Tenney norm: Difference between revisions

Revision as of 23:20, 6 February 2015

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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 ||~ [[monzo|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 ||

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 />
<!-- ws:start:WikiTextCodeRule:0:
&lt;pre class=&quot;text&quot;&gt;|| |e2 e3 ... ep&amp;gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * 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| = log2(2^|e2| * 3^|e3| * ... * 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>


<table class="wiki_table">
    <tr>
        <th>Interval names<br />
</th>
        <th>Frequency ratio<br />
</th>
        <th><a class="wiki_link" href="/monzo">ket vector</a><br />
</th>
        <th>log2 (Benedetti height)<br />
</th>
    </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>

</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:Omegatron|Omegatron]] and made on <tt>2014-09-11 13:25:09 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2015-02-06 23:20:42 UTC</tt>.<br>
-: The original revision id was <tt>521820608</tt>.<br>
+: The original revision id was <tt>540068698</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 20: / Line 20: @@
 || just major third || 5/4 || |-2 0 1&gt; || log2(20) = 4.322 ||
 || harmonic seventh || 7/4 || |-2 0 0 1&gt; || log2(28) = 4.807 ||
+</pre></div>
-The name //Tenney height// stems from the fact that [[James Tenney]] [[http://www.plainsound.org/pdfs/JC&amp;ToH.pdf|proposed it, using the name "harmonic distance"]] . 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 , which defines it as sqrt(n*d) instead//</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tenney Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If p/q is a positive rational number reduced to its lowest terms, then the &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt; is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 (&lt;a class="wiki_link" href="/log2"&gt;log2&lt;/a&gt;), of the &lt;a class="wiki_link" href="/Benedetti%20height"&gt;Benedetti height&lt;/a&gt;, leading to Tenney &lt;a class="wiki_link" href="/height"&gt;height&lt;/a&gt;. In either form it is widely used as a &lt;a class="wiki_link" href="/measure%20of%20inharmonicity"&gt;measure of inharmonicity&lt;/a&gt; and/or complexity for intervals.&lt;br /&gt;
@@ Line 110: / Line 107: @@
 &lt;/table&gt;
-&lt;br /&gt;
+&lt;/body&gt;&lt;/html&gt;</pre></div>
-The name &lt;em&gt;Tenney height&lt;/em&gt; stems from the fact that &lt;a class="wiki_link" href="/James%20Tenney"&gt;James Tenney&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.plainsound.org/pdfs/JC&amp;amp;ToH.pdf" rel="nofollow"&gt;proposed it, using the name &amp;quot;harmonic distance&amp;quot;&lt;/a&gt; . The &lt;em&gt;Benedetti height&lt;/em&gt;, the product of the numerator and denominator, was first proposed as a consonance measure by the Renaissance scientist and mathematician &lt;a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow"&gt;Giovanni Battista Benedetti&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;See also, discussion at &lt;!-- ws:start:WikiTextUrlRule:106:http://lumma.org/tuning/faq/#heights --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/faq/#heights" rel="nofollow"&gt;http://lumma.org/tuning/faq/#heights&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:106 --&gt; , which defines it as sqrt(n*d) instead&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Tenney norm: Difference between revisions

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