Gene Ward Smith
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**Gene Ward Smith** (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of [[http://en.wikipedia.org/wiki/Galois_theory|Galois theory]] and [[http://en.wikipedia.org/wiki/Monstrous_moonshine|Moonshine theory]]. In music theory, he is noted for a number of innovations in the theory of musical tuning, such as the introduction of [[http://en.wikipedia.org/wiki/Multilinear_algebra|multilinear algebra]] and for being the first to write music in a number of exotic intonation systems. A boyhood friend of [[http://en.wikipedia.org/wiki/Steven_Spielberg|Steven Spielberg]], a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.<ref>{{citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1}}.</ref> While a graduate student at [[http://en.wikipedia.org/wiki/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 [[http://en.wikipedia.org/wiki/Usenet|Usenet]], causing them to be nicknamed the [[http://en.wikipedia.org/wiki/Brahms_Gang|Brahms Gang]] (because ''brahms.berkeley.edu'' was the name of the server they posted from).
==Music theory==
Smith introduced [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]]s as a way of classifying [[regular temperament]]s, and of dealing with the problem of [[http://en.wikipedia.org/wiki/Torsion_%28abstract_algebra%29|torsion]]. In this system, a temperament is specified by means of a ''wedgie'', which technically may be identified as a point on a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]].
Smith has long been drawing attention to the relationship between [[equal division of the octave|equal divisions of the octave]] and the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]].<ref>[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]].</ref>
Smith was among the first to consider extending the [[http://en.wikipedia.org/wiki/Tonnetz|Tonnetz]] of [[http://en.wikipedia.org/wiki/Hugo_Riemann|Hugo Riemann]] beyond the 5-limit and hence into higher dimensional [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattices]]. In three dimensions, the [[http://en.wikipedia.org/wiki/Hexagonal_lattice|hexagonal lattice]] of [[Harmonic Limit|5-limit harmony]] extends to a lattice of type A3 ~ D3.
==Mathematics==
In mathematics, Smith's most notable achievement is the construction of what has been called the Smith [[http://en.wikipedia.org/wiki/Generic_polynomial|generic cyclic polynomial]].<ref>{{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}}.</ref> 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 [[http://en.wikipedia.org/wiki/Characteristic_%28algebra%29|characteristic (algebra)|characteristic]] prime to ''n''. This can then be extended to [[http://en.wikipedia.org/wiki/Metacyclic_group|metacyclic]] extensions, such as [[http://en.wikipedia.org/wiki/Dihedral_group|dihedral group]]s.
Smith was also a member of the ''Amdahl Six'' group which held the largest prime record from 1989-1992.<ref>[http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html The Amdahl Six].</ref>
==External links==
*[http://lumma.org/tuning/gws/home.htm Smith's old website]
Original HTML content:
<html><head><title>Gene Ward Smith</title></head><body><strong>Gene Ward Smith</strong> (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Galois_theory" rel="nofollow">Galois theory</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monstrous_moonshine" rel="nofollow">Moonshine theory</a>. In music theory, he is noted for a number of innovations in the theory of musical tuning, such as the introduction of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multilinear_algebra" rel="nofollow">multilinear algebra</a> and for being the first to write music in a number of exotic intonation systems. A boyhood friend of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Steven_Spielberg" rel="nofollow">Steven Spielberg</a>, a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.<!-- ws:start:WikiTextRefRule:2:&lt;ref&gt;<tt>citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1</tt>.&lt;/ref&gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:2 --> While a graduate student at <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/University_of_California,_Berkeley" rel="nofollow">Berkeley</a>, he and fellow mathematician Matthew P. Wiener gained online notoriety for fierce debating and frequent participation in flame wars on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Usenet" rel="nofollow">Usenet</a>, causing them to be nicknamed the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Brahms_Gang" rel="nofollow">Brahms Gang</a> (because ''brahms.berkeley.edu'' was the name of the server they posted from).<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc0"><a name="x-Music theory"></a><!