Tenney norm: Difference between revisions

Revision as of 15:51, 25 July 2011

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2011-07-25 15:51:14 UTC.

The original revision id was 242780659.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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 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(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 on [[http://www.webcitation.org/607KmOwoI|http://lumma.org/music/theory/TuningFAQ.html]] 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 <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(|e2| * |e3| * ... * |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(|e2| * |e3| * ... * |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>
<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(Benedetti Height)</strong><br />
</td>
    </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>

<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 on <a class="wiki_link_ext" href="http://www.webcitation.org/607KmOwoI" rel="nofollow">http://lumma.org/music/theory/TuningFAQ.html</a> section CONSONANCE AND DISSONANCE</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:genewardsmith|genewardsmith]] and made on <tt>2011-07-12 04:01:17 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 15:51:14 UTC</tt>.<br>
-: The original revision id was <tt>240952357</tt>.<br>
+: The original revision id was <tt>242780659</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>
-<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;white-space: pre-wrap ! important" class="old-revision-html">If p/q is a positive rational number reduced to its lowest terms, then the **Tenney height** 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.
+<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;white-space: pre-wrap ! important" class="old-revision-html">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|Tenney height]] of a [[monzo]] is given by
+The //Tenney height// of a [[monzo]] is given by
 [[code type=text]]
 || |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |ep|)
@@ Line 15: / Line 15: @@
 ==Examples==
-|| **interval name** || **frequency ratio** || **ket vector** || **log2(Tenney Height)** ||
+|| **interval name** || **frequency ratio** || **ket vector** || **log2(Benedetti Height)** ||
 || prime || 1/1 || |0&gt; ||  0 ||
 || octave || 2/1 || |1&gt; ||  1 ||
@@ Line 22: / Line 22: @@
 || harmonic seventh || 7/4 || |-2 0 0 1&gt; || log2(28) = 4.807 ||
-The name //Tenney height// stems from the fact that [[James Tenney]] proposed it. However, it has been proposed that it should be called //Benedetti height//, since the first person to propose it as a consonance measure seems to have been Renaissance scientist and mathematician [[http://www.webcitation.org/6076Lm8r4|Giovanni Battista Benedetti]].
+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 on [[http://www.webcitation.org/607KmOwoI|http://lumma.org/music/theory/TuningFAQ.html]] section CONSONANCE AND DISSONANCE//
 </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;strong&gt;Tenney height&lt;/strong&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 height. 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;
+<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 Benedetti height, leading to Tenney height. 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;
 &lt;br /&gt;
-The &lt;a class="wiki_link" href="/Tenney%20Height"&gt;Tenney height&lt;/a&gt; of a &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is given by&lt;br /&gt;
+The &lt;em&gt;Tenney height&lt;/em&gt; of a &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is given by&lt;br /&gt;
 &lt;!-- ws:start:WikiTextCodeRule:0:
 &amp;lt;pre class=&amp;quot;text&amp;quot;&amp;gt;|| |e2 e3 ... ep&amp;amp;gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |ep|)&amp;lt;/pre&amp;gt;
@@ Line 58: / Line 58: @@
          &lt;td&gt;&lt;strong&gt;ket vector&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;log2(Tenney Height)&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;log2(Benedetti Height)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 114: / Line 114: @@
 &lt;br /&gt;
-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; proposed it. However, it has been proposed that it should be called &lt;em&gt;Benedetti height&lt;/em&gt;, since the first person to propose it as a consonance measure seems to have been 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;
+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; proposed it. 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 on &lt;a class="wiki_link_ext" href="http://www.webcitation.org/607KmOwoI" rel="nofollow"&gt;http://lumma.org/music/theory/TuningFAQ.html&lt;/a&gt; section CONSONANCE AND DISSONANCE&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Tenney norm: Difference between revisions

Revision as of 15:51, 25 July 2011

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

Navigation menu

Search