Kees semi-height: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 480002862 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 480002960 - 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>2013-12-31 10: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-31 10:29:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>480002960</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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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<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;white-space: pre-wrap ! important" class="old-revision-html">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. | <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;white-space: pre-wrap ! important" class="old-revision-html">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 | 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 | 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]] | [[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]] | ||
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<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"><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 &quot;expressibility&quot; 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&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &quot;KE&quot; denotes Kees expressibility and |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space.<br /> | <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"><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 &quot;expressibility&quot; 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&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &quot;KE&quot; denotes Kees expressibility and |m2 m3 m5 ... mp&gt; is a vector with weighted coordinates in interval space.<br /> | ||
<br /> | <br /> | ||
The set of JI intervals with | 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 /> | ||
<br /> | <br /> | ||
The point of | 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 /> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br /> | <a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br /> | ||
Revision as of 10:29, 31 December 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2013-12-31 10:29:47 UTC.
- The original revision id was 480002960.
- 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. 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 />
<br />
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 />
<br />
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 />
<br />
<a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 --> Examples </h2>
<table class="wiki_table">
<tr>
<td style="text-align: center;"><strong>interval</strong><br />
</td>
<td style="text-align: center;"><strong>kees height</strong><br />
</td>
</tr>
<tr>
<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>