Gene Ward Smith: Difference between revisions

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**Gene Ward Smith** (b. 1947) is a mathematician, music theorist, and composer.

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 introduced [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge products]] as a way of classifying [[regular temperaments]].  In this system, a temperament is specified by means of a [[Wedgies and Multivals|wedgie]], which may technically be identified as a point on a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]].  He has long drawn attention to the relationship between [[Equal Temperaments|equal divisions of the octave]] and the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]].<ref>Rusin, Dave "Why 12 tones per octave?" http://www.math.niu.edu/~rusin/uses-math/music/12</ref><ref>Increasingly large peaks of the Riemann zeta function on the critical line http://oeis.org/A117536</ref><ref>Increasingly large integrals of the Z function between zeros http://oeis.org/A117538</ref>

Gene was among the first to consider extending the [[http://en.wikipedia.org/wiki/Tonnetz|Tonnetz]] of 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.  He is also the first to write music in a number of exotic intonation systems.  See [[Microtonal Music by Gene Ward Smith]].

Original HTML content:

<html><head><title>Gene Ward Smith</title></head><body><strong>Gene Ward Smith</strong> (b. 1947) is a mathematician, music theorist, and composer.<br />
<br />
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>.<br />
<br />
In music theory, he introduced <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge products</a> as a way of classifying <a class="wiki_link" href="/regular%20temperaments">regular temperaments</a>.  In this system, a temperament is specified by means of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, which may technically be identified as a point on a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>.  He has long drawn attention to the relationship between <a class="wiki_link" href="/Equal%20Temperaments">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:1:&amp;lt;ref&amp;gt;Rusin, Dave &amp;quot;Why 12 tones per octave?&amp;quot; http://www.math.niu.edu/~rusin/uses-math/music/12&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:1 --><!-- ws:start:WikiTextRefRule:3:&amp;lt;ref&amp;gt;Increasingly large peaks of the Riemann zeta function on the critical line http://oeis.org/A117536&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:3 --><!-- ws:start:WikiTextRefRule:5:&amp;lt;ref&amp;gt;Increasingly large integrals of the Z function between zeros http://oeis.org/A117538&amp;lt;/ref&amp;gt; --><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup><!-- ws:end:WikiTextRefRule:5 --><br />
<br />
Gene 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 Hugo Riemann 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.  He is also the first to write music in a number of exotic intonation systems.  See <a class="wiki_link" href="/Microtonal%20Music%20by%20Gene%20Ward%20Smith">Microtonal Music by Gene Ward Smith</a>.<!-- ws:start:WikiTextReferencesRule:27: --><hr class="references" /><ol class="references">
<li id="cite_note-1"><a href="#cite_ref-1">^</a> Rusin, Dave &quot;Why 12 tones per octave?&quot; <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></li>
<li id="cite_note-2"><a href="#cite_ref-2">^</a> Increasingly large peaks of the Riemann zeta function on the critical line <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">http://oeis.org/A117536</a></li>
<li id="cite_note-3"><a href="#cite_ref-3">^</a> Increasingly large integrals of the Z function between zeros <a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow">http://oeis.org/A117538</a></li>
</ol><!-- ws:end:WikiTextReferencesRule:27 --></body></html>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-06-04 17:53:55 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-08-13 15:12:43 UTC</tt>.<br>
-: The original revision id was <tt>147075935</tt>.<br>
+: The original revision id was <tt>245783633</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Rewriting wikipedia content as appropriate for xenwiki</tt><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>
-<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 [[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.&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 [[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).
+<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** (b. 1947) is a mathematician, music theorist, and composer.
-==Music theory==
+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]].
-Smith introduced [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]]s as a way of classifying [[regular temperaments]], 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 Temperaments|equal divisions of the octave]] and the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|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'', [[http://en.wikipedia.org/wiki/Online_Encyclopedia_of_Integer_Sequences|On-Line Encyclopedia of Integer Sequences]].&lt;/ref&gt;
