Tenney norm: Difference between revisions

Revision as of 07:42, 1 June 2010

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This revision was by author genewardsmith and made on 2010-06-01 07:42:43 UTC.

The original revision id was 146185801.

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Original Wikitext content:

If p/q is a positive rational number reduced to its lowest terms, then the **Tenney height**, named for [[James Tenney]] who proposed it, is the integer pq. Often it is more convenient instead to take the logarithm (usually base 2) of the height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.

The [[Tenney Height|Tenney height]] of a [[monzo]] is given by
[[code type=text]]
|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep|
[[code]]

==Examples==

|| **interval name** || **frequency ratio** || **ket vector** || **Tenney Height** ||
|| prime || 1/1 || |0> ||  0 ||
|| octave || 2/1 || |1> ||  1 ||
|| just perfect fifth || 3/2 || |-1 1> || log2(6) = 2.585 ||
|| just major third || 5/4 || |-2 0 1> || log2(20) = 4.322 ||
|| harmonic seventh || 7/4 || |-2 0 0 1> || log2(28) = 4.807 ||


//see also discussion on http://lumma.org/music/theory/TuningFAQ.txt section CONSONANCE AND DISSONANCE//

Original HTML content:

<html><head><title>Tenney Height</title></head><body>If p/q is a positive rational number reduced to its lowest terms, then the <strong>Tenney height</strong>, named for <a class="wiki_link" href="/James%20Tenney">James Tenney</a> who proposed it, is the integer pq. Often it is more convenient instead to take the logarithm (usually base 2) of the height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.<br />
<br />
The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of a <a class="wiki_link" href="/monzo">monzo</a> is given by<br />
<!-- ws:start:WikiTextCodeRule:0:
&lt;pre class=&quot;text&quot;&gt;|| |e2 e3 ... ep&amp;gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep|&lt;/pre&gt;
 -->
<style type="text/css"><!--
/**
 * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
 * (http://qbnz.com/highlighter/ and http://geshi.org/)
 */
.text  {font-family:monospace;}
.text .imp {font-weight: bold; color: red;}
.text span.xtra { display:block; }

-->
</style><pre class="text">|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep|</pre>

<!-- ws:end:WikiTextCodeRule:0 --><br />
<!-- ws:start:WikiTextHeadingRule:1:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h2>
<br />


<table class="wiki_table">
    <tr>
        <td><strong>interval name</strong><br />
</td>
        <td><strong>frequency ratio</strong><br />
</td>
        <td><strong>ket vector</strong><br />
</td>
        <td><strong>Tenney Height</strong><br />
</td>
    </tr>
    <tr>
        <td>prime<br />
</td>
        <td>1/1<br />
</td>
        <td>|0&gt;<br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>octave<br />
</td>
        <td>2/1<br />
</td>
        <td>|1&gt;<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>just perfect fifth<br />
</td>
        <td>3/2<br />
</td>
        <td>|-1 1&gt;<br />
</td>
        <td>log2(6) = 2.585<br />
</td>
    </tr>
    <tr>
        <td>just major third<br />
</td>
        <td>5/4<br />
</td>
        <td>|-2 0 1&gt;<br />
</td>
        <td>log2(20) = 4.322<br />
</td>
    </tr>
    <tr>
        <td>harmonic seventh<br />
</td>
        <td>7/4<br />
</td>
        <td>|-2 0 0 1&gt;<br />
</td>
        <td>log2(28) = 4.807<br />
</td>
    </tr>
</table>

<br />
<br />
<em>see also discussion on <!-- ws:start:WikiTextUrlRule:99:http://lumma.org/music/theory/TuningFAQ.txt --><a class="wiki_link_ext" href="http://lumma.org/music/theory/TuningFAQ.txt" rel="nofollow">http://lumma.org/music/theory/TuningFAQ.txt</a><!-- ws:end:WikiTextUrlRule:99 --> section CONSONANCE AND DISSONANCE</em></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:xenwolf|xenwolf]] and made on <tt>2010-06-01 03:27:49 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-01 07:42:43 UTC</tt>.<br>
-: The original revision id was <tt>146159623</tt>.<br>
+: The original revision id was <tt>146185801</tt>.<br>
-: The revision comment was: <tt>added example</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>
@@ Line 13: / Line 13: @@
 [[code]]
-==examples==
+==Examples==
-I tried to make it more understabable (for me?) using a few examples (maybe not perfect)
 || **interval name** || **frequency ratio** || **ket vector** || **Tenney Height** ||
-|| prime || 1/1 || |.&gt; ||  0 ||
+|| prime || 1/1 || |0&gt; ||  0 ||
 || octave || 2/1 || |1&gt; ||  1 ||
-|| just perfect fifth || 3/2 || |-1 1&gt; || ca 2.585 ||
+|| just perfect fifth || 3/2 || |-1 1&gt; || log2(6) = 2.585 ||
-|| just major third || 5/4 || |-2 0 1&gt; || ca 4.322 ||
+|| just major third || 5/4 || |-2 0 1&gt; || log2(20) = 4.322 ||
-|| harmonic seventh || 7/4 || |-2 0 0 1&gt; || ca 4.807 ||
+|| harmonic seventh || 7/4 || |-2 0 0 1&gt; || log2(28) = 4.807 ||
@@ Line 46: / Line 45: @@
 &lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;examples&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Examples&lt;/h2&gt;
-I tried to make it more understabable (for me?) using a few examples (maybe not perfect)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 67: / Line 65: @@
          &lt;td&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;|.&amp;gt;&lt;br /&gt;
+         &lt;td&gt;|0&amp;gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;0&lt;br /&gt;
@@ Line 89: / Line 87: @@
          &lt;td&gt;|-1 1&amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;ca 2.585&lt;br /&gt;
+         &lt;td&gt;log2(6) = 2.585&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 99: / Line 97: @@
          &lt;td&gt;|-2 0 1&amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;ca 4.322&lt;br /&gt;
+         &lt;td&gt;log2(20) = 4.322&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 109: / Line 107: @@
          &lt;td&gt;|-2 0 0 1&amp;gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;ca 4.807&lt;br /&gt;
+         &lt;td&gt;log2(28) = 4.807&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 116: / Line 114: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;see also discussion on &lt;!-- ws:start:WikiTextUrlRule:100:http://lumma.org/music/theory/TuningFAQ.txt --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/music/theory/TuningFAQ.txt" rel="nofollow"&gt;http://lumma.org/music/theory/TuningFAQ.txt&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:100 --&gt; section CONSONANCE AND DISSONANCE&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;em&gt;see also discussion on &lt;!-- ws:start:WikiTextUrlRule:99:http://lumma.org/music/theory/TuningFAQ.txt --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/music/theory/TuningFAQ.txt" rel="nofollow"&gt;http://lumma.org/music/theory/TuningFAQ.txt&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:99 --&gt; section CONSONANCE AND DISSONANCE&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Tenney norm: Difference between revisions

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