Tenney norm
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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 ([[log2]]), 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| = log2(|e2| * |e3| * ... * |ep|) [[code]] ==Examples== || **interval name** || **frequency ratio** || **ket vector** || **log2(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// **<-- Error 404 - File not Found**
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 (<a class="wiki_link" href="/log2">log2</a>), of the height. In either form it is widely used as a <a class="wiki_link" href="/measure%20of%20inharmonicity">measure of inharmonicity</a> 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 />
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<pre class="text">|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |ep|)</pre>
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</style><pre class="text">|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |ep|)</pre>
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<!-- ws:start:WikiTextHeadingRule:1:<h2> --><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>log2(Tenney Height)</strong><br />
</td>
</tr>
<tr>
<td>prime<br />
</td>
<td>1/1<br />
</td>
<td>|0><br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>octave<br />
</td>
<td>2/1<br />
</td>
<td>|1><br />
</td>
<td>1<br />
</td>
</tr>
<tr>
<td>just perfect fifth<br />
</td>
<td>3/2<br />
</td>
<td>|-1 1><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><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><br />
</td>
<td>log2(28) = 4.807<br />
</td>
</tr>
</table>
<br />
<br />
<em>see also discussion on <!-- ws:start:WikiTextUrlRule:101: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:101 --> section CONSONANCE AND DISSONANCE</em> <strong><-- Error 404 - File not Found</strong></body></html>