefinite form we obtain an associated [[http://en.wikipedia.org/wiki/Definite_quadratic_form|semidefinite quadratic form]] b**P**b* and from this the [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|seminorm]] sqrt(b**P**b*). | Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[RMS tuning|TOP-RMS]] tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a [[http://en.wikipedia.org/wiki/Positive-definite_matrix|positive semidefinite matrix]], so it defines a [[http://en.wikipedia.org/wiki/Definite_bilinear_form|positive semidefinite bilinear form]]. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W^(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = **P**. From the semidefinite form we obtain an associated [[http://en.wikipedia.org/wiki/Definite_quadratic_form|semidefinite quadratic form]] b**P**b* and from this the [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|seminorm]] sqrt(b**P**b*). | ||
Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The [[http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[http://en.wikipedia.org/wiki/Normed_vector_space|normed vector space]] with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the //temperamental complexity// of the intervals of the regular temperament. | Denoting this temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The [[http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[http://en.wikipedia.org/wiki/Normed_vector_space|normed vector space]] with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the //temperamental complexity// of the intervals of the regular temperament. | ||
==OE complexity== | ==OE complexity== | ||
| Line 27: | Line 27: | ||
Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The <a class="wiki_link" href="/RMS%20tuning">TOP-RMS</a> tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow">positive semidefinite matrix</a>, so it defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_bilinear_form" rel="nofollow">positive semidefinite bilinear form</a>. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W^(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = <strong>P</strong>. From the semidefinite form we obtain an associated <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_quadratic_form" rel="nofollow">semidefinite quadratic form</a> b<strong>P</strong>b* and from this the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow">seminorm</a> sqrt(b<strong>P</strong>b*). <br /> | Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The <a class="wiki_link" href="/RMS%20tuning">TOP-RMS</a> tuning matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow">positive semidefinite matrix</a>, so it defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_bilinear_form" rel="nofollow">positive semidefinite bilinear form</a>. In terms of weighted monzos m1 and m2, m1Pm2* defines the semidefinite form on weighted monzos, and hence b1W^(-1)PW^(-1)b2* defines a semidefinite form on unweighted monzos, in terms of the matrix W^(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = <strong>P</strong>. From the semidefinite form we obtain an associated <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_quadratic_form" rel="nofollow">semidefinite quadratic form</a> b<strong>P</strong>b* and from this the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow">seminorm</a> sqrt(b<strong>P</strong>b*). <br /> | ||
<br /> | <br /> | ||
Denoting this temperament-defined seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29" rel="nofollow">quotient space</a> of the full vector space by the commatic subspace such that T(x) = 0 is now a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">normed vector space</a> with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the <em>temperamental complexity</em> of the intervals of the regular temperament.<br /> | Denoting this temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a sublattice of the lattice of monzos consisting of the commas of the temperament. The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29" rel="nofollow">quotient space</a> of the full vector space by the commatic subspace such that T(x) = 0 is now a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">normed vector space</a> with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the <em>temperamental complexity</em> of the intervals of the regular temperament.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-OE complexity"></a><!-- ws:end:WikiTextHeadingRule:4 -->OE complexity</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-OE complexity"></a><!-- ws:end:WikiTextHeadingRule:4 -->OE complexity</h2> | ||