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:~~Omegatron~~|~~Omegatron~~]] and made on <tt>~~2014~~-09-~~04 17~~:51:38 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2016-03-05 22:19:28 UTC</tt>.<br>

: The original revision id was <tt>~~520874412~~</tt>.<br>

: The original revision id was <tt>576671173</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.

The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd limit|q odd limit]]

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]].

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[[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]]

== Examples ==

==Examples==

||= **interval** ||= **kees height** ||

||= 5/3 ||= 5 ||

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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"><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 />

<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.<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 />

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. 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 />

<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 />

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<a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow">Kees tuning pages</a><br />

<br />

<h2 id="toc0"><a name="x-Examples"></a> Examples </h2>

<h2 id="toc0"><a name="x-Examples"></a>Examples</h2>

@@ 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:Omegatron|Omegatron]] and made on <tt>2014-09-04 17:51:38 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2016-03-05 22:19:28 UTC</tt>.<br>
-: The original revision id was <tt>520874412</tt>.<br>
+: The original revision id was <tt>576671173</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.
-The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd limit|q odd limit]]
+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. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can&lt;span style="line-height: 1.5;"&gt; be extended to tempered intervals using the quotient norm.&lt;/span&gt;
+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]].
@@ Line 14: / Line 17: @@
 [[http://www.kees.cc/tuning/perbl.html|Kees tuning pages]]
-== Examples ==
+==Examples==
 ||= **interval** ||= **kees height** ||
 ||= 5/3 ||= 5 ||
@@ Line 20: / Line 23: @@
 ||= 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 &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;
+<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.&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;
+&lt;br /&gt;
+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. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can&lt;span style="line-height: 1.5;"&gt; be extended to tempered intervals using the quotient norm.&lt;/span&gt;&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;
 &lt;br /&gt;
 The point of Kees height is to serve as a metric/height on &lt;a class="wiki_link" href="/Pitch%20class"&gt;JI pitch classes&lt;/a&gt; corresponding to &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt; on pitches. The measure was proposed by &lt;a class="wiki_link" href="/Kees%20van%20Prooijen"&gt;Kees van Prooijen&lt;/a&gt;.&lt;br /&gt;
@@ Line 28: / Line 34: @@
 &lt;a class="wiki_link_ext" href="http://www.kees.cc/tuning/perbl.html" rel="nofollow"&gt;Kees tuning pages&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; Examples &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h2&gt;
 &lt;table class="wiki_table"&gt;