Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 567492207 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 567492373 - 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 12: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-23 12:35:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>567492373</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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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, [[pentadekany2]], which is Cps([2,3,5,7,9,11], 3), 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. | 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, [[pentadekany2]], which is Cps([2,3,5,7,9,11], 3), 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. | ||
An invariant related to sparcity is //spread//. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^d(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function f = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with f(0) = **P**, the number of notes in the scale and therefore points in the space, and f(1) = 1. We can think of f(0) as the highest magnification, with each of the points showing clearly, and f(1) as the lowest, where all points have merged together. In between, | An invariant related to sparcity is //spread//. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^d(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function f = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with f(0) = **P**, the number of notes in the scale and therefore points in the space, and f(1) = 1. We can think of f(0) as the highest magnification, with each of the points showing clearly, and f(1) as the lowest, where all points have merged together. In between, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparcity measure.</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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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, <a class="wiki_link" href="/pentadekany2">pentadekany2</a>, which is Cps([2,3,5,7,9,11], 3), 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.<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. To give another example, <a class="wiki_link" href="/pentadekany2">pentadekany2</a>, which is Cps([2,3,5,7,9,11], 3), 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.<br /> | ||
<br /> | <br /> | ||
An invariant related to sparcity is <em>spread</em>. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^d(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function f = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with f(0) = <strong>P</strong>, the number of notes in the scale and therefore points in the space, and f(1) = 1. We can think of f(0) as the highest magnification, with each of the points showing clearly, and f(1) as the lowest, where all points have merged together. In between, | An invariant related to sparcity is <em>spread</em>. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^d(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function f = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with f(0) = <strong>P</strong>, the number of notes in the scale and therefore points in the space, and f(1) = 1. We can think of f(0) as the highest magnification, with each of the points showing clearly, and f(1) as the lowest, where all points have merged together. In between, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparcity measure.</body></html></pre></div> | ||