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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | 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|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|height]] function ever defined for any purpose. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:08:57 UTC</tt>.<br>
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| : The original revision id was <tt>588205047</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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.
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| See also [[Kees Height|Kees Height.]] | | See also [[Kees_Height|Kees Height.]] |
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| =Examples= | | =Examples= |
| ||= Interval ||= Benedetti height ||= Tenney height ||
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| || 3/2 || 6 || 2.585 ||
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| || 6/5 || 30 || 4.907 ||
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| || 9/7 || 63 || 5.977 ||
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| || 13/11 || 143 || 7.160 ||
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| </pre></div>
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| <h4>Original HTML content:</h4>
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| <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 <a class="wiki_link" href="/height">height</a> function ever defined for any purpose.<br />
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| <br />
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| See also <a class="wiki_link" href="/Kees%20Height">Kees Height.</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h1>
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| <table class="wiki_table">
| | {| class="wikitable" |
| <tr>
| | |- |
| <td style="text-align: center;">Interval<br />
| | | style="text-align:center;" | Interval |
| </td>
| | | style="text-align:center;" | Benedetti height |
| <td style="text-align: center;">Benedetti height<br />
| | | style="text-align:center;" | Tenney height |
| </td>
| | |- |
| <td style="text-align: center;">Tenney height<br />
| | | | 3/2 |
| </td>
| | | | 6 |
| </tr>
| | | | 2.585 |
| <tr>
| | |- |
| <td>3/2<br />
| | | | 6/5 |
| </td>
| | | | 30 |
| <td>6<br />
| | | | 4.907 |
| </td>
| | |- |
| <td>2.585<br />
| | | | 9/7 |
| </td>
| | | | 63 |
| </tr>
| | | | 5.977 |
| <tr>
| | |- |
| <td>6/5<br />
| | | | 13/11 |
| </td>
| | | | 143 |
| <td>30<br />
| | | | 7.160 |
| </td>
| | |} |
| <td>4.907<br />
| | [[Category:benedetti]] |
| </td>
| | [[Category:definition]] |
| </tr>
| | [[Category:height]] |
| <tr>
| | [[Category:measure]] |
| <td>9/7<br />
| | [[Category:psychoacoustics]] |
| </td>
| | [[Category:tenney]] |
| <td>63<br />
| | [[Category:theory]] |
| </td>
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| <td>5.977<br />
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| </td>
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| </tr>
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| <tr>
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| <td>13/11<br />
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| </td>
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| <td>143<br />
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| </td>
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| <td>7.160<br />
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| </td>
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| </tr>
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| </table>
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| </body></html></pre></div>
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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 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.
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
|