Tenney–Euclidean metrics: Difference between revisions
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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: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-27 21:04:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>355130806</tt>.<br> | ||
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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> | ||
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=Temperamental 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 [[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, 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 [[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, 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 [[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, 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. | ||
Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains 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* | ||
=The OETES= | =The OETES= | ||
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To define the OETES, or Tenney-Euclidean 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. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of the p-limit regular temperament given by A. | To define the OETES, or Tenney-Euclidean 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. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of the p-limit regular temperament given by A. | ||
=Logflat TE badness= | =Logflat TE badness= | ||
Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity. | Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then //logflat badness// is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity. | ||
=Examples= | =Examples= | ||
Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [<15 24 35|, <22 35 51|]. From this we may obtain the matrix **P** as A*(AW^2A*)^(-1)A, approximately | Consider the temperament defined by the 5-limit [[Patent val|patent vals]] for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [<15 24 35|, <22 35 51|]. From this we may obtain the matrix **P** as A*(AW^2A*)^(-1)A, approximately | ||
[0.9911 0.1118 -0.1440] | [0.9911 0.1118 -0.1440] | ||
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If instead we want the OETES, we may remove the first row of [<1 2 3|, <0 -3 -5|], leaving just [<0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [<0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps. | If instead we want the OETES, we may remove the first row of [<1 2 3|, <0 -3 -5|], leaving just [<0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [<0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps. | ||
For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0>, |-5 2 2 -1>]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0>, |-5 2p3 2p5 -p7>], and P = I - M`M = [|1 0 0 0>, |0 4p5^2+p7^2 -4p3p5 2p3p7>/H, | For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0>, |-5 2 2 -1>]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0>, |-5 2p3 2p5 -p7>], and P = I - M`M = [|1 0 0 0>, |0 4p5^2+p7^2 -4p3p5 2p3p7>/H, | ||
|0 -4p3p5 4p3^2+p7^2 2p5p7>/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)>/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]. Removing the first row gives [<0 1 0 2|, <0 0 1 2], and val weighting this gives C = [<0 1/p3 0 2/p7|, <0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before. | |0 -4p3p5 4p3^2+p7^2 2p5p7>/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)>/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]. Removing the first row gives [<0 1 0 2|, <0 0 1 2], and val weighting this gives C = [<0 1/p3 0 2/p7|, <0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before. | ||
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. This seminorm applied to monzos gives the OE 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. This seminorm applied to monzos gives the OE complexity. | ||
If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW^2R*)^(-1). In the case of marvel, we obtain S = [[p3^2(4p5^2+p7^2) -4p3^2p5^2], [-4p3^2p5^2 p5^2(4p3^2+p7^2)]]/H. If k = [a b] is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt(kSk*) gives the OE complexity of the note class. | If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW^2R*)^(-1). In the case of marvel, we obtain S = [[p3^2(4p5^2+p7^2) -4p3^2p5^2], [-4p3^2p5^2 p5^2(4p3^2+p7^2)]]/H. If k = [a b] is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt(kSk*) gives the OE complexity of the note class.</pre></div> | ||
</pre></div> | |||
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tenney-Euclidean metrics</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#The weighting matrix">The weighting matrix</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Temperamental complexity">Temperamental complexity</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#The OETES">The OETES</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Logflat TE badness">Logflat TE badness</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tenney-Euclidean metrics</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#The weighting matrix">The weighting matrix</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Temperamental complexity">Temperamental complexity</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#The OETES">The OETES</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Logflat TE badness">Logflat TE badness</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperamental complexity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperamental complexity</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperamental complexity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperamental complexity</h1> | ||
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, 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="/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, 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 /> | ||
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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 <strong>P</strong> 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 <em>P</em> = (AW^2A*)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.<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="/Tenney-Euclidean%20temperament%20measures">TE complexity</a>, and hence is the corresponding Gram matrix for the dual space. Hence <strong>P</strong> 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 <em>P</em> = (AW^2A*)^(-1) defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.<br /> | ||
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Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains 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 <em>temperamental norm</em> or <em>temperamental complexity</em> of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt(t<em>P</em>t*) where t is the image of a monzo b by t = bA*<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The OETES"></a><!-- ws:end:WikiTextHeadingRule:4 -->The OETES</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The OETES"></a><!-- ws:end:WikiTextHeadingRule:4 -->The OETES</h1> | ||
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Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then <em>logflat badness</em> is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.<br /> | Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank r of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if p is the largest such prime, then the dimension d is pi(p), the number of primes to p. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then <em>logflat badness</em> is defined by the formula S(A) C(A)^(r/(d-r)). If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:8 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:8 -->Examples</h1> | ||
Consider the temperament defined by the 5-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [&lt;15 24 35|, &lt;22 35 51|]. From this we may obtain the matrix <strong>P</strong> as A*(AW^2A*)^(-1)A, approximately <br /> | Consider the temperament defined by the 5-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> for 15 and 22 equal. From the vals, we may contruct a 2x3 matrix A = [&lt;15 24 35|, &lt;22 35 51|]. From this we may obtain the matrix <strong>P</strong> as A*(AW^2A*)^(-1)A, approximately<br /> | ||
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[0.9911 0.1118 -0.1440]<br /> | [0.9911 0.1118 -0.1440]<br /> | ||
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If instead we want the OETES, we may remove the first row of [&lt;1 2 3|, &lt;0 -3 -5|], leaving just [&lt;0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [&lt;0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.<br /> | If instead we want the OETES, we may remove the first row of [&lt;1 2 3|, &lt;0 -3 -5|], leaving just [&lt;0 -3 -5|]. If we now call this 1x3 matrix A, then (AW^2A*)^(-1) is a 1x1 matrix; in effect a scalar, with value [&lt;0.1215588|]. Multiplying a monzo b times A* gives a 1x1 matrix bA* whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which b belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of steps.<br /> | ||
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For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H, <br /> | For a more substantial example we need to consider at least a rank three temperament, so let us turn to marvel, the 7-limit temperament tempering out 225/224. The 2x4 matrix of monzos whose first row represents 2 and whose second row 225/224 is [|1 0 0 0&gt;, |-5 2 2 -1&gt;]. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is M = [|1 0 0 0&gt;, |-5 2p3 2p5 -p7&gt;], and P = I - M`M = [|1 0 0 0&gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&gt;/H,<br /> | ||
|0 -4p3p5 4p3^2+p7^2 2p5p7&gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&gt;/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]. Removing the first row gives [&lt;0 1 0 2|, &lt;0 0 1 2], and val weighting this gives C = [&lt;0 1/p3 0 2/p7|, &lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.<br /> | |0 -4p3p5 4p3^2+p7^2 2p5p7&gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&gt;/H], where H = 4p3^2+4p5^2+p7^2. On the other hand, we may start from the normal val list for the temperament, which is [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]. Removing the first row gives [&lt;0 1 0 2|, &lt;0 0 1 2], and val weighting this gives C = [&lt;0 1/p3 0 2/p7|, &lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.<br /> | ||
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