+In music theory, he introduced [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge products]] as a way of classifying [[regular temperaments]].  In this system, a temperament is specified by means of a [[Wedgies and Multivals|wedgie]], which may technically be identified as a point on a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]].  He has long drawn attention to the relationship between [[Equal Temperaments|equal divisions of the octave]] and the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]].&lt;ref&gt;Rusin, Dave "Why 12 tones per octave?" http://www.math.niu.edu/~rusin/uses-math/music/12&lt;/ref&gt;&lt;ref&gt;Increasingly large peaks of the Riemann zeta function on the critical line http://oeis.org/A117536&lt;/ref&gt;&lt;ref&gt;Increasingly large integrals of the Z function between zeros http://oeis.org/A117538&lt;/ref&gt;
-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.
+Gene was among the first to consider extending the [[http://en.wikipedia.org/wiki/Tonnetz|Tonnetz]] of 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.  He is also the first to write music in a number of exotic intonation systems.  See [[Microtonal Music by Gene Ward Smith]].</pre></div>
-==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]].&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 [[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.&lt;ref&gt;[http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html The Amdahl Six].&lt;/ref&gt;
-==Music by Gene Ward Smith==
-[[Microtonal Music by Gene Ward Smith]]
-==External links==
-*[http://lumma.org/tuning/gws/home.htm Smith's old website]
-</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Gene Ward Smith&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Gene Ward Smith&lt;/strong&gt; (born 1947) is an American mathematician and music theorist. In mathematics he has worked in the areas of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Galois_theory" rel="nofollow"&gt;Galois theory&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monstrous_moonshine" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/Multilinear_algebra" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/Steven_Spielberg" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/University_of_California,_Berkeley" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/Usenet" rel="nofollow"&gt;Usenet&lt;/a&gt;, causing them to be nicknamed the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Brahms_Gang" rel="nofollow"&gt;Brahms Gang&lt;/a&gt; (because ''brahms.berkeley.edu'' was the name of the server they posted from).&lt;br /&gt;
+<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;&lt;strong&gt;Gene Ward Smith&lt;/strong&gt; (b. 1947) is a mathematician, music theorist, and composer.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Music theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Music theory&lt;/h2&gt;
-Smith introduced &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow"&gt;wedge product&lt;/a&gt;s as a way of classifying &lt;a class="wiki_link" href="/regular%20temperaments"&gt;regular temperaments&lt;/a&gt;, and of dealing with the problem of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Torsion_%28abstract_algebra%29" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow"&gt;Grassmannian&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-Smith has long been drawing attention to the relationship between &lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;equal divisions of the octave&lt;/a&gt; and the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow"&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_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Online_Encyclopedia_of_Integer_Sequences&amp;quot; rel=&amp;quot;nofollow&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;
-&lt;br /&gt;
-Smith was among the first to consider extending the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonnetz" rel="nofollow"&gt;Tonnetz&lt;/a&gt; of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hugo_Riemann" rel="nofollow"&gt;Hugo Riemann&lt;/a&gt; beyond the 5-limit and hence into higher dimensional &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattices&lt;/a&gt;. In three dimensions, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; of &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit harmony&lt;/a&gt;  extends to a lattice of type A3 ~ D3.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Mathematics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Mathematics&lt;/h2&gt;
-In mathematics, Smith's most notable achievement is the construction of what has been called the Smith &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Generic_polynomial" rel="nofollow"&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_ext" href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29" rel="nofollow"&gt;characteristic (algebra)|characteristic&lt;/a&gt; prime to ''n''. This can then be extended to &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metacyclic_group" rel="nofollow"&gt;metacyclic&lt;/a&gt; extensions, such as &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dihedral_group" rel="nofollow"&gt;dihedral group&lt;/a&gt;s.&lt;br /&gt;
-&lt;br /&gt;
-Smith was also a member of the ''Amdahl Six'' group which held the largest prime record from 1989-1992.&lt;!-- ws:start:WikiTextRefRule:15:&amp;amp;lt;ref&amp;amp;gt;[http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html The Amdahl Six].&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-4" class="reference"&gt;&lt;a href="#cite_note-4"&gt;[4]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:15 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Music by Gene Ward Smith"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Music by Gene Ward Smith&lt;/h2&gt;
