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) 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** || **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 />
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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>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: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>