Gene Ward Smith: Difference between revisions

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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">Gene Ward Smith (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of [[Galois theory]] and [[Moonshine theory]]. In music theory, he is noted for a number of innovations in the theory of musical tuning, such as the introduction of [[multilinear algebra]] and for being the first to write music in a number of exotic intonation systems. A boyhood friend of [[Steven Spielberg]], a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.&lt;ref&gt;{{citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1}}.&lt;/ref&gt; While a graduate student at [[University of California, Berkeley|Berkeley]], he and fellow mathematician Matthew P. Wiener gained online notoriety for fierce debating and frequent participation in flame wars on [[Usenet]], causing them to be nicknamed the [[Brahms Gang]] (because ''brahms.berkeley.edu'' was the name of the server they posted from).

Smith introduced [[exterior algebra|wedge product]]s as a way of classifying [[regular temperament]]s, and of dealing with the problem of [[Torsion (abstract algebra)|torsion]]. In this system, a temperament is specified by means of a ''wedgie'', which technically may be identified as a point on a [[Grassmannian]].

Smith has long been drawing attention to the relationship between [[equal division of the octave|equal divisions of the octave]] and the [[Riemann zeta function]].&lt;ref&gt;[http://www.math.niu.edu/~rusin/uses-math/music/12 Why 12 tones per octave?], Dave Rusin. Sequence {{OEIS2C|A117536}} ''Increasingly large peaks of the Riemann zeta function on the critical line'' and {{OEIS2C|A117538}} ''Increasingly large integrals of the Z function between zeros'', [[On-Line Encyclopedia of Integer Sequences]].&lt;/ref&gt;

Smith was among the first to consider extending the [[Tonnetz]] of [[Hugo Riemann]] beyond the [[limit (music)|5-limit]] and hence into higher dimensional [[lattice (group)|lattices]]. In three dimensions, the [[hexagonal lattice]] of 5-limit harmony extends to a lattice of type A&lt;sub&gt;3&lt;/sub&gt; ~ D&lt;sub&gt;3&lt;/sub&gt;.

In mathematics, Smith's most notable achievement is the construction of what has been called the Smith [[generic polynomial|generic cyclic polynomial]].&lt;ref&gt;{{citation|first1=Christian U.|last1=Jensen|first2=Arne|last2=Ledet|first3=Noriko|last3=Yui|title=Generic Polynomials: Constructive Aspects of the Inverse Galois Problem|publisher=Cambridge University Press|location=Cambridge|year=2002|isbn=0-521-81998-9|url=http://www.msri.org/communications/books/Book45/files/book45.pdf}}.&lt;/ref&gt; For any integer ''n'' not divisible by eight, this constructs a polynomial which, upon specializing the values, gives all of the cyclic extensions of any given base field with [[characteristic (algebra)|characteristic]] prime to ''n''. This can then be extended to [[metacyclic]] extensions, such as [[dihedral group]]s.

 

 

 

 

 

read on Wikipedia http://en.wikipedia.org/wiki/Gene_Ward_Smith</pre></div>

<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;Gene Ward Smith&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Gene Ward Smith (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of &lt;a class="wiki_link" href="/Galois%20theory"&gt;Galois theory&lt;/a&gt; and &lt;a class="wiki_link" href="/Moonshine%20theory"&gt;Moonshine theory&lt;/a&gt;. In music theory, he is noted for a number of innovations in the theory of musical tuning, such as the introduction of &lt;a class="wiki_link" href="/multilinear%20algebra"&gt;multilinear algebra&lt;/a&gt; and for being the first to write music in a number of exotic intonation systems. A boyhood friend of &lt;a class="wiki_link" href="/Steven%20Spielberg"&gt;Steven Spielberg&lt;/a&gt;, a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.&lt;!-- ws:start:WikiTextRefRule:2:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;tt&amp;gt;citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1&amp;lt;/tt&amp;gt;.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:2 --&gt; While a graduate student at &lt;a class="wiki_link" href="/University%20of%20California%2C%20Berkeley"&gt;Berkeley&lt;/a&gt;, he and fellow mathematician Matthew P. Wiener gained online notoriety for fierce debating and frequent participation in flame wars on &lt;a class="wiki_link" href="/Usenet"&gt;Usenet&lt;/a&gt;, causing them to be nicknamed the &lt;a class="wiki_link" href="/Brahms%20Gang"&gt;Brahms Gang&lt;/a&gt; (because ''brahms.berkeley.edu'' was the name of the server they posted from).&lt;br /&gt;

