Nearest just interval: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701,955000865... cents):

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<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;Nearest just interval&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An irrational interval or ratio of frequencies given by a real number r has an infinite list of &lt;em&gt;nearest just intervals&lt;/em&gt;; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call &lt;em&gt;best rational approximations&lt;/em&gt;. A ratio of integers p/q with q &amp;gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &amp;lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.&lt;br /&gt;

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