Tenney norm: Difference between revisions

Revision as of 15:23, 9 September 2013

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This revision was by author Hyacinth3 and made on 2013-09-09 15:23:05 UTC.

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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 Benedetti height, leading to Tenney 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 <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 />
==Examples==|| <strong>Interval names</strong> || <strong>Frequency ratio</strong> || <strong>ket vector</strong> || <strong>log2 (Benedetti height)</strong> ||<br />


<table class="wiki_table">
    <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 />
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:87: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:87 --></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:Hyacinth3|Hyacinth3]] and made on <tt>2013-09-09 15:21:34 UTC</tt>.<br>
+: This revision was by author [[User:Hyacinth3|Hyacinth3]] and made on <tt>2013-09-09 15:23:05 UTC</tt>.<br>
-: The original revision id was <tt>449744508</tt>.<br>
+: The original revision id was <tt>449744880</tt>.<br>
-: The revision comment was: <tt>Attempt to fix table</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==||~ Interval names ||~ Frequency ratio ||~ ket vector ||~ log2 (Benedetti height) ||
+==Examples==|| **Interval names** || **Frequency ratio** || **ket vector** || **log2 (Benedetti height)** ||
 || prime || 1/1 || |0&gt; || 0 ||
 || octave || 2/1 || |1&gt; || 1 ||
@@ Line 43: / Line 43: @@
 &lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;br /&gt;
-==Examples==||~ Interval names ||~ Frequency ratio ||~ ket vector ||~ log2 (Benedetti height) ||&lt;br /&gt;
+==Examples==|| &lt;strong&gt;Interval names&lt;/strong&gt; || &lt;strong&gt;Frequency ratio&lt;/strong&gt; || &lt;strong&gt;ket vector&lt;/strong&gt; || &lt;strong&gt;log2 (Benedetti height)&lt;/strong&gt; ||&lt;br /&gt;

Tenney norm: Difference between revisions

Revision as of 15:23, 9 September 2013

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