Smith introduced &lt;a class="wiki_link" href="/exterior%20algebra"&gt;wedge product&lt;/a&gt;s as a way of classifying &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&gt;s, and of dealing with the problem of &lt;a class="wiki_link" href="/Torsion%20%28abstract%20algebra%29"&gt;torsion&lt;/a&gt;. In this system, a temperament is specified by means of a ''wedgie'', which technically may be identified as a point on a &lt;a class="wiki_link" href="/Grassmannian"&gt;Grassmannian&lt;/a&gt;.&lt;br /&gt;

Smith has long been drawing attention to the relationship between &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal divisions of the octave&lt;/a&gt; and the &lt;a class="wiki_link" href="/Riemann%20zeta%20function"&gt;Riemann zeta function&lt;/a&gt;.&lt;!-- ws:start:WikiTextRefRule:9:&amp;amp;lt;ref&amp;amp;gt;[http://www.math.niu.edu/~rusin/uses-math/music/12 Why 12 tones per octave?], Dave Rusin. Sequence &amp;lt;tt&amp;gt;OEIS2C|A117536&amp;lt;/tt&amp;gt; ''Increasingly large peaks of the Riemann zeta function on the critical line'' and &amp;lt;tt&amp;gt;OEIS2C|A117538&amp;lt;/tt&amp;gt; ''Increasingly large integrals of the Z function between zeros'', &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/On-Line%20Encyclopedia%20of%20Integer%20Sequences&amp;quot;&amp;gt;On-Line Encyclopedia of Integer Sequences&amp;lt;/a&amp;gt;.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-2" class="reference"&gt;&lt;a href="#cite_note-2"&gt;[2]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:9 --&gt;&lt;br /&gt;

Smith was among the first to consider extending the &lt;a class="wiki_link" href="/Tonnetz"&gt;Tonnetz&lt;/a&gt; of &lt;a class="wiki_link" href="/Hugo%20Riemann"&gt;Hugo Riemann&lt;/a&gt; beyond the &lt;a class="wiki_link" href="/limit%20%28music%29"&gt;5-limit&lt;/a&gt; and hence into higher dimensional &lt;a class="wiki_link" href="/lattice%20%28group%29"&gt;lattices&lt;/a&gt;. In three dimensions, the &lt;a class="wiki_link" href="/hexagonal%20lattice"&gt;hexagonal lattice&lt;/a&gt; of 5-limit harmony extends to a lattice of type A&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; ~ D&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;.&lt;br /&gt;

In mathematics, Smith's most notable achievement is the construction of what has been called the Smith &lt;a class="wiki_link" href="/generic%20polynomial"&gt;generic cyclic polynomial&lt;/a&gt;.&lt;!-- ws:start:WikiTextRefRule:13:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;tt&amp;gt;citation|first1=Christian U.|last1=Jensen|first2=Arne|last2=Ledet|first3=Noriko|last3=Yui|title=Generic Polynomials: Constructive Aspects of the Inverse Galois Problem|publisher=Cambridge University Press|location=Cambridge|year=2002|isbn=0-521-81998-9|url=http://www.msri.org/communications/books/Book45/files/book45.pdf&amp;lt;/tt&amp;gt;.&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-3" class="reference"&gt;&lt;a href="#cite_note-3"&gt;[3]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:13 --&gt; For any integer ''n'' not divisible by eight, this constructs a polynomial which, upon specializing the values, gives all of the cyclic extensions of any given base field with &lt;a class="wiki_link" href="/characteristic%20%28algebra%29"&gt;characteristic&lt;/a&gt; prime to ''n''. This can then be extended to &lt;a class="wiki_link" href="/metacyclic"&gt;metacyclic&lt;/a&gt; extensions, such as &lt;a class="wiki_link" href="/dihedral%20group"&gt;dihedral group&lt;/a&gt;s.&lt;br /&gt;

Homepage: &lt;!-- ws:start:WikiTextUrlRule:80:http://lumma.org/tuning/gws/home.htm --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/home.htm" rel="nofollow"&gt;http://lumma.org/tuning/gws/home.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:80 --&gt;&lt;br /&gt;

read on Wikipedia &lt;!-- ws:start:WikiTextUrlRule:81:http://en.wikipedia.org/wiki/Gene_Ward_Smith --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gene_Ward_Smith" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Gene_Ward_Smith&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:81 --&gt;&lt;!-- ws:start:WikiTextReferencesRule:86: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;

&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:86 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>