Tenney norm
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Original Wikitext content:
If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the [[Benedetti height]], leading to Tenney [[height]]. In either form it is widely used as a [[measure of inharmonicity]] and/or complexity for intervals. The //Tenney height// of a [[monzo]] is given by [[code]] || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|) [[code]] ==Examples== ||~ Interval names ||~ Frequency ratio ||~ ket vector ||~ log2 (Benedetti 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 || The name //Tenney height// stems from the fact that [[James Tenney]] proposed it. The //Benedetti height//, the product of the numerator and denominator, was first proposed as a consonance measure by the Renaissance scientist and mathematician [[http://www.webcitation.org/6076Lm8r4|Giovanni Battista Benedetti]]. //See also, discussion at http://lumma.org/tuning/faq/#heights//
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 <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a> 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 <a class="wiki_link" href="/Benedetti%20height">Benedetti height</a>, leading to Tenney <a class="wiki_link" href="/height">height</a>. 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 <em>Tenney height</em> 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(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>
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</style><pre class="text">|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>
<!-- ws:end:WikiTextCodeRule:0 --><br />
<!-- ws:start:WikiTextHeadingRule:1:<h2> --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h2>
<table class="wiki_table">
<tr>
<th>Interval names<br />
</th>
<th>Frequency ratio<br />
</th>
<th>ket vector<br />
</th>
<th>log2 (Benedetti height)<br />
</th>
</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 />
The name <em>Tenney height</em> stems from the fact that <a class="wiki_link" href="/James%20Tenney">James Tenney</a> proposed it. The <em>Benedetti height</em>, the product of the numerator and denominator, was first proposed as a consonance measure by the Renaissance scientist and mathematician <a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow">Giovanni Battista Benedetti</a>.<br />
<br />
<em>See also, discussion at <!-- ws:start:WikiTextUrlRule:104:http://lumma.org/tuning/faq/#heights --><a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#heights" rel="nofollow">http://lumma.org/tuning/faq/#heights</a><!-- ws:end:WikiTextUrlRule:104 --></em></body></html>