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>2015-10-~~19 11~~:39:18 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-10-23 13:26:57 UTC</tt>.

: The original revision id was <tt>~~562964941~~</tt>.

: The original revision id was <tt>563659663</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.

<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<**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] ||.

<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<**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]).

</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>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="/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 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Structure metric</title></head><body>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="/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 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:

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 &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].

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])

~~The structure metric has~~ the ~~following properties:~~</body></html></pre></div>

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]).</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>2015-10-19 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>