Structure metric: 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>2015-11-~~05 17~~:13:42 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-06 03:23:58 UTC</tt>.<br>

: The original revision id was <tt>~~565373211~~</tt>.<br>

: The original revision id was <tt>565410779</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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The [[https://en.wikipedia.org/wiki/Gromov_product|Gromov product]] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.

If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q<p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.</pre></div>

If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q<p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.

p = 1.1135814 [[duodene]], [[novadene]], [[marveldene]]; these are isometric

p = 1.2651510 [[zeus8tri]], [[star]], [[nova]]; these are isometric

p = 1.3404363 [[thirteendene]]

p = 1.3652790 [[centaur]]

p = 1.5709365 [[zarlino]]

p = 1.8501138 [[raven]]

p = 1.9855771 [[blue]]

p = 2 exactly all MOS scales

p = 3.1062837 [[hexany]]

p = 4.4843144 otonal and utonal pentad

p = 6.9477267 otonal and utonal heptad

p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

</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>Structure metric</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

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The <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Gromov_product" rel="nofollow">Gromov product</a> is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.<br />

<br />

If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is &quot;rounder&quot;, and with a lower one &quot;flatter&quot;. Below is a listing of some scales (either JI or in some edo) by increasing roundness.</body></html></pre></div>

If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is &quot;rounder&quot;, and with a lower one &quot;flatter&quot;. Below is a listing of some scales (either JI or in some edo) by increasing roundness.<br />

<br />

p = 1.1135814 <a class="wiki_link" href="/duodene">duodene</a>, <a class="wiki_link" href="/novadene">novadene</a>, <a class="wiki_link" href="/marveldene">marveldene</a>; these are isometric<br />

p = 1.2651510 <a class="wiki_link" href="/zeus8tri">zeus8tri</a>, <a class="wiki_link" href="/star">star</a>, <a class="wiki_link" href="/nova">nova</a>; these are isometric<br />

p = 1.3404363 <a class="wiki_link" href="/thirteendene">thirteendene</a><br />

p = 1.3652790 <a class="wiki_link" href="/centaur">centaur</a><br />

p = 1.5709365 <a class="wiki_link" href="/zarlino">zarlino</a><br />

p = 1.8501138 <a class="wiki_link" href="/raven">raven</a><br />

p = 1.9855771 <a class="wiki_link" href="/blue">blue</a><br />

p = 2 exactly all MOS scales<br />

p = 3.1062837 <a class="wiki_link" href="/hexany">hexany</a><br />

p = 4.4843144 otonal and utonal pentad<br />

p = 6.9477267 otonal and utonal heptad<br />

p = ∞ otonal and utonal tetrad; this implies the space is ultrametric</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>2015-11-05 17:13:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-06 03:23:58 UTC</tt>.<br>
-: The original revision id was <tt>565373211</tt>.<br>
+: The original revision id was <tt>565410779</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 47: / Line 47: @@
 The [[https://en.wikipedia.org/wiki/Gromov_product|Gromov product]] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.
-If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.</pre></div>
+If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.
+p = 1.1135814 [[duodene]],  [[novadene]], [[marveldene]]; these are isometric
+p = 1.2651510 [[zeus8tri]], [[star]], [[nova]]; these are isometric
+p = 1.3404363 [[thirteendene]]
+p = 1.3652790 [[centaur]]
+p = 1.5709365 [[zarlino]]
+p = 1.8501138 [[raven]]
+p = 1.9855771 [[blue]]
+p = 2 exactly all MOS scales
+p = 3.1062837 [[hexany]]
+p = 4.4843144 otonal and utonal pentad
+p = 6.9477267 otonal and utonal heptad
+p = ∞ otonal and utonal tetrad; this implies the space is ultrametric
+</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;Structure metric&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
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 The &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Gromov_product" rel="nofollow"&gt;Gromov product&lt;/a&gt; is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.&lt;br /&gt;
 &lt;br /&gt;
-If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&amp;lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is &amp;quot;rounder&amp;quot;, and with a lower one &amp;quot;flatter&amp;quot;. Below is a listing of some scales (either JI or in some edo) by increasing roundness.&lt;/body&gt;&lt;/html&gt;</pre></div>
+If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&amp;lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is &amp;quot;rounder&amp;quot;, and with a lower one &amp;quot;flatter&amp;quot;. Below is a listing of some scales (either JI or in some edo) by increasing roundness.&lt;br /&gt;
+&lt;br /&gt;
+p = 1.1135814 &lt;a class="wiki_link" href="/duodene"&gt;duodene&lt;/a&gt;,  &lt;a class="wiki_link" href="/novadene"&gt;novadene&lt;/a&gt;, &lt;a class="wiki_link" href="/marveldene"&gt;marveldene&lt;/a&gt;; these are isometric&lt;br /&gt;
+p = 1.2651510 &lt;a class="wiki_link" href="/zeus8tri"&gt;zeus8tri&lt;/a&gt;, &lt;a class="wiki_link" href="/star"&gt;star&lt;/a&gt;, &lt;a class="wiki_link" href="/nova"&gt;nova&lt;/a&gt;; these are isometric&lt;br /&gt;
+p = 1.3404363 &lt;a class="wiki_link" href="/thirteendene"&gt;thirteendene&lt;/a&gt;&lt;br /&gt;
+p = 1.3652790 &lt;a class="wiki_link" href="/centaur"&gt;centaur&lt;/a&gt;&lt;br /&gt;
+p = 1.5709365 &lt;a class="wiki_link" href="/zarlino"&gt;zarlino&lt;/a&gt;&lt;br /&gt;
+p = 1.8501138 &lt;a class="wiki_link" href="/raven"&gt;raven&lt;/a&gt;&lt;br /&gt;
+p = 1.9855771 &lt;a class="wiki_link" href="/blue"&gt;blue&lt;/a&gt;&lt;br /&gt;
+p = 2 exactly all MOS scales&lt;br /&gt;
+p = 3.1062837 &lt;a class="wiki_link" href="/hexany"&gt;hexany&lt;/a&gt;&lt;br /&gt;
+p = 4.4843144 otonal and utonal pentad&lt;br /&gt;
+p = 6.9477267 otonal and utonal heptad&lt;br /&gt;
+p = ∞ otonal and utonal tetrad; this implies the space is ultrametric&lt;/body&gt;&lt;/html&gt;</pre></div>