Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 565324029 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 565330479 - Original comment: ** |
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<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>2015-11-05 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-05 12:19:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>565330479</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> | ||
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==Centrality== | ==Centrality== | ||
The //eccentricity// of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center.</pre></div> | The //eccentricity// of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. | ||
==Roundness== | |||
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. </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>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule: | <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><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 1em;"><a href="#Properties">Properties</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Isometry">Isometry</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#Invariants">Invariants</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#Invariants-Centrality">Centrality</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Invariants-Roundness">Roundness</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div> | ||
<!-- ws:end:WikiTextTocRule:19 --><br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
The <em>structure metric</em> is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within the period, which give to it the property of being a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow">finite metric space</a>. (In academic theory, constant structure is called the <em>partitioning property</em>.) If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if <strong>s</strong>[i] with 0≤i&lt;<strong>P</strong> is a note of <strong>s</strong> within the period <strong>P</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i], 0≤j&lt;<strong>P</strong>}. These have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. In other words, there are <strong>P</strong>-n intervals, counting multiplicities, in the class of <strong>s</strong>[i] other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| ||.<br /> | The <em>structure metric</em> is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within the period, which give to it the property of being a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow">finite metric space</a>. (In academic theory, constant structure is called the <em>partitioning property</em>.) If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if <strong>s</strong>[i] with 0≤i&lt;<strong>P</strong> is a note of <strong>s</strong> within the period <strong>P</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i], 0≤j&lt;<strong>P</strong>}. These have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. In other words, there are <strong>P</strong>-n intervals, counting multiplicities, in the class of <strong>s</strong>[i] other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| ||.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Invariants-Centrality"></a><!-- ws:end:WikiTextHeadingRule:8 -->Centrality</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Invariants-Centrality"></a><!-- ws:end:WikiTextHeadingRule:8 -->Centrality</h2> | ||
The <em>eccentricity</em> of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center.</body></html></pre></div> | The <em>eccentricity</em> of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Invariants-Roundness"></a><!-- ws:end:WikiTextHeadingRule:10 -->Roundness</h2> | |||
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.</body></html></pre></div> | |||