Tenney–Euclidean metrics

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==The weighting matrix==
Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || <a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

==Temperamental complexity==
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.

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.

==OE complexity==
Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I -  M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for **P** in terms of the matrix of monzos B.

To define the OE, or octave equivalent seminorm, we simply add a row |1 0 0 ... 0> representing 2 to the matrix B. An alternative proceedure is to find the [[normal lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity.

Octaves are now projected to the origin as well as commas. We can as before form the quotient space with respect to the seminorm, and obtain a normed space in which octave-equivalent interval classes of the intervals of the temperament are the lattice points. The seminorm applied to monzos gives the OE complexity.

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<html><head><title>Tenney-Euclidean metrics</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-The weighting matrix"></a><!-- ws:end:WikiTextHeadingRule:0 -->The weighting matrix</h2>
Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val &quot;a&quot; expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| || = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2). Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2) as the norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Temperamental complexity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperamental complexity</h2>
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 />
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 <strong>P</strong> defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.<br />
<br />
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 <em>temperamental complexity</em> of the intervals of the regular temperament.<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>
Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I -  M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for <strong>P</strong> in terms of the matrix of monzos B.<br />
<br />
To define the OE, or octave equivalent seminorm, we simply add a row |1 0 0 ... 0&gt; representing 2 to the matrix B. An alternative proceedure is to find the <a class="wiki_link" href="/normal%20lists">normal val list</a>, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity.<br />
<br />
Octaves are now projected to the origin as well as commas. We can as before form the quotient space with respect to the seminorm, and obtain a normed space in which octave-equivalent interval classes of the intervals of the temperament are the lattice points. The seminorm applied to monzos gives the OE complexity.</body></html>