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>2018-02-~~19 19~~:50:40 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2018-02-20 05:32:30 UTC</tt>.

: The original revision id was <tt>~~626630809~~</tt>.

: The original revision id was <tt>626642279</tt>.

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

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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 [[Tenney-Euclidean tuning|TE]] tuning projection matrix is then V`V, where V` is the [[Tenney-Euclidean Tuning|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, m1*Pm2 defines the semidefinite form on weighted monzos, and hence b1*W^(-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 [[Tenney-Euclidean temperament measures|TE 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 [[http://en.wikipedia.org/wiki/Inner_product_space|inner product]] on the result. Given a monzo b, bA* represents the tempered interval corresponding to b in a basis defined by the mapping A, and //P// = (AW^2A*)^(-1) 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 [[Tenney-Euclidean temperament measures|TE 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 [[http://en.wikipedia.org/wiki/Inner_product_space|inner product]] on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and //P// = (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 lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t//P//t*) where t is the image of a monzo b by t = bA*

Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t*//P//t) where t is the image of a monzo b by t = Ab

=The OETES=

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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="/Tenney-Euclidean%20tuning">TE</a> tuning projection matrix is then V`V, where V` is the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">pseudoinverse</a>. 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, m1*Pm2 defines the semidefinite form on weighted monzos, and hence b1*W^(-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> b*Pb and from this the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29" rel="nofollow">seminorm</a> sqrt(b*Pb).

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="/Tenney-Euclidean%20temperament%20measures">TE 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Inner_product_space" rel="nofollow">inner product</a> on the result. Given a monzo b, bA* represents the tempered interval corresponding to b in a basis defined by the mapping A, and P = (AW^2A*)^(-1) 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="/Tenney-Euclidean%20temperament%20measures">TE 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Inner_product_space" rel="nofollow">inner product</a> on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and P = (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 lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm or temperamental complexity of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(tPt*) where t is the image of a monzo b by t = bA*

Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm or temperamental complexity of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t*Pt) where t is the image of a monzo b by t = Ab

<h1 id="toc2"><a name="The OETES"></a>The OETES</h1>

@@ 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>2018-02-19 19:50:40 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2018-02-20 05:32:30 UTC</tt>.<br>
-: The original revision id was <tt>626630809</tt>.<br>
+: The original revision id was <tt>626642279</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 14: / Line 14: @@
 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 [[Tenney-Euclidean tuning|TE]] tuning projection matrix is then V`V, where V` is the [[Tenney-Euclidean Tuning|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, m1*Pm2 defines the semidefinite form on weighted monzos, and hence b1*W^(-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 [[Tenney-Euclidean temperament measures|TE 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 [[http://en.wikipedia.org/wiki/Inner_product_space|inner product]] on the result. Given a monzo b, bA* represents the tempered interval corresponding to b in a basis defined by the mapping A, and //P// = (AW^2A*)^(-1) 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 [[Tenney-Euclidean temperament measures|TE 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 [[http://en.wikipedia.org/wiki/Inner_product_space|inner product]] on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and //P// = (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 lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t//P//t*) where t is the image of a monzo b by t = bA*
+Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm// or //temperamental complexity// of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t*//P//t) where t is the image of a monzo b by t = Ab
 =The OETES=
@@ Line 61: / Line 61: @@
   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="/Tenney-Euclidean%20tuning"&gt;TE&lt;/a&gt; tuning projection matrix is then V`V, where V` is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning"&gt;pseudoinverse&lt;/a&gt;. 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, m1*Pm2 defines the semidefinite form on weighted monzos, and hence b1*W^(-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="/Tenney-Euclidean%20temperament%20measures"&gt;TE complexity&lt;/a&gt;, and hence is the corresponding Gram matrix for the dual space. Hence &lt;strong&gt;P&lt;/strong&gt; represents a change of basis defined by the mapping given by the vals combined with an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Inner_product_space" rel="nofollow"&gt;inner product&lt;/a&gt; on the result. Given a monzo b, bA* represents the tempered interval corresponding to b in a basis defined by the mapping A, and &lt;em&gt;P&lt;/em&gt; = (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;
+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="/Tenney-Euclidean%20temperament%20measures"&gt;TE complexity&lt;/a&gt;, and hence is the corresponding Gram matrix for the dual space. Hence &lt;strong&gt;P&lt;/strong&gt; represents a change of basis defined by the mapping given by the vals combined with an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Inner_product_space" rel="nofollow"&gt;inner product&lt;/a&gt; on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and &lt;em&gt;P&lt;/em&gt; = (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 lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm&lt;/em&gt; or &lt;em&gt;temperamental complexity&lt;/em&gt; of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t&lt;em&gt;P&lt;/em&gt;t*) where t is the image of a monzo b by t = bA*&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 lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. 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 norm&lt;/em&gt; or &lt;em&gt;temperamental complexity&lt;/em&gt; of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t*&lt;em&gt;P&lt;/em&gt;t) where t is the image of a monzo b by t = Ab&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The OETES"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The OETES&lt;/h1&gt;