Superparticular ratio: Difference between revisions

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This revision was by author Sarzadoce and made on 2011-08-09 01:34:17 UTC.

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Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as "above a part."

These ratios have some peculiar properties:
* The difference tone of the dyad is also the virtual fundamental.
* The first 7 such ratios ([[Octave|2/1]], [[3_2|3/2]], [[4_3|4/3]], [[5_4|5/4]], [[6_5|6/5]], [[7_6|7/6]], [[8_7|8/7]]) are notable [[harmonic entropy]] minima.
* The difference between two successive epimoric ratios is always an epimoric ratio.
* The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.

Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number.


<span style="background-color: initial;">[[http://en.wikipedia.org/wiki/Superparticular_number]]</span>

Original HTML content:

<html><head><title>superparticular</title></head><body>Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as &quot;above a part.&quot;<br />
<br />
These ratios have some peculiar properties:<br />
<ul><li>The difference tone of the dyad is also the virtual fundamental.</li><li>The first 7 such ratios (<a class="wiki_link" href="/Octave">2/1</a>, <a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/8_7">8/7</a>) are notable <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> minima.</li><li>The difference between two successive epimoric ratios is always an epimoric ratio.</li><li>The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.</li></ul><br />
Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number.<br />
<br />
<br />
<span style="background-color: initial;"><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a></span></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:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 01:33:25 UTC</tt>.<br>
+: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 01:34:17 UTC</tt>.<br>
-: The original revision id was <tt>244979913</tt>.<br>
+: The original revision id was <tt>244980011</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">[[http://en.wikipedia.org/wiki/Superparticular_number]]
+<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">Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as "above a part."
-Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as "above a part."
 These ratios have some peculiar properties:
@@ Line 15: / Line 14: @@
 * The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.
-Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number. </pre></div>
+Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number.
+&lt;span style="background-color: initial;"&gt;[[http://en.wikipedia.org/wiki/Superparticular_number]]&lt;/span&gt;</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;superparticular&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Superparticular_number&lt;/a&gt;&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;superparticular&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as &amp;quot;above a part.&amp;quot;&lt;br /&gt;
-Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n. In ancient Greece they were known as Epimoric (επιμοριοσ) ratios, which is literally translated as &amp;quot;above a part.&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 These ratios have some peculiar properties:&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;The difference tone of the dyad is also the virtual fundamental.&lt;/li&gt;&lt;li&gt;The first 7 such ratios (&lt;a class="wiki_link" href="/Octave"&gt;2/1&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;, &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;) are notable &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; minima.&lt;/li&gt;&lt;li&gt;The difference between two successive epimoric ratios is always an epimoric ratio.&lt;/li&gt;&lt;li&gt;The sum of two successive epimoric ratios is either an epimoric ratio or an epimeric ratio.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Curiously enough, the ancient Greeks considered 2/1 a superparticular number even though 1 was not considered to be a true number.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;span style="background-color: initial;"&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Superparticular_number&lt;/a&gt;&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Superparticular ratio: Difference between revisions

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