Structure metric: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-~~07 13~~:47:02 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-08 11:18:27 UTC</tt>.

: The original revision id was <tt>~~565550045~~</tt>.

: The original revision id was <tt>565607599</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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=Definition=

The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[periodic scale]] within a single period, 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] with 0≤i<**P** is a ~~note of~~ **s** ~~within the period~~ **P**, ~~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], 0≤j<**P**}. These 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 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]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.

The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[periodic scale]] within a single period, 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 c is an interval **s**[i+j] - **s**[i] with 0≤i<i+j<**P**, then we may define the specific interval set S(c, j) to be the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(**s**[a], **s**[b]), which we will abbreviate as d(a, b), to be **P** - #S(|**s**[a] - **s**[b]|, |a-b|). base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i], 0≤j<**P**}. These 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 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]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.

=Properties=

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<h1 id="toc0"><a name="Definition"></a>Definition</h1>

The structure metric is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within a single period, 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 s is a periodic scale with quasiperiod P, and if s[i] with 0≤i&lt;P is a ~~note of~~ s ~~within the period~~ P, ~~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], 0≤j&lt;P}. These 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 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]. In other words, there are P-n intervals, counting multiplicities, in the class of s[i] other than s[i]. Then the structure complexity || i || of s[i] is defined to be P-n, and the structure metric is defined as d(s[i], s[j]) = || |i - j| ||.

The structure metric is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within a single period, 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 s is a periodic scale with quasiperiod P, and if c is an interval s[i+j] - s[i] with 0≤i&lt;i+j&lt;P, then we may define the specific interval set S(c, j) to be the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(s[a], s[b]), which we will abbreviate as d(a, b), to be P - #S(|s[a] - s[b]|, |a-b|). base points set base(s[i]) to be the set of integers {j | s[j+i] - s[j] = s[i], 0≤j&lt;P}. These 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 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]. In other words, there are P-n intervals, counting multiplicities, in the class of s[i] other than s[i]. Then the structure complexity || i || of s[i] is defined to be P-n, and the structure metric is defined as d(s[i], s[j]) = || |i - j| ||.

<h1 id="toc1"><a name="Properties"></a>Properties</h1>

@@ 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>2015-11-07 13:47:02 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-08 11:18:27 UTC</tt>.<br>
-: The original revision id was <tt>565550045</tt>.<br>
+: The original revision id was <tt>565607599</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 9: / Line 9: @@
 =Definition=
-The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[periodic scale]] within a single period, 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] with  0≤i&lt;**P** is a note of **s** within the period **P**, 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], 0≤j&lt;**P**}. These 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 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]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.
+The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[periodic scale]] within a single period, 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 c is an interval **s**[i+j] - **s**[i] with 0≤i&lt;i+j&lt;**P**, then we may define the specific interval set S(c, j) to be the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(**s**[a], **s**[b]), which we will abbreviate as d(a, b), to be **P** - #S(|**s**[a] - **s**[b]|, |a-b|). base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i], 0≤j&lt;**P**}. These 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 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]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.
 =Properties=
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 &lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-The &lt;em&gt;structure metric&lt;/em&gt; is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow"&gt;distance function&lt;/a&gt; on the notes of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within a single period, which give to it the property of being a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;finite metric space&lt;/a&gt;. If &lt;strong&gt;s&lt;/strong&gt; is a periodic scale with quasiperiod &lt;strong&gt;P&lt;/strong&gt;, and if &lt;strong&gt;s&lt;/strong&gt;[i] with  0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt; is a note of &lt;strong&gt;s&lt;/strong&gt; within the period &lt;strong&gt;P&lt;/strong&gt;, then we may define the base points set base(&lt;strong&gt;s&lt;/strong&gt;[i]) to be the set of integers {j | &lt;strong&gt;s&lt;/strong&gt;[j+i] - &lt;strong&gt;s&lt;/strong&gt;[j] = &lt;strong&gt;s&lt;/strong&gt;[i], 0≤j&amp;lt;&lt;strong&gt;P&lt;/strong&gt;}. These have the property that the interval between the base note &lt;strong&gt;s&lt;/strong&gt;[j] and the note i steps away, &lt;strong&gt;s&lt;/strong&gt;[j+i], is in class(i), the interval class to which &lt;strong&gt;s&lt;/strong&gt;[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of &lt;strong&gt;s&lt;/strong&gt;[i], and &lt;strong&gt;P&lt;/strong&gt;-n which correspond to indicies of intervals other than &lt;strong&gt;s&lt;/strong&gt;[i]. In other words, there are &lt;strong&gt;P&lt;/strong&gt;-n intervals, counting multiplicities, in the class of &lt;strong&gt;s&lt;/strong&gt;[i] other than &lt;strong&gt;s&lt;/strong&gt;[i]. Then the &lt;em&gt;structure complexity&lt;/em&gt; || i || of &lt;strong&gt;s&lt;/strong&gt;[i] is defined to be &lt;strong&gt;P&lt;/strong&gt;-n, and the structure metric is defined as d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = || |i - j| ||.&lt;br /&gt;
+The &lt;em&gt;structure metric&lt;/em&gt; is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow"&gt;distance function&lt;/a&gt; on the notes of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within a single period, which give to it the property of being a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;finite metric space&lt;/a&gt;. If &lt;strong&gt;s&lt;/strong&gt; is a periodic scale with quasiperiod &lt;strong&gt;P&lt;/strong&gt;, and if c is an interval &lt;strong&gt;s&lt;/strong&gt;[i+j] - &lt;strong&gt;s&lt;/strong&gt;[i] with 0≤i&amp;lt;i+j&amp;lt;&lt;strong&gt;P&lt;/strong&gt;, then we may define the specific interval set S(c, j) to be the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(&lt;strong&gt;s&lt;/strong&gt;[a], &lt;strong&gt;s&lt;/strong&gt;[b]), which we will abbreviate as d(a, b), to be &lt;strong&gt;P&lt;/strong&gt; - #S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[b]|, |a-b|). base points set base(&lt;strong&gt;s&lt;/strong&gt;[i]) to be the set of integers {j | &lt;strong&gt;s&lt;/strong&gt;[j+i] - &lt;strong&gt;s&lt;/strong&gt;[j] = &lt;strong&gt;s&lt;/strong&gt;[i], 0≤j&amp;lt;&lt;strong&gt;P&lt;/strong&gt;}. These have the property that the interval between the base note &lt;strong&gt;s&lt;/strong&gt;[j] and the note i steps away, &lt;strong&gt;s&lt;/strong&gt;[j+i], is in class(i), the interval class to which &lt;strong&gt;s&lt;/strong&gt;[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of &lt;strong&gt;s&lt;/strong&gt;[i], and &lt;strong&gt;P&lt;/strong&gt;-n which correspond to indicies of intervals other than &lt;strong&gt;s&lt;/strong&gt;[i]. In other words, there are &lt;strong&gt;P&lt;/strong&gt;-n intervals, counting multiplicities, in the class of &lt;strong&gt;s&lt;/strong&gt;[i] other than &lt;strong&gt;s&lt;/strong&gt;[i]. Then the &lt;em&gt;structure complexity&lt;/em&gt; || i || of &lt;strong&gt;s&lt;/strong&gt;[i] is defined to be &lt;strong&gt;P&lt;/strong&gt;-n, and the structure metric is defined as d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = || |i - j| ||.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Properties&lt;/h1&gt;