Kees semi-height: Difference between revisions

Revision as of 08:48, 6 March 2016

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This revision was by author xenwolf and made on 2016-03-06 08:48:32 UTC.

The original revision id was 576689845.

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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. 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== 
||= **intervals** ||= **kees height** ||
||= 7/4, 7/5, 7/6, 8/7 ||= 7 ||
||= 5/3, 8/5, 5/4, 6/5 ||= 5 ||
||= 4/3, 3/2 ||= 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 &quot;expressibility&quot; is then the logarithm base two of the Kees height.<br />
<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 />
<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:&lt;h2&gt; --><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>intervals</strong><br />
</td>
        <td style="text-align: center;"><strong>kees height</strong><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7/4, 7/5, 7/6, 8/7<br />
</td>
        <td style="text-align: center;">7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5/3, 8/5, 5/4, 6/5<br />
</td>
        <td style="text-align: center;">5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4/3, 3/2<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>

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2016-03-05 22:19:28 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-03-06 08:48:32 UTC</tt>.<br>
-: The original revision id was <tt>576671173</tt>.<br>
+: The original revision id was <tt>576689845</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>
@@ Line 18: / Line 18: @@
 ==Examples==
-||= **interval** ||= **kees height** ||
+||= **intervals** ||= **kees height** ||
-||= 5/3 ||= 5 ||
+||= 7/4, 7/5, 7/6, 8/7 ||= 7 ||
-||= 4/3 ||= 3 ||
+||= 5/3, 8/5, 5/4, 6/5 ||= 5 ||
+||= 4/3, 3/2 ||= 3 ||
 ||= 2/1 ||= 1 ||</pre></div>
 <h4>Original HTML content:</h4>
@@ Line 39: / Line 40: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;interval&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;intervals&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;strong&gt;kees height&lt;/strong&gt;&lt;br /&gt;
@@ Line 45: / Line 46: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;5/3&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;7/4, 7/5, 7/6, 8/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;5/3, 8/5, 5/4, 6/5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
@@ Line 51: / Line 58: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;4/3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4/3, 3/2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;3&lt;br /&gt;

Kees semi-height: Difference between revisions

Revision as of 08:48, 6 March 2016

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