Kees semi-height
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2013-12-31 10:28:49 UTC.
- The original revision id was 480002862.
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Original Wikitext content:
Given a ratio of positive integers p/q, the //Kees height// is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height. Expressibility can be extended to all vectors in [[Monzos and Interval Space|interval space]], by means of the formula KE(|m2 m3 m5... mp>) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space. The set of JI intervals with kees height less than or equal to an odd integer q comprises the [[Odd limit|q odd limit]] The point of kees height is to serve as a metric/height on [[Pitch class|JI pitch classes]] corresponding to [[Benedetti height]] on pitches. The measure was proposed by [[Kees van Prooijen]]. [[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]] == Examples == ||= **interval** ||= **kees height** || ||= 5/3 ||= 5 || ||= 4/3 ||= 3 || ||= 2/1 ||= 1 ||
Original HTML content:
<html><head><title>Kees Height</title></head><body>Given a ratio of positive integers p/q, the <em>Kees height</em> is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height. Expressibility can be extended to all vectors in <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, by means of the formula KE(|m2 m3 m5... mp>) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space.<br />
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The set of JI intervals with kees height less than or equal to an odd integer q comprises the <a class="wiki_link" href="/Odd%20limit">q odd limit</a><br />
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The point of kees height is to serve as a metric/height on <a class="wiki_link" href="/Pitch%20class">JI pitch classes</a> corresponding to <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a> on pitches. The measure was proposed by <a class="wiki_link" href="/Kees%20van%20Prooijen">Kees van Prooijen</a>.<br />
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<a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br />
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<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 --> Examples </h2>
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<td style="text-align: center;"><strong>interval</strong><br />
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<td style="text-align: center;"><strong>kees height</strong><br />
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<td style="text-align: center;">5/3<br />
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<td style="text-align: center;">5<br />
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<td style="text-align: center;">4/3<br />
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<td style="text-align: center;">3<br />
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<td style="text-align: center;">2/1<br />
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<td style="text-align: center;">1<br />
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