Gene Ward Smith: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-10 23:53:58 UTC</tt>.<br>

: The original revision id was <tt>333349522</tt>.<br>

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;OEIS. Increasingly large peaks of the Riemann zeta function on the critical line http://oeis.org/A117536&lt;/ref&gt;&lt;ref&gt;OEIS. Increasingly large integrals of the Z function between zeros http://oeis.org/A117538&lt;/ref&gt; He [[http://www.webcitation.org/67ZUSajSK|early on]] identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the [[http://x31eq.com/paradigm.html|regular mapping paradigm]].

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;OEIS. 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;OEIS. 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; He &lt;a class="wiki_link_ext" href="http://www.webcitation.org/67ZUSajSK" rel="nofollow"&gt;early on&lt;/a&gt; identified and emphasized free abelian groups of finite rank and their homomorphisms, and it was from that perspective that he contributed to the creation of the &lt;a class="wiki_link_ext" href="http://x31eq.com/paradigm.html" rel="nofollow"&gt;regular mapping paradigm&lt;/a&gt;.&lt;br /&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 hexagonal lattice 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:28: --&gt;&lt;hr class="references" /&gt;&lt;ol class="references"&gt;

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