Structure metric: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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Original Wikitext content:

The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[constant structure]] [[periodic scale]] within the 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// ||**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]| ||.

The structure metric has the following properties:

1. d(**s**[i], **s**[i]) = 0
|| **s**[i] - **s**[i] || = ||0|| which equals 0.

2. d(**s**[i], **s**[j]) ≥ 0
This is so since the cardinality n of the base point set is less than or equal to **P**.

3. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]
Suppose ||**s**[**I**]|| equals 0 with 0 < **I** < **P**. Then **s**[j+**I**] - **s**[j] = **s**[**I**], so that **s** is periodic with quasiperiod **I**. But by assumption, **P** is the least quasiperiod of **s**. Hence, ||**s**[**I**]|| equals 0 implies **I** equals 0. It follows that if d(**s**[i], **s**[j]) equals || **s**[i] - **s**[j] || equals 0, then **s**[i] - **s**[j] equals 0 and **s**[i] equals **s**[j]. 

4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
d(**s**[i], **s**[j]) equals || |**s**[i] - **s**[j]| || equals || |**s**[j] - **s**[i]| || equals d(**s**[j], **s**[i])

5. d(**s**[i], **s**[k]) ≤ d(**s**[i], **s**[j]) + d(**s**[j], **s**[k])
First, || **s**[i + j] mod **O** || ≤ ||**s**[i]|| + ||**s**[j]|| where **O** is the interval of equivalence. If an interval in the interval class of **s**[i] equals **s**[i] and an interval in the interval class of **s**[j] equals **s**[j], then their product, reduced modulo the interval of equivalence **O** equals **s**[**P**], will be **s**[i + j] mod **O**. Hence to get an interval in the class of **s**[i + j] mod **O** other than **s**[i + j] mod **O** as a product, either the interval in the class of **s**[i] must be other than **s**[i], or the interval in the class of **s**[j] must be other than **s**[j]. If always only one of the intervals is different than the defining interval for its class, then || **s**[i + j] mod **O** || equals ||**s**[i]|| + ||**s**[j]||. However, there may be overlap, so that the first interval is not in the class for **s**[i] and the second not in the class for **s**[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(**s**[i], **s**[j]) + d(**s**[j], **s**[k]) = || |**s**[i] - **s**[j]| || + || |**s**[j] - **s**[k]| || ≥ || |**s**[i] - **s**[k]| || = d(**s**[i], **s**[k]).

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 notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within the 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 <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if <strong>s</strong>[i] with  0≤i&lt;<strong>P</strong> is a note of <strong>s</strong> within the period <strong>P</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], 0≤j&lt;<strong>P</strong>}. These 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 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]. In other words, there are <strong>P</strong>-n intervals, counting multiplicities, in the class of <strong>s</strong>[i] 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]| ||.<br />
<br />
The structure metric has the following properties:<br />
<br />
1. d(<strong>s</strong>[i], <strong>s</strong>[i]) = 0<br />
|| <strong>s</strong>[i] - <strong>s</strong>[i] || = ||0|| which equals 0.<br />
<br />
2. d(<strong>s</strong>[i], <strong>s</strong>[j]) ≥ 0<br />
This is so since the cardinality n of the base point set is less than or equal to <strong>P</strong>.<br />
<br />
3. d(<strong>s</strong>[i], <strong>s</strong>[j]) = 0 implies <strong>s</strong>[i] equals <strong>s</strong>[j]<br />
Suppose ||<strong>s</strong>[<strong>I</strong>]|| equals 0 with 0 &lt; <strong>I</strong> &lt; <strong>P</strong>. Then <strong>s</strong>[j+<strong>I</strong>] - <strong>s</strong>[j] = <strong>s</strong>[<strong>I</strong>], so that <strong>s</strong> is periodic with quasiperiod <strong>I</strong>. But by assumption, <strong>P</strong> is the least quasiperiod of <strong>s</strong>. Hence, ||<strong>s</strong>[<strong>I</strong>]|| equals 0 implies <strong>I</strong> equals 0. It follows that if d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || <strong>s</strong>[i] - <strong>s</strong>[j] || equals 0, then <strong>s</strong>[i] - <strong>s</strong>[j] equals 0 and <strong>s</strong>[i] equals <strong>s</strong>[j]. <br />
<br />
4. d(<strong>s</strong>[i], <strong>s</strong>[j]) = d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || |<strong>s</strong>[i] - <strong>s</strong>[j]| || equals || |<strong>s</strong>[j] - <strong>s</strong>[i]| || equals d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
<br />
5. d(<strong>s</strong>[i], <strong>s</strong>[k]) ≤ d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k])<br />
First, || <strong>s</strong>[i + j] mod <strong>O</strong> || ≤ ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]|| where <strong>O</strong> is the interval of equivalence. If an interval in the interval class of <strong>s</strong>[i] equals <strong>s</strong>[i] and an interval in the interval class of <strong>s</strong>[j] equals <strong>s</strong>[j], then their product, reduced modulo the interval of equivalence <strong>O</strong> equals <strong>s</strong>[<strong>P</strong>], will be <strong>s</strong>[i + j] mod <strong>O</strong>. Hence to get an interval in the class of <strong>s</strong>[i + j] mod <strong>O</strong> other than <strong>s</strong>[i + j] mod <strong>O</strong> as a product, either the interval in the class of <strong>s</strong>[i] must be other than <strong>s</strong>[i], or the interval in the class of <strong>s</strong>[j] must be other than <strong>s</strong>[j]. If always only one of the intervals is different than the defining interval for its class, then || <strong>s</strong>[i + j] mod <strong>O</strong> || equals ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]||. However, there may be overlap, so that the first interval is not in the class for <strong>s</strong>[i] and the second not in the class for <strong>s</strong>[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| || + || |<strong>s</strong>[j] - <strong>s</strong>[k]| || ≥ || |<strong>s</strong>[i] - <strong>s</strong>[k]| || = d(<strong>s</strong>[i], <strong>s</strong>[k]).</body></html>

