Tenney norm: Difference between revisions
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Wikispaces>xenwolf **Imported revision 515617060 - Original comment: ** |
Wikispaces>Omegatron **Imported revision 521734486 - 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: | : This revision was by author [[User:Omegatron|Omegatron]] and made on <tt>2014-09-11 00:14:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>521734486</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. 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]]. | 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//</pre></div> | //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> | <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"><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 /> | <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"><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 /> | ||
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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. 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 /> | 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 /> | <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></pre></div> | <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 --> , which defines it as sqrt(n*d) instead</em></body></html></pre></div> | ||
Revision as of 00:14, 11 September 2014
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Omegatron and made on 2014-09-11 00:14:24 UTC.
- The original revision id was 521734486.
- 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]] || |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 , which defines it as sqrt(n*d) instead//
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:
<pre class="text">|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>
-->
<style type="text/css"><!--
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* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
* (http://qbnz.com/highlighter/ and http://geshi.org/)
*/
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.text .imp {font-weight: bold; color: red;}
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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 --> , which defines it as sqrt(n*d) instead</em></body></html>