Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 568143457 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 568198177 - 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>2015-11-29 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-29 19:27:04 UTC</tt>.<br> | ||
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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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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 spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = **P**, the number of notes in the scale and therefore points in the space, and spread(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 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. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. | 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 spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = **P**, the number of notes in the scale and therefore points in the space, and spread(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 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. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. | ||
In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, spread(Euler(3*5)) = 4/(t^3 + 2t^2 +1), spread(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).</pre></div> | In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, spread(Euler(3*5)) = 4/(t^3 + 2t^2 +1), spread(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).</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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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 spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = <strong>P</strong>, the number of notes in the scale and therefore points in the space, and spread(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 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. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. <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 spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = <strong>P</strong>, the number of notes in the scale and therefore points in the space, and spread(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 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. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. <br /> | ||
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In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, spread(Euler(3*5)) = 4/(t^3 + 2t^2 +1), spread(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).</body></html></pre></div> | In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, spread(Euler(3*5)) = 4/(t^3 + 2t^2 +1), spread(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).</body></html></pre></div> | ||