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Benedetti height: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.
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>2011-07-25 17:07:51 UTC</tt>.<br>
The [[logarithm base two]] of the Benedetti height is the Tenney height, or [[Tenney norm]].
: The original revision id was <tt>242791315</tt>.<br>
 
: The revision comment was: <tt></tt><br>
== Computation ==
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
=== Ratio form ===
<h4>Original Wikitext content:</h4>
The Benedetti height of a ratio ''n''/''d'' is given by
<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.</pre></div>
 
<h4>Original HTML content:</h4>
$$ nd $$
<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;Benedetti height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;Benedetti height&lt;/em&gt; 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 &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;, or Tenney norm. The name is based on the fact that the scientist, mathematician and music theorist &lt;a class="wiki_link_ext" href="http://www.webcitation.org/6076Lm8r4" rel="nofollow"&gt;Giovanni Battista Benedetti&lt;/a&gt; first proposed it as a measure of inharmonicity. It may be the first number theoretic height function ever defined for any purpose.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Vector form ===
The Benedetti height of a [[Harmonic limit|''p''-limit]] [[monzo]] '''m''' = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by
 
$$ 2^{\lVert H \vec m \rVert_1} = 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|} $$
 
where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},
 
$$ H = \operatorname {diag} (\log_2 (Q)) $$
 
== Examples ==
{| class="wikitable center-1 center-3"
! Ratio
! Monzo
! Benedetti height
|-
| [[1/1]]
| {{Monzo| 0 }}
| 1
|-
| [[2/1]]
| {{Monzo| 1 }}
| 2
|-
| [[3/2]]
| {{Monzo| -1 1 }}
| 6
|-
| [[6/5]]
| {{Monzo| 1 1 -1 }}
| 30
|-
| [[9/7]]
| {{Monzo| 0 2 0 -1 }}
| 63
|-
| [[13/11]]
| {{Monzo| 0 0 0 0 -1 1 }}
| 143
|}
 
== History and terminology ==
Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. 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.
 
Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956 Yahoo! Tuning Group | ''Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?'']</ref>
 
== See also ==
* [[Kees semi-height]]
* [[Wikipedia: Giambattista Benedetti]]
 
== References ==
<references/>
 
[[Category:Interval complexity measures]]
[[Category:Tenney-weighted measures]]
Retrieved from "https://en.xen.wiki/w/Benedetti_height"