Superparticular ratio: Difference between revisions
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Wikispaces>Sarzadoce **Imported revision 362668662 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 573345705 - 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:xenwolf|xenwolf]] and made on <tt>2016-01-29 06:58:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>573345705</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">Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole 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, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part." | ||
These ratios have some peculiar properties: | These ratios have some peculiar properties: | ||
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See: [[List of Superparticular Intervals]] and the Wikipedia page for [[http://en.wikipedia.org/wiki/Superparticular_number|Superparticular number]].</pre></div> | See: [[List of Superparticular Intervals]] and the Wikipedia page for [[http://en.wikipedia.org/wiki/Superparticular_number|Superparticular 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>Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>superparticular</title></head><body>Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as &quot;above a part.&quot;<br /> | ||
<br /> | <br /> | ||
These ratios have some peculiar properties:<br /> | These ratios have some peculiar properties:<br /> | ||
Revision as of 06:58, 29 January 2016
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2016-01-29 06:58:13 UTC.
- The original revision id was 573345705.
- 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:
Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) 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 6 such ratios ([[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 (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio. * The sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]]. * Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), but more than one such splitting method may exist. * If a/b and c/d are Farey neighbors, that is if a/b < c/d and bc - ad = 1, then (c/d)/(a/b) = bc/ad is epimoric. Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a [[Harmonic|multiple of the fundamental]] (the same rule applies to all natural harmonics in the Greek system). See: [[List of Superparticular Intervals]] and the Wikipedia page for [[http://en.wikipedia.org/wiki/Superparticular_number|Superparticular number]].
Original HTML content:
<html><head><title>superparticular</title></head><body>Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) 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 6 such ratios (<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 (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.</li><li>The sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an <a class="wiki_link" href="/Superpartient">epimeric ratio</a>.</li><li>Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), but more than one such splitting method may exist.</li><li>If a/b and c/d are Farey neighbors, that is if a/b < c/d and bc - ad = 1, then (c/d)/(a/b) = bc/ad is epimoric.</li></ul><br /> Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a <a class="wiki_link" href="/Harmonic">multiple of the fundamental</a> (the same rule applies to all natural harmonics in the Greek system).<br /> <br /> See: <a class="wiki_link" href="/List%20of%20Superparticular%20Intervals">List of Superparticular Intervals</a> and the Wikipedia page for <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Superparticular_number" rel="nofollow">Superparticular number</a>.</body></html>