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Structure metric: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 565612683 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 565735207 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2015-11-08 12:25:09 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-09 11:58:55 UTC</tt>.<br>
: The original revision id was <tt>565612683</tt>.<br>
: The original revision id was <tt>565735207</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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1. d(a, a) = 0
1. d(a, a) = 0
#S(|**s**[a] - **s**[a]|, |a - b|) = #S(0, 0) = **P**.
#S(|**s**[a] - **s**[a]|, |a - a|) = #S(0, 0) = **P**.


2. d(a, b) ≥ 0
2. d(a, b) ≥ 0
The cardinality of #S(c, j) cannot exceed ***P**, since 0≤i&lt;**P**.
The cardinality of #S(c, j) cannot exceed **P**, since 0≤i&lt;**P**.


3. d(a, b) = 0 implies a equals b.
3. d(a, b) = 0 implies a equals b.
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].
If d(a, b) = 0 then #S(|**s**[a] - **s**[b]|, |a - b|)) = **P**


4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
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p = 1.9855771 [[blue-ji|blue]]
p = 1.9855771 [[blue-ji|blue]]
p = 2 exactly all MOS scales
p = 2 exactly all MOS scales
p = 2.7580875 Cps([2,3,5,7,11], 2) and Cps([2,3,5,7,11], 3), the 2)5 and 3)5 dekanys
p = 3.1062837 [[hexany]]
p = 3.1062837 [[hexany]]
p = 4.4843144 otonal and utonal pentad
p = 4.4843144 otonal and utonal pentad
p = 6.9477267 otonal and utonal heptad
p = 6.9477267 otonal and utonal heptad
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric
</pre></div>
</pre></div>
<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
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&lt;br /&gt;
&lt;br /&gt;
1. d(a, a) = 0&lt;br /&gt;
1. d(a, a) = 0&lt;br /&gt;
#S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[a]|, |a - b|) = #S(0, 0) = &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
#S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[a]|, |a - a|) = #S(0, 0) = &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
2. d(a, b) ≥ 0&lt;br /&gt;
2. d(a, b) ≥ 0&lt;br /&gt;
The cardinality of #S(c, j) cannot exceed &lt;strong&gt;*P&lt;/strong&gt;, since 0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
The cardinality of #S(c, j) cannot exceed &lt;strong&gt;P&lt;/strong&gt;, since 0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
3. d(a, b) = 0 implies a equals b.&lt;br /&gt;
3. d(a, b) = 0 implies a equals b.&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;
If d(a, b) = 0 then #S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[b]|, |a - b|)) = &lt;strong&gt;P&lt;/strong&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
4. 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;
4. 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;
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p = 1.9855771 &lt;a class="wiki_link" href="/blue-ji"&gt;blue&lt;/a&gt;&lt;br /&gt;
p = 1.9855771 &lt;a class="wiki_link" href="/blue-ji"&gt;blue&lt;/a&gt;&lt;br /&gt;
p = 2 exactly all MOS scales&lt;br /&gt;
p = 2 exactly all MOS scales&lt;br /&gt;
p = 2.7580875 Cps([2,3,5,7,11], 2) and Cps([2,3,5,7,11], 3), the 2)5 and 3)5 dekanys&lt;br /&gt;
p = 3.1062837 &lt;a class="wiki_link" href="/hexany"&gt;hexany&lt;/a&gt;&lt;br /&gt;
p = 3.1062837 &lt;a class="wiki_link" href="/hexany"&gt;hexany&lt;/a&gt;&lt;br /&gt;
p = 4.4843144 otonal and utonal pentad&lt;br /&gt;
p = 4.4843144 otonal and utonal pentad&lt;br /&gt;
p = 6.9477267 otonal and utonal heptad&lt;br /&gt;
p = 6.9477267 otonal and utonal heptad&lt;br /&gt;
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric&lt;/body&gt;&lt;/html&gt;</pre></div>
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric&lt;/body&gt;&lt;/html&gt;</pre></div>
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