Kees semi-height: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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~~:29:47 UTC</tt>.<br>

: This revision was by author [[User:Omegatron|Omegatron]] and made on <tt>2014-09-04 17:51:38 UTC</tt>.<br>

: The original revision id was <tt>~~480002960~~</tt>.<br>

: The original revision id was <tt>520874412</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>

<h4>Original Wikitext 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;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 Kees height less than or equal to an odd integer q comprises the [[Odd limit|q odd limit]]

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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"><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 <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 &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 />

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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 10:29:47 UTC</tt>.<br>
+: This revision was by author [[User:Omegatron|Omegatron]] and made on <tt>2014-09-04 17:51:38 UTC</tt>.<br>
-: The original revision id was <tt>480002960</tt>.<br>
+: The original revision id was <tt>520874412</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>
 <h4>Original Wikitext 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;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&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp&gt; 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&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp&gt; 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]]
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 ||= 2/1 ||= 1 ||</pre></div>
 <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;Kees Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given a ratio of positive integers p/q, the &lt;em&gt;Kees height&lt;/em&gt; 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 &amp;quot;expressibility&amp;quot; is then the logarithm base two of the Kees height. Expressibility can be extended to all vectors in &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, by means of the formula KE(|m2 m3 m5... mp&amp;gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &amp;quot;KE&amp;quot; denotes Kees expressibility and |m2 m3 m5 ... mp&amp;gt; is a vector with weighted coordinates in interval space.&lt;br /&gt;
+<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;Kees Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Given a ratio of positive integers p/q, the &lt;em&gt;Kees &lt;a class="wiki_link" href="/height"&gt;height&lt;/a&gt; &lt;/em&gt; 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 &amp;quot;expressibility&amp;quot; is then the logarithm base two of the Kees height. Expressibility can be extended to all vectors in &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, by means of the formula KE(|m2 m3 m5... mp&amp;gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where &amp;quot;KE&amp;quot; denotes Kees expressibility and |m2 m3 m5 ... mp&amp;gt; is a vector with weighted coordinates in interval space.&lt;br /&gt;
 &lt;br /&gt;
 The set of JI intervals with Kees height less than or equal to an odd integer q comprises the &lt;a class="wiki_link" href="/Odd%20limit"&gt;q odd limit&lt;/a&gt;&lt;br /&gt;