Structure metric: Difference between revisions

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

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

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| ||.

|| i - i || = ||0|| which equals 0.

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 || |i - j| || equals 0, then i - j equals 0 and so **s**[i] equals **s**[j].  

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

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** which is  **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]) = || |i - j| || + || |j - k| || ≥ || |i - k| || = d(**s**[i], **s**[k]).

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;

|| i - i || = ||0|| which equals 0.&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 || |i - j| || equals 0, then i - j equals 0 and so &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[j]. &lt;br /&gt;

d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) equals || |i - j| || equals || |j - i| || equals d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[i])&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; which is  &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]) = || |i - j| || + || |j - k| || ≥ || |i - k| || = d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[k]).&lt;br /&gt;