Structure metric: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 562824955 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 562854845 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2015-10- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-10-18 12:50:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>562854845</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the period-equivalenced notes of a [[constant structure]] [[periodic scale]] which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. If **s** is a periodic scale with quasiperiod **P**, and if **s**[i] is a note of **s**, then we may define the base points set base(**s**[i]) to be the set of integers {j|**s**[j+i] - **s**[j] = **s**[i]}. Reducing these modulo **P** to the range 0 ... **P**-1 gives a finite set of period-equivalenced notes.</pre></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;white-space: pre-wrap ! important" class="old-revision-html">The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the period-equivalenced notes of a [[constant structure]] [[periodic scale]] which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. If **s** is a periodic scale with quasiperiod **P**, and if **s**[i] is a note of **s**, then we may define the base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i]}. Reducing these modulo **P** to the range 0 ... **P**-1 gives a finite set of indicies of period-equivalenced notes, which have the property that the interval between the base note **s**[j] and the note i steps away, **s**[j+i], is in class(i), the interval class to which **s**[i] belongs. If the cardinality of this mod **P** reduced set is n, there are n indicies which correspond to intervals of **s**[i], and **P**-n which correspond to indicies of intervals other than **s**[i]. Then the //structure complexity// ||**s**[i]|| of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = ||**s**[i] - **s**[j]||.</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>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 period-equivalenced notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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 <strong>s</strong>[i] is a note of <strong>s</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j|<strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i]}. Reducing these modulo <strong>P</strong> to the range 0 ... <strong>P</strong>-1 gives a finite set of period-equivalenced notes.</body></html></pre></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>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 period-equivalenced notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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 <strong>s</strong>[i] is a note of <strong>s</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i]}. Reducing these modulo <strong>P</strong> to the range 0 ... <strong>P</strong>-1 gives a finite set of indicies of period-equivalenced notes, which have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this mod <strong>P</strong> reduced set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = ||<strong>s</strong>[i] - <strong>s</strong>[j]||.</body></html></pre></div> | ||
Revision as of 12:50, 18 October 2015
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2015-10-18 12:50:07 UTC.
- The original revision id was 562854845.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the period-equivalenced notes of a [[constant structure]] [[periodic scale]] which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. If **s** is a periodic scale with quasiperiod **P**, and if **s**[i] is a note of **s**, then we may define the base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i]}. Reducing these modulo **P** to the range 0 ... **P**-1 gives a finite set of indicies of period-equivalenced notes, which have the property that the interval between the base note **s**[j] and the note i steps away, **s**[j+i], is in class(i), the interval class to which **s**[i] belongs. If the cardinality of this mod **P** reduced set is n, there are n indicies which correspond to intervals of **s**[i], and **P**-n which correspond to indicies of intervals other than **s**[i]. Then the //structure complexity// ||**s**[i]|| of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = ||**s**[i] - **s**[j]||.Original HTML content:
<html><head><title>Structure metric</title></head><body>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 period-equivalenced notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> 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 <strong>s</strong>[i] is a note of <strong>s</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i]}. Reducing these modulo <strong>P</strong> to the range 0 ... <strong>P</strong>-1 gives a finite set of indicies of period-equivalenced notes, which have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this mod <strong>P</strong> reduced set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = ||<strong>s</strong>[i] - <strong>s</strong>[j]||.</body></html>