Structure metric: Difference between revisions

: 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>562854845</tt>.<br>

<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>

<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 period-equivalenced 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; 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] is a note of &lt;strong&gt;s&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]}. Reducing these modulo &lt;strong&gt;P&lt;/strong&gt; to the range 0 ... &lt;strong&gt;P&lt;/strong&gt;-1 gives a finite set of indicies of period-equivalenced notes, which 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 mod &lt;strong&gt;P&lt;/strong&gt; reduced 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]. 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;/body&gt;&lt;/html&gt;</pre></div>