Tenney–Euclidean metrics: Difference between revisions
Wikispaces>genewardsmith **Imported revision 209349086 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 209349882 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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>2011-03-10 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 15:00:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>209349882</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 20: | Line 20: | ||
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 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 | 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= | |||
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 | ||
| Line 53: | Line 53: | ||
</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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: --><!-- 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: --> | ||
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The weighting matrix"></a><!-- ws:end:WikiTextHeadingRule:0 -->The weighting matrix</h1> | <!-- ws:end:WikiTextTocRule:16 --><!-- 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 /> | ||
| Line 67: | Line 67: | ||
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 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 seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of | 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 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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Logflat TE badness"></a><!-- ws:end:WikiTextHeadingRule:6 -->Logflat TE badness</h1> | <!-- 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt; | <!-- 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 /> | ||
<br /> | <br /> | ||