Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 567479529 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 567486981 - 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>2015-11-23 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-23 12:07:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>567486981</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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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. | 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. | ||
Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by [[srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.</pre></div> | Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by [[srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0. To give another example, the [[eikosany]] has a sparcity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440 and 1029/1024 as well as 3025/3024) lowers that to 0.4521, miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282.</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: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> | <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> | ||
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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. <br /> | 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. <br /> | ||
<br /> | <br /> | ||
Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by <a class="wiki_link" href="/srutal">srutal</a>, where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.</body></html></pre></div> | Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by <a class="wiki_link" href="/srutal">srutal</a>, where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0. To give another example, the <a class="wiki_link" href="/eikosany">eikosany</a> has a sparcity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440 and 1029/1024 as well as 3025/3024) lowers that to 0.4521, miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282.</body></html></pre></div> | ||