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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:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-10-19 11:39:18 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-10-23 13:26:57 UTC</tt>.<br>
-: The original revision id was <tt>562964941</tt>.<br>
+: The original revision id was <tt>563659663</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>
 <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 notes of a [[constant structure]] [[periodic scale]] within the 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// ||**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] ||.
+<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 notes of a [[constant structure]] [[periodic scale]] within the 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// ||**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]| ||.
 The structure metric has the following properties:
+. d(**s**[i], **s**[i]) = 0
+|| **s**[i] - **s**[i] || = ||0|| which equals 0.
+. d(**s**[i], **s**[j]) ≥ 0
+This is so since the cardinality n of the base point set is less than or equal to **P**.
+. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]
+Suppose ||**s**[**I**]|| equals 0 with 0 &lt; **I** &lt; **P**. Then **s**[j+**I**] - **s**[j] = **s**[**I**], so that **s** is periodic with quasiperiod **I**. But by assumption, **P** is the least quasiperiod of **s**. Hence, ||**s**[**I**]|| equals 0 implies **I** equals 0. It follows that if d(**s**[i], **s**[j]) equals || **s**[i] - **s**[j] || equals 0, then **s**[i] - **s**[j] equals 0 and **s**[i] equals **s**[j].
+. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
+d(**s**[i], **s**[j]) equals || |**s**[i] - **s**[j]| || equals || |**s**[j] - **s**[i]| || equals d(**s**[j], **s**[i])
+. d(**s**[i], **s**[k]) ≤ d(**s**[i], **s**[j]) + d(**s**[j], **s**[k])
+First, || **s**[i + j] mod **O** || ≤ ||**s**[i]|| + ||**s**[j]|| where **O** is the interval of equivalence. If an interval in the interval class of **s**[i] equals **s**[i] and an interval in the interval class of **s**[j] equals **s**[j], then their product, reduced modulo the interval of equivalence **O** equals **s**[**P**], will be **s**[i + j] mod **O**. Hence to get an interval in the class of **s**[i + j] mod **O** other than **s**[i + j] mod **O** as a product, either the interval in the class of **s**[i] must be other than **s**[i], or the interval in the class of **s**[j] must be other than **s**[j]. If always only one of the intervals is different than the defining interval for its class, then || **s**[i + j] mod **O** || equals ||**s**[i]|| + ||**s**[j]||. However, there may be overlap, so that the first interval is not in the class for **s**[i] and the second not in the class for **s**[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(**s**[i], **s**[j]) + d(**s**[j], **s**[k]) = || |**s**[i] - **s**[j]| || + || |**s**[j] - **s**[k]| || ≥ || |**s**[i] - **s**[k]| || = d(**s**[i], **s**[k]).
 </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;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="/constant%20structure"&gt;constant structure&lt;/a&gt; &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within the 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; ||&lt;strong&gt;s&lt;/strong&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]) = || &lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j] ||.&lt;br /&gt;
+<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;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="/constant%20structure"&gt;constant structure&lt;/a&gt; &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within the 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; ||&lt;strong&gt;s&lt;/strong&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]) = || |&lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j]| ||.&lt;br /&gt;
+&lt;br /&gt;
+The structure metric has the following properties:&lt;br /&gt;
+&lt;br /&gt;
+. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[i]) = 0&lt;br /&gt;
