Superparticular ratio
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Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. 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 (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]]. * 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) = 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). See: [[List of Superparticular Intervals]] and the Wikipedia page for [[http://en.wikipedia.org/wiki/Superparticular_number|Superparticular number]].
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<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 other than 1. 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 (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 < 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).<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>