Benedetti height: Difference between revisions
Wikispaces>xenwolf **Imported revision 344977304 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 504155460 - Original comment: added examples** |
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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:spt3125|spt3125]] and made on <tt>2014-04-23 17:51:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>504155460</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>added examples</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> | ||
<h4>Original Wikitext content:</h4> | <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">The //Benedetti height// of a positive rational number N/D reduced to lowest terms (no common factor between N and D) is equal to N*D, the product of the numerator and denominator. The logarithm base two of the Benedetti height is the [[Tenney height]], or Tenney norm. The name is based on the fact that the scientist, mathematician and music theorist [[http://www.webcitation.org/6076Lm8r4|Giovanni Battista Benedetti]] first proposed it as a measure of inharmonicity. It may be the first number theoretic height function ever defined for any purpose. | <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">The //Benedetti height// of a positive rational number N/D reduced to lowest terms (no common factor between N and D) is equal to N*D, the product of the numerator and denominator. The logarithm base two of the Benedetti height is the [[Tenney height]], or Tenney norm. The name is based on the fact that the scientist, mathematician and music theorist [[http://www.webcitation.org/6076Lm8r4|Giovanni Battista Benedetti]] first proposed it as a measure of inharmonicity. It may be the first number theoretic height function ever defined for any purpose. | ||
See also [[Kees Height|Kees Height.]] | |||
=Examples= | |||
||~ interval ||~ Benedetti height ||~ Tenney height || | |||
|| 3/2 || 6 || 2.585 || | |||
|| 6/5 || 30 || 4.907 || | |||
|| 9/7 || 63 || 5.977 || | |||
|| 13/11 || 143 || 7.160 ||</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>Benedetti height</title></head><body>The <em>Benedetti height</em> of a positive rational number N/D reduced to lowest terms (no common factor between N and D) is equal to N*D, the product of the numerator and denominator. The logarithm base two of the Benedetti height is the <a class="wiki_link" href="/Tenney%20height">Tenney height</a>, or Tenney norm. The name is based on the fact that the scientist, mathematician and music theorist <a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow">Giovanni Battista Benedetti</a> first proposed it as a measure of inharmonicity. It may be the first number theoretic height function ever defined for any purpose.<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>Benedetti height</title></head><body>The <em>Benedetti height</em> of a positive rational number N/D reduced to lowest terms (no common factor between N and D) is equal to N*D, the product of the numerator and denominator. The logarithm base two of the Benedetti height is the <a class="wiki_link" href="/Tenney%20height">Tenney height</a>, or Tenney norm. The name is based on the fact that the scientist, mathematician and music theorist <a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow">Giovanni Battista Benedetti</a> first proposed it as a measure of inharmonicity. It may be the first number theoretic height function ever defined for any purpose.<br /> | ||
<br /> | <br /> | ||
See also <a class="wiki_link" href="/Kees%20Height">Kees Height.</a><br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h1> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>interval<br /> | |||
</th> | |||
<th>Benedetti height<br /> | |||
</th> | |||
<th>Tenney height<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>3/2<br /> | |||
</td> | |||
<td>6<br /> | |||
</td> | |||
<td>2.585<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6/5<br /> | |||
</td> | |||
<td>30<br /> | |||
</td> | |||
<td>4.907<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/7<br /> | |||
</td> | |||
<td>63<br /> | |||
</td> | |||
<td>5.977<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/11<br /> | |||
</td> | |||
<td>143<br /> | |||
</td> | |||
<td>7.160<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
</body></html></pre></div> | |||