Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 567312639 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 567379759 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-22 14:10:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>567379759</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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p = 4.4843144 otonal and utonal pentad; isometric | p = 4.4843144 otonal and utonal pentad; isometric | ||
p = 6.9477267 otonal and utonal heptad; isometric | p = 6.9477267 otonal and utonal heptad; isometric | ||
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric</pre></div> | p = ∞ otonal and utonal tetrad; this implies the space is ultrametric | ||
==Sparcity== | |||
If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the //average distance// in the space is S/(2(n-1)(n-2)), and the //sparcity// of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less. </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:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#Properties">Properties</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#Isometry">Isometry</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 1em;"><a href="#Invariants">Invariants</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#Invariants-Centrality">Centrality</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 2em;"><a href="#Invariants-Roundness">Roundness</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 2em;"><a href="#Invariants-Sparcity">Sparcity</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --></div> | ||
<!-- ws:end:WikiTextTocRule:22 --><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="/periodic%20scale">periodic scale</a> within a single 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>. If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if c is an interval <strong>s</strong>[i+j] - <strong>s</strong>[i] with 0≤i&lt;<strong>P</strong>, then we may define the specific interval set S(c, j) to be {i|<strong>s</strong>[i+j] - <strong>s</strong>[i] = c} with 0≤i&lt;<strong>P</strong>, that is, indicies for the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(<strong>s</strong>[a], <strong>s</strong>[b]), which we will abbreviate as d(a, b), to be <strong>P</strong> - #S(|<strong>s</strong>[a] - <strong>s</strong>[b]|, |a - b|). <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="/periodic%20scale">periodic scale</a> within a single 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>. If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if c is an interval <strong>s</strong>[i+j] - <strong>s</strong>[i] with 0≤i&lt;<strong>P</strong>, then we may define the specific interval set S(c, j) to be {i|<strong>s</strong>[i+j] - <strong>s</strong>[i] = c} with 0≤i&lt;<strong>P</strong>, that is, indicies for the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(<strong>s</strong>[a], <strong>s</strong>[b]), which we will abbreviate as d(a, b), to be <strong>P</strong> - #S(|<strong>s</strong>[a] - <strong>s</strong>[b]|, |a - b|). <br /> | ||
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p = 4.4843144 otonal and utonal pentad; isometric<br /> | p = 4.4843144 otonal and utonal pentad; isometric<br /> | ||
p = 6.9477267 otonal and utonal heptad; isometric<br /> | p = 6.9477267 otonal and utonal heptad; isometric<br /> | ||
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric</body></html></pre></div> | p = ∞ otonal and utonal tetrad; this implies the space is ultrametric<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Invariants-Sparcity"></a><!-- ws:end:WikiTextHeadingRule:12 -->Sparcity</h2> | |||
If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the <em>average distance</em> in the space is S/(2(n-1)(n-2)), and the <em>sparcity</em> of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less.</body></html></pre></div> | |||