+|| &lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[i] || = ||0|| which equals 0.&lt;br /&gt;
+&lt;br /&gt;
+. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) ≥ 0&lt;br /&gt;
+This is so since the cardinality n of the base point set is less than or equal to &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
+&lt;br /&gt;
+. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = 0 implies &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[j]&lt;br /&gt;
+Suppose ||&lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;]|| equals 0 with 0 &amp;lt; &lt;strong&gt;I&lt;/strong&gt; &amp;lt; &lt;strong&gt;P&lt;/strong&gt;. Then &lt;strong&gt;s&lt;/strong&gt;[j+&lt;strong&gt;I&lt;/strong&gt;] - &lt;strong&gt;s&lt;/strong&gt;[j] = &lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;], so that &lt;strong&gt;s&lt;/strong&gt; is periodic with quasiperiod &lt;strong&gt;I&lt;/strong&gt;. But by assumption, &lt;strong&gt;P&lt;/strong&gt; is the least quasiperiod of &lt;strong&gt;s&lt;/strong&gt;. Hence, ||&lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;]|| equals 0 implies &lt;strong&gt;I&lt;/strong&gt; equals 0. It follows that if d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) equals || &lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j] || equals 0, then &lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j] equals 0 and &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[j]. &lt;br /&gt;
+&lt;br /&gt;
+. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[i])&lt;br /&gt;
+d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) equals || |&lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j]| || equals || |&lt;strong&gt;s&lt;/strong&gt;[j] - &lt;strong&gt;s&lt;/strong&gt;[i]| || equals d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[i])&lt;br /&gt;
 &lt;br /&gt;
-The structure metric has the following properties:&lt;/body&gt;&lt;/html&gt;</pre></div>
+. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[k]) ≤ d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) + d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[k])&lt;br /&gt;
+First, || &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; || ≤ ||&lt;strong&gt;s&lt;/strong&gt;[i]|| + ||&lt;strong&gt;s&lt;/strong&gt;[j]|| where &lt;strong&gt;O&lt;/strong&gt; is the interval of equivalence. If an interval in the interval class of &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[i] and an interval in the interval class of &lt;strong&gt;s&lt;/strong&gt;[j] equals &lt;strong&gt;s&lt;/strong&gt;[j], then their product, reduced modulo the interval of equivalence &lt;strong&gt;O&lt;/strong&gt; equals &lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;P&lt;/strong&gt;], will be &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt;. Hence to get an interval in the class of &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; other than &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; as a product, either the interval in the class of &lt;strong&gt;s&lt;/strong&gt;[i] must be other than &lt;strong&gt;s&lt;/strong&gt;[i], or the interval in the class of &lt;strong&gt;s&lt;/strong&gt;[j] must be other than &lt;strong&gt;s&lt;/strong&gt;[j]. If always only one of the intervals is different than the defining interval for its class, then || &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; || equals ||&lt;strong&gt;s&lt;/strong&gt;[i]|| + ||&lt;strong&gt;s&lt;/strong&gt;[j]||. However, there may be overlap, so that the first interval is not in the class for &lt;strong&gt;s&lt;/strong&gt;[i] and the second not in the class for &lt;strong&gt;s&lt;/strong&gt;[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) + d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[k]) = || |&lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[j]| || + || |&lt;strong&gt;s&lt;/strong&gt;[j] - &lt;strong&gt;s&lt;/strong&gt;[k]| || ≥ || |&lt;strong&gt;s&lt;/strong&gt;[i] - &lt;strong&gt;s&lt;/strong&gt;[k]| || = d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[k]).&lt;/body&gt;&lt;/html&gt;</pre></div>

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