+In mathematics, he has worked in the areas of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Galois_theory" rel="nofollow"&gt;Galois theory&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monstrous_moonshine" rel="nofollow"&gt;Moonshine theory&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;a class="wiki_link" href="/Microtonal%20Music%20by%20Gene%20Ward%20Smith"&gt;Microtonal Music by Gene Ward Smith&lt;/a&gt;&lt;br /&gt;
+In music theory, he introduced &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow"&gt;wedge products&lt;/a&gt; as a way of classifying &lt;a class="wiki_link" href="/regular%20temperaments"&gt;regular temperaments&lt;/a&gt;.  In this system, a temperament is specified by means of a &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, which may technically be identified as a point on a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow"&gt;Grassmannian&lt;/a&gt;.  He has long drawn attention to the relationship between &lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;equal divisions of the octave&lt;/a&gt; and the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow"&gt;Riemann zeta function&lt;/a&gt;.&lt;!-- ws:start:WikiTextRefRule:1:&amp;amp;lt;ref&amp;amp;gt;Rusin, Dave &amp;amp;quot;Why 12 tones per octave?&amp;amp;quot; http://www.math.niu.edu/~rusin/uses-math/music/12&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:1 --&gt;&lt;!-- ws:start:WikiTextRefRule:3:&amp;amp;lt;ref&amp;amp;gt;Increasingly large peaks of the Riemann zeta function on the critical line http://oeis.org/A117536&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:3 --&gt;&lt;!-- ws:start:WikiTextRefRule:5:&amp;amp;lt;ref&amp;amp;gt;Increasingly large integrals of the Z function between zeros http://oeis.org/A117538&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:5 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;External links&lt;/h2&gt;
+Gene was among the first to consider extending the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonnetz" rel="nofollow"&gt;Tonnetz&lt;/a&gt; of Hugo Riemann beyond the 5-limit and hence into higher dimensional &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattices&lt;/a&gt;.  In three dimensions, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow"&gt;hexagonal lattice&lt;/a&gt; of &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit harmony&lt;/a&gt; extends to a lattice of type A3 ~ D3.  He is also the first to write music in a number of exotic intonation systems.  See &lt;a class="wiki_link" href="/Microtonal%20Music%20by%20Gene%20Ward%20Smith"&gt;Microtonal Music by Gene Ward Smith&lt;/a&gt;.&lt;!-- ws:start:WikiTextReferencesRule:27: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;
-*[&lt;!-- ws:start:WikiTextUrlRule:62: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:62 --&gt; Smith's old website]&lt;!-- ws:start:WikiTextReferencesRule:65: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;
+&lt;li id="cite_note-1"&gt;&lt;a href="#cite_ref-1"&gt;^&lt;/a&gt; Rusin, Dave &amp;quot;Why 12 tones per octave?&amp;quot; &lt;a class="wiki_link_ext" href="http://www.math.niu.edu/~rusin/uses-math/music/12" rel="nofollow"&gt;http://www.math.niu.edu/~rusin/uses-math/music/12&lt;/a&gt;&lt;/li&gt;
-&lt;li id="cite_note-1"&gt;&lt;a href="#cite_ref-1"&gt;^&lt;/a&gt; &lt;tt&gt;citation|first=Joseph|last=McBride|title=Steven Spielberg: A Biography|publisher=Da Capo Press|year=1999|isbn=0-306-80900-1&lt;/tt&gt;.&lt;/li&gt;
+&lt;li id="cite_note-2"&gt;&lt;a href="#cite_ref-2"&gt;^&lt;/a&gt; Increasingly large peaks of the Riemann zeta function on the critical line &lt;a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow"&gt;http://oeis.org/A117536&lt;/a&gt;&lt;/li&gt;
-&lt;li id="cite_note-2"&gt;&lt;a href="#cite_ref-2"&gt;^&lt;/a&gt; [&lt;a class="wiki_link_ext" href="http://www.math.niu.edu/~rusin/uses-math/music/12" rel="nofollow"&gt;http://www.math.niu.edu/~rusin/uses-math/music/12&lt;/a&gt; Why 12 tones per octave?], Dave Rusin. Sequence &lt;tt&gt;OEIS2C|A117536&lt;/tt&gt; ''Increasingly large peaks of the Riemann zeta function on the critical line'' and &lt;tt&gt;OEIS2C|A117538&lt;/tt&gt; ''Increasingly large integrals of the Z function between zeros'', &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Online_Encyclopedia_of_Integer_Sequences" rel="nofollow"&gt;On-Line Encyclopedia of Integer Sequences&lt;/a&gt;.&lt;/li&gt;
+&lt;li id="cite_note-3"&gt;&lt;a href="#cite_ref-3"&gt;^&lt;/a&gt; Increasingly large integrals of the Z function between zeros &lt;a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow"&gt;http://oeis.org/A117538&lt;/a&gt;&lt;/li&gt;
-&lt;li id="cite_note-3"&gt;&lt;a href="#cite_ref-3"&gt;^&lt;/a&gt; &lt;tt&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=&lt;a class="wiki_link_ext" href="http://www.msri.org/communications/books/Book45/files/book45.pdf" rel="nofollow"&gt;http://www.msri.org/communications/books/Book45/files/book45.pdf&lt;/a&gt;&lt;/tt&gt;.&lt;/li&gt;
+&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:27 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
-&lt;li id="cite_note-4"&gt;&lt;a href="#cite_ref-4"&gt;^&lt;/a&gt; [&lt;a class="wiki_link_ext" href="http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html" rel="nofollow"&gt;http://www.isthe.com:/chongo/tech/math/prime/amdahl6.html&lt;/a&gt; The Amdahl Six].&lt;/li&gt;
-&lt;/ol&gt;&lt;!-- ws:end:WikiTextReferencesRule:65 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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