Kees semi-height
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author mbattaglia1 and made on 2016-03-05 22:19:28 UTC.
- The original revision id was 576671173.
- 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:
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. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.</span> 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 <a class="wiki_link" href="/height">height</a></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.<br />
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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. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.</span><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>
<table class="wiki_table">
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<td style="text-align: center;"><strong>interval</strong><br />
</td>
<td style="text-align: center;"><strong>kees height</strong><br />
</td>
</tr>
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<td style="text-align: center;">5/3<br />
</td>
<td style="text-align: center;">5<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4/3<br />
</td>
<td style="text-align: center;">3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2/1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
</body></html>