Superparticular ratio: Difference between revisions
Wikispaces>genewardsmith **Imported revision 244989655 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 244990441 - 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>2011-08-09 03: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-09 03:41:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>244990441</tt>.<br> | ||
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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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* The difference (ie quotient) between two successive epimoric ratios is always an epimoric ratio. | * The difference (ie quotient) between two successive epimoric ratios is always an epimoric ratio. | ||
* The sum of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]]. | * The sum 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 via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)). | * 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 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) = cb/ad is epimoric. | * 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) = cb/ad is epimoric. | ||
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These ratios have some peculiar properties:<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 (ie quotient) 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 <a class="wiki_link" href="/Superpartient">epimeric ratio</a>.</li><li>Every epimoric ratio can be split into the product of two epimoric ratios via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)).</li><li>If a/b and c/d are Farey neighbors, that is if a/b &lt; c/d and bc - ad = 1, then (c/d)/(a/b) = cb/ad is epimoric.</li></ul><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 (ie quotient) 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 <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 may exist.</li><li>If a/b and c/d are Farey neighbors, that is if a/b &lt; c/d and bc - ad = 1, then (c/d)/(a/b) = cb/ad is epimoric.</li></ul><br /> | ||
Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a multiple of the fundamental (the same rule applies to all natural harmonics).<br /> | Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a multiple of the fundamental (the same rule applies to all natural harmonics).<br /> | ||
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<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></pre></div> | <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></pre></div> | ||