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:genewardsmith|genewardsmith]] and made on <tt>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 14:32:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>209334948</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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<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;white-space: pre-wrap ! important" class="old-revision-html"> | <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;white-space: pre-wrap ! important" class="old-revision-html"> | ||
=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. | 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 [[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*). | ||
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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 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 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 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 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* | ||
=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. | 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 | 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 is equivalent to finding the matrix **P** = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining **S**, and defining the seminorm on monzos as sqrt(b**S**b*) for monzos b. | ||
=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. | ||
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Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]//P//[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while **P** does not depend on the choice of basis vals for the temperament, //P// does; if we choose [<1 2 3|, <0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and //P// changes coordinates to produce the same final result of temperamental complexity. | Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]//P//[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while **P** does not depend on the choice of basis vals for the temperament, //P// does; if we choose [<1 2 3|, <0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and //P// changes coordinates to produce the same final result of temperamental complexity. | ||
If instead we want | 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, | ||
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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:WikiTextHeadingRule:0:&lt; | <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><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The weighting matrix"></a><!-- ws:end:WikiTextHeadingRule:0 -->The weighting matrix</h1> | |||
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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- 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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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 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 /> | 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 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; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="OE complexity"></a><!-- ws:end:WikiTextHeadingRule:4 -->OE complexity</h1> | ||
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 /> | 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 /> | <br /> | ||
To define the | To define the OETES, or Tenney-Euclidean 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. This is equivalent to finding the matrix <strong>P</strong> = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining <strong>S</strong>, and defining the seminorm on monzos as sqrt(b<strong>S</strong>b*) for monzos b.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt; | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Logflat TE badness"></a><!-- ws:end:WikiTextHeadingRule:6 -->Logflat TE badness</h1> | ||
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;h2&gt; --><h2 id="toc4"><a name=" | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Logflat TE badness-Examples"></a><!-- ws:end:WikiTextHeadingRule:8 -->Examples</h2> | ||
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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Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]<em>P</em>[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while <strong>P</strong> does not depend on the choice of basis vals for the temperament, <em>P</em> does; if we choose [&lt;1 2 3|, &lt;0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and <em>P</em> changes coordinates to produce the same final result of temperamental complexity.<br /> | Using this, we find the temperamental norm of [9 13] to be sqrt([9 13]<em>P</em>[9 13]*) ~ sqrt(1.875) ~ 1.3693, identical to the temperamental seminorm of 3/2. Note however that while <strong>P</strong> does not depend on the choice of basis vals for the temperament, <em>P</em> does; if we choose [&lt;1 2 3|, &lt;0 -3 -5|] for our basis instead, then 3/2 is represented by [1 -3] and <em>P</em> changes coordinates to produce the same final result of temperamental complexity.<br /> | ||
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If instead we want | 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 /> | ||