Gene Ward Smith: Difference between revisions

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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 17:27:51 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-04 17:31:19 UTC</tt>.<br>

: The original revision id was <tt>~~147073187~~</tt>.<br>

: The original revision id was <tt>147073595</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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==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 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 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 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]].<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.

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<br />

<h2 id="toc0"><a name="x-Music theory"></a>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 />

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%20temperaments">regular temperaments</a>, 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>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><br />

Smith has long been drawing 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>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><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 />

@@ Line 1: / Line 1: @@
 <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 17:27:51 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-04 17:31:19 UTC</tt>.<br>
-: The original revision id was <tt>147073187</tt>.<br>
+: The original revision id was <tt>147073595</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 9: / Line 9: @@
 ==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 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 division of the octave|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'', [[On-Line Encyclopedia of Integer Sequences]].&lt;/ref&gt;
+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'', [[On-Line Encyclopedia of Integer Sequences]].&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.
@@ Line 28: / Line 28: @@
 &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%20temperament"&gt;regular temperament&lt;/a&gt;s, 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;
+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%20division%20of%20the%20octave"&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&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 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&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;
 &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;