Tenney norm: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 240952277 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 240952357 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-12 04:01:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>240952357</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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. 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]]. | ||
//see also discussion on [[http://lumma.org/music/theory/TuningFAQ.html | //see also discussion on [[http://www.webcitation.org/607KmOwoI|http://lumma.org/music/theory/TuningFAQ.html]] section CONSONANCE AND DISSONANCE// | ||
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The name <em>Tenney height</em> stems from the fact that <a class="wiki_link" href="/James%20Tenney">James Tenney</a> proposed it. However, it has been proposed that it should be called <em>Benedetti height</em>, since the first person to propose it as a consonance measure seems to have been Renaissance scientist and mathematician <a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow">Giovanni Battista Benedetti</a>.<br /> | The name <em>Tenney height</em> stems from the fact that <a class="wiki_link" href="/James%20Tenney">James Tenney</a> proposed it. However, it has been proposed that it should be called <em>Benedetti height</em>, since the first person to propose it as a consonance measure seems to have been Renaissance scientist and mathematician <a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow">Giovanni Battista Benedetti</a>.<br /> | ||
<br /> | <br /> | ||
<em>see also discussion on | <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></pre></div> | ||
Revision as of 04:01, 12 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 2011-07-12 04:01:17 UTC.
- The original revision id was 240952357.
- 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 **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. 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** || **log2(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 || 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]]. //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 <strong>Tenney height</strong> 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 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 <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of a <a class="wiki_link" href="/monzo">monzo</a> is given by<br />
<!-- ws:start:WikiTextCodeRule:0:
<pre class="text">|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |ep|)</pre>
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* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
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</style><pre class="text">|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep| = log2(|e2| * |e3| * ... * |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>
<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(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 />
The name <em>Tenney height</em> stems from the fact that <a class="wiki_link" href="/James%20Tenney">James Tenney</a> proposed it. However, it has been proposed that it should be called <em>Benedetti height</em>, since the first person to propose it as a consonance measure seems to have been 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>