Structure metric: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-~~23 21~~:30:41 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-29 11:31:51 UTC</tt>.<br>

: The original revision id was <tt>~~567561495~~</tt>.<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. If A and B are two metric matricies for the same set of points, then A //dominates// B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparcity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.

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. ~~The sparcity~~ could use more study as it applies to scales; one notable fact for example is that most scales seem to have a ~~sparcity~~ inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a ~~sparcity~~ function with such an inflection point. </pre></div>

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>

<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><div id="toc"><h1 class="nopad">Table of Contents</h1><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. If A and B are two metric matricies for the same set of points, then A <em>dominates</em> B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparcity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.<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, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparcity measure. ~~The sparcity~~ could use more study as it applies to scales; one notable fact for example is that most scales seem to have a ~~sparcity~~ inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a ~~sparcity~~ function with such an inflection point.</body></html></pre></div>

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 />

<br />

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>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 21:30:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-29 11:31:51 UTC</tt>.<br>
-: The original revision id was <tt>567561495</tt>.<br>
+: The original revision id was <tt>568143457</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>
@@ Line 77: / Line 77: @@
 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. If A and B are two metric matricies for the same set of points, then A //dominates// B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparcity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.
-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. The sparcity could use more study as it applies to scales; one notable fact for example is that most scales seem to have a sparcity inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a sparcity function with such an inflection point. </pre></div>
+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>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Structure metric&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
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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 &lt;a class="wiki_link" href="/srutal"&gt;srutal&lt;/a&gt;, 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, &lt;a class="wiki_link" href="/pentadekany2"&gt;pentadekany2&lt;/a&gt;, 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. If A and B are two metric matricies for the same set of points, then A &lt;em&gt;dominates&lt;/em&gt; B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparcity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.&lt;br /&gt;
 &lt;br /&gt;
-An invariant related to sparcity is &lt;em&gt;spread&lt;/em&gt;. 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) = &lt;strong&gt;P&lt;/strong&gt;, 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. The sparcity could use more study as it applies to scales; one notable fact for example is that most scales seem to have a sparcity inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a sparcity function with such an inflection point.&lt;/body&gt;&lt;/html&gt;</pre></div>
+An invariant related to sparcity is &lt;em&gt;spread&lt;/em&gt;. 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) = &lt;strong&gt;P&lt;/strong&gt;, 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. &lt;br /&gt;
+&lt;br /&gt;
+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).&lt;/body&gt;&lt;/html&gt;</pre></div>