-- ws:end:WikiTextHeadingRule:16 -->Music theory</h2> Smith introduced <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a>s as a way of classifying <a class="wiki_link" href="/regular%20temperament">regular temperament</a>s, and of dealing with the problem of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Torsion_%28abstract_algebra%29" rel="nofollow">torsion</a>. In this system, a temperament is specified by means of a ''wedgie'', which technically may be identified as a point on a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>.<br /> <br /> Smith has long been drawing attention to the relationship between <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal divisions of the octave</a> and the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>.<!-- ws:start:WikiTextRefRule:9:&lt;ref&gt;[http://www.math.niu.edu/~rusin/uses-math/music/12 Why 12 tones per octave?], Dave Rusin. Sequence <tt>OEIS2C|A117536</tt> ''Increasingly large peaks of the Riemann zeta function on the critical line'' and <tt>OEIS2C|A117538</tt> ''Increasingly large integrals of the Z function between zeros'', <a class="wiki_link" href="/On-Line%20Encyclopedia%20of%20Integer%20Sequences">On-Line Encyclopedia of Integer Sequences</a>.&lt;/ref&gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:9 --><br /> <br /> Smith was among the first to consider extending the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonnetz" rel="nofollow">Tonnetz</a> of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hugo_Riemann" rel="nofollow">Hugo Riemann</a> beyond the 5-limit and hence into higher dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattices</a>. In three dimensions, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow">hexagonal lattice</a> of <a class="wiki_link" href="/Harmonic%20Limit">5-limit harmony</a> extends to a lattice of type A3 ~ D3.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc1"><a name="x-Mathematics"></a><!-- ws:end:WikiTextHeadingRule:18 -->Mathematics</h2> In mathematics, Smith's most notable achievement is the construction of what has been called the Smith <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Generic_polynomial" rel="nofollow">generic cyclic polynomial</a>.<!-- ws:start:WikiTextRefRule:13:&lt;ref&gt;<tt>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</tt>.&lt;/ref&gt; --><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup><!-- ws:end:WikiTextRefRule:13 --> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29" rel="nofollow">characteristic (algebra)|characteristic</a> prime to ''n''. This can then be extended to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metacyclic_group" rel="nofollow">metacyclic</a> extensions, such as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dihedral_group" rel="nofollow">dihedral group</a>s.<br /> <br /> Smith was also a member of the ''Amdahl Six'' group which held the largest prime record from 1989-1992.<!-- ws:start:WikiTextRefRule:15:&lt;ref&gt;[http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html The Amdahl Six].&lt;/ref&gt; --><sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup><!-- ws:end:WikiTextRefRule:15 --><br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc2"><a name="x-External links"></a><!-- ws:end:WikiTextHeadingRule:20 -->External links</h2> *[<!-- ws:start:WikiTextUrlRule:57:http://lumma.org/tuning/gws/home.htm --><a class="wiki_link_ext" href="http://lumma.org/tuning/gws/home.htm" rel="nofollow">http://lumma.org/tuning/gws/home.htm</a><!-- ws:end:WikiTextUrlRule:57 --> Smith's old website]<!-- ws:start:WikiTextReferencesRule:60: --><hr class="references" /><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <tt>citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1</tt>.</li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> [<a class="wiki_link_ext" href="http://www.math.niu.edu/~rusin/uses-math/music/12" rel="nofollow">http://www.math.niu.edu/~rusin/uses-math/music/12</a> Why 12 tones per octave?], Dave Rusin. Sequence <tt>OEIS2C|A117536</tt> ''Increasingly large peaks of the Riemann zeta function on the critical line'' and <tt>OEIS2C|A117538</tt> ''Increasingly large integrals of the Z function between zeros'', <a class="wiki_link" href="/On-Line%20Encyclopedia%20of%20Integer%20Sequences">On-Line Encyclopedia of Integer Sequences</a>.</li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <tt>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=<a class="wiki_link_ext" href="http://www.msri.org/communications/books/Book45/files/book45.pdf" rel="nofollow">http://www.msri.org/communications/books/Book45/files/book45.pdf</a></tt>.</li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> [<a class="wiki_link_ext" href="http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html" rel="nofollow">http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html</a> The Amdahl Six].</li> </ol><!-- ws:end:WikiTextReferencesRule:60 --></body></html>