Tenney–Euclidean metrics: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-30 18:06:44 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-30 18:13:25 UTC</tt>.

: The original revision id was <tt>~~175009631~~</tt>.

: The original revision id was <tt>175010293</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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*).

It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gram matrix]] used to compute [[RMS temperament measures|wedgie complexity]], and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents b in a basis defined by the mapping A, and ~~**P**~~ defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.

It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gram matrix]] used to compute [[RMS temperament measures|wedgie complexity]], and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents the tempering of b in a basis defined by the mapping A, and (AW^2A)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.

Denoting the 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.

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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 = P. 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> bPb* and from this the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow">seminorm</a> sqrt(bPb*).

It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow">Gram matrix</a> used to compute <a class="wiki_link" href="/RMS%20temperament%20measures">wedgie complexity</a>, and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents b in a basis defined by the mapping A, and ~~P~~ defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.

It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow">Gram matrix</a> used to compute <a class="wiki_link" href="/RMS%20temperament%20measures">wedgie complexity</a>, and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents the tempering of b in a basis defined by the mapping A, and (AW^2A)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.

Denoting the 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 temperamental complexity of the intervals of the regular temperament.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-10-30 18:06:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-30 18:13:25 UTC</tt>.<br>
-: The original revision id was <tt>175009631</tt>.<br>
+: The original revision id was <tt>175010293</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>
@@ 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*).
-It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gram matrix]] used to compute [[RMS temperament measures|wedgie complexity]], and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents b in a basis defined by the mapping A, and **P** defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.
+It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gram matrix]] used to compute [[RMS temperament measures|wedgie complexity]], and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents the tempering of b in a basis defined by the mapping A, and (AW^2A)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.
 Denoting the 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.
@@ Line 29: / Line 29: @@
 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 &lt;a class="wiki_link" href="/RMS%20tuning"&gt;TOP-RMS&lt;/a&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow"&gt;positive semidefinite matrix&lt;/a&gt;, so it defines a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_bilinear_form" rel="nofollow"&gt;positive semidefinite bilinear form&lt;/a&gt;. 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 = &lt;strong&gt;P&lt;/strong&gt;. From the semidefinite form we obtain an associated &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Definite_quadratic_form" rel="nofollow"&gt;semidefinite quadratic form&lt;/a&gt; b&lt;strong&gt;P&lt;/strong&gt;b* and from this the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow"&gt;seminorm&lt;/a&gt; sqrt(b&lt;strong&gt;P&lt;/strong&gt;b*). &lt;br /&gt;
 &lt;br /&gt;
-It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow"&gt;Gram matrix&lt;/a&gt; used to compute &lt;a class="wiki_link" href="/RMS%20temperament%20measures"&gt;wedgie complexity&lt;/a&gt;, and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents b in a basis defined by the mapping A, and &lt;strong&gt;P&lt;/strong&gt; defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.&lt;br /&gt;
+It may be noted that (VV*)^(-1) = (AW^2A)^(-1) is the inverse of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow"&gt;Gram matrix&lt;/a&gt; used to compute &lt;a class="wiki_link" href="/RMS%20temperament%20measures"&gt;wedgie complexity&lt;/a&gt;, and hence is the corresponding Gram matrix for the dual space. Hence P represents a change of basis defined by the mapping given by the vals combined with an inner product on the result. Given a monzo b, bA* represents the tempering of b in a basis defined by the mapping A, and (AW^2A)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.&lt;br /&gt;
 &lt;br /&gt;
 Denoting the 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29" rel="nofollow"&gt;quotient space&lt;/a&gt; of the full vector space by the commatic subspace such that T(x) = 0 is now a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;normed vector space&lt;/a&gt; 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 &lt;em&gt;temperamental complexity&lt;/em&gt; of the intervals of the regular temperament.&lt;br /&gt;