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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-02 03:01:12 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-02 18:06:19 UTC</tt>.<br>

: The original revision id was <tt>~~175585333~~</tt>.<br>

: The original revision id was <tt>175849481</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>

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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.

For example, consider marvel temperament, tempering out 225/224. If we add a row for 2, we get [|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.

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 C, then the inner product on note classes in this basis is defined by the symmetric matrix S = (CW^~~(-2)C~~*)^(-1). ~~For~~ example, starting from the normal val list for gamelismic, the 1029/1024 temperament, which is [<1 1 0 3|, <0 3 0 -1|, <0 0 1 0|], we may remove the first val to obtain C = [<0 3 0 -1|, <0 0 1 0|]. From this we obtain S = [[~~0.26958 0~~], [~~0 5.39135~~]]. ~~Since 3/2 maps to |3 0> by C, the~~ note class defined by 3~~/2 is represented by k = |3 0>,~~ and ~~the OE complexity~~, ~~or length, of this class is~~ sqrt(kSk*)~~, which is 1.55762~~. </pre></div>

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). Using the marvel example just considered, 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>

<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><h2 id="toc0"><a name="x-The weighting matrix"></a>The weighting matrix</h2>

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<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 />

For example, consider marvel temperament, tempering out 225/224. If we add a row for 2, we get [|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 />

<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. This seminorm applied to monzos gives the OE complexity.<br />

<br />

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 C, then the inner product on note classes in this basis is defined by the symmetric matrix S = (CW^~~(-2)C~~*)^(-1). ~~For~~ example, starting from the normal val list for gamelismic, the 1029/1024 temperament, which is [&lt;1 1 0 3|, &lt;0 3 0 -1|, &lt;0 0 1 0|], we may remove the first val to obtain C = [&lt;0 3 0 -1|, &lt;0 0 1 0|]. From this we obtain S = [[~~0.26958 0~~], [~~0 5.39135~~]]. ~~Since 3/2 maps to |3 0&gt; by C, the~~ note class defined by 3~~/2 is represented by k = |3 0&gt;,~~ and ~~the OE complexity~~, ~~or length, of this class is~~ sqrt(kSk*)~~, which is 1.55762~~.</body></html></pre></div>

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). Using the marvel example just considered, 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.</body></html></pre></div>

@@ 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>2010-11-02 03:01:12 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-02 18:06:19 UTC</tt>.<br>
-: The original revision id was <tt>175585333</tt>.<br>
+: The original revision id was <tt>175849481</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 20: / Line 20: @@
 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 [[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.
+For example, consider marvel temperament, tempering out 225/224. If we add a row for 2, we get [|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,
+|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.
 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 C, then the inner product on note classes in this basis is defined by the symmetric matrix S = (CW^(-2)C*)^(-1). For example, starting from the normal val list for gamelismic, the 1029/1024 temperament, which is  [&lt;1 1 0 3|, &lt;0 3 0 -1|, &lt;0 0 1 0|], we may remove the first val to obtain C = [&lt;0 3 0 -1|, &lt;0 0 1 0|]. From this we obtain S = [[0.26958 0], [0 5.39135]]. Since 3/2 maps to |3 0&gt; by C, the note class defined by 3/2 is represented by k = |3 0&gt;, and the OE complexity, or length, of this class is sqrt(kSk*), which is 1.55762.  </pre></div>
+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). Using the marvel example just considered, 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>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tenney-Euclidean metrics&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-The weighting matrix"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;The weighting matrix&lt;/h2&gt;
@@ Line 39: / Line 43: @@
 &lt;br /&gt;
 To define the OE, or octave equivalent seminorm, we simply add a row |1 0 0 ... 0&amp;gt; representing 2 to the matrix B. An alternative proceedure is to find the &lt;a class="wiki_link" href="/normal%20lists"&gt;normal val list&lt;/a&gt;, 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.&lt;br /&gt;
+&lt;br /&gt;
+For example, consider marvel temperament, tempering out 225/224. If we add a row for 2, we get [|1 0 0 0&amp;gt;, |-5 2 2 -1&amp;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&amp;gt;, |-5 2p3 2p5 -p7&amp;gt;], and P = I - M`M = [|1 0 0 0&amp;gt;, |0 4p5^2+p7^2 -4p3p5 2p3p7&amp;gt;/H, &lt;br /&gt;
+|0 -4p3p5 4p3^2+p7^2 2p5p7&amp;gt;/H, |0 2p3p7 2p5p7 4(p3^2+p5^2)&amp;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 [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]. Removing the first row gives [&amp;lt;0 1 0 2|, &amp;lt;0 0 1 2], and val weighting this gives C = [&amp;lt;0 1/p3 0 2/p7|, &amp;lt;0 0 1/p5 2/p7|]. Then P = C`C is precisely the same matrix we obtained before.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;
 &lt;br /&gt;
-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 C, then the inner product on note classes in this basis is defined by the symmetric matrix S = (CW^(-2)C*)^(-1). For example, starting from the normal val list for gamelismic, the 1029/1024 temperament, which is  [&amp;lt;1 1 0 3|, &amp;lt;0 3 0 -1|, &amp;lt;0 0 1 0|], we may remove the first val to obtain C = [&amp;lt;0 3 0 -1|, &amp;lt;0 0 1 0|]. From this we obtain S = [[0.26958 0], [0 5.39135]]. Since 3/2 maps to |3 0&amp;gt; by C, the note class defined by 3/2 is represented by k = |3 0&amp;gt;, and the OE complexity, or length, of this class is sqrt(kSk*), which is 1.55762.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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). Using the marvel example just considered, 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.&lt;/body&gt;&lt;/html&gt;</pre></div>