Superparticular ratio: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 244979571 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 244979913 - 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: | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 01:33:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>244979913</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> | ||
<h4>Original Wikitext content:</h4> | <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]]</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;white-space: pre-wrap ! important" class="old-revision-html">[[http://en.wikipedia.org/wiki/Superparticular_number]] | ||
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. </pre></div> | |||
<h4>Original HTML content:</h4> | <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>superparticular</title></head><body><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a></body></html></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"><html><head><title>superparticular</title></head><body><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a><br /> | ||
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.</body></html></pre></div> | |||
Revision as of 01:33, 9 August 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Sarzadoce and made on 2011-08-09 01:33:25 UTC.
- The original revision id was 244979913.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[http://en.wikipedia.org/wiki/Superparticular_number]] 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.
Original HTML content:
<html><head><title>superparticular</title></head><body><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">http://en.wikipedia.org/wiki/Superparticular_number</a><br /> 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."<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.</body></html>