Tenney norm: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Wikispaces>clumma
**Imported revision 540068860 - Original comment: **
Wikispaces>FREEZE
No edit summary
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti_height|Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2|log2]]), of the [[Benedetti_height|Benedetti height]], leading to Tenney [[Height|height]]. In either form it is widely used as a [[measure_of_inharmonicity|measure of inharmonicity]] and/or complexity for intervals.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:clumma|clumma]] and made on <tt>2015-02-06 23:25:34 UTC</tt>.<br>
: The original revision id was <tt>540068860</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 [[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
The ''Tenney height'' of a [[monzo|monzo]] is given by
[[code]]
 
|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)
<pre>|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>
[[code]]


==Examples==
==Examples==
||= Interval name ||= Frequency ratio ||= monzo ||= log2(Benedetti height) ||
|| unison || 1/1 || |0&gt; || 0 ||
|| octave || 2/1 || |1&gt; || 1 ||
|| just perfect fifth || 3/2 || |-1 1&gt; || log2(6) = 2.585 ||
|| 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>
<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;
&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(2^|e2| * 3^|e3| * ... * p^|ep|)&amp;lt;/pre&amp;gt;
--&gt;
&lt;style type="text/css"&gt;&lt;!--
/**
* 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; }
--&gt;
&lt;/style&gt;&lt;pre class="text"&gt;|| |e2 e3 ... ep&amp;gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)&lt;/pre&gt;
&lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Examples&lt;/h2&gt;
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;Interval name&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Frequency ratio&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;monzo&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;log2(Benedetti height)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|1&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;just perfect fifth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|-1 1&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;log2(6) = 2.585&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;just major third&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|-2 0 1&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;log2(20) = 4.322&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;harmonic seventh&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;|-2 0 0 1&amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;log2(28) = 4.807&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable"
|-
| style="text-align:center;" | Interval name
| style="text-align:center;" | Frequency ratio
| style="text-align:center;" | monzo
| style="text-align:center;" | log2(Benedetti height)
|-
| | unison
| | 1/1
| | |0&gt;
| | 0
|-
| | octave
| | 2/1
| | |1&gt;
| | 1
|-
| | just perfect fifth
| | 3/2
| | |-1 1&gt;
| | log2(6) = 2.585
|-
| | 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
|}
[[Category:benedetti]]
[[Category:consonance]]
[[Category:dissonance]]
[[Category:harmonic_entropy]]
[[Category:height]]
[[Category:measure]]
[[Category:psychoacoustics]]
[[Category:theory]]

Revision as of 00:00, 17 July 2018

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 

|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)

Examples

Interval name Frequency ratio monzo log2(Benedetti height)
unison 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
Retrieved from "https://en.xen.wiki/index.php?title=Tenney_norm&oldid=3647"