Star and Nova: 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>2012-09-04 11:39:19 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-04 13:15:00 UTC</tt>.<br>

: The original revision id was <tt>~~361929750~~</tt>.<br>

: The original revision id was <tt>361963034</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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=Transformations=

Star has many permutations of its notes which send dyadic chords to other dyadic chords. ~~If we~~ use the standard [[http://en.wikipedia.org/wiki/Cycle_notation|cycle notation]] used with permutation groups to mean </pre></div>

Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard [[http://en.wikipedia.org/wiki/Cycle_notation|cycle notation]] used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an [[http://en.wikipedia.org/wiki/Elementary_abelian_group|elementary abelian 2-group]] isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory [[Graph-theoretic properties of scales#Examples-Star|here]].</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"><html><head><title>Star and Nova</title></head><body><a href="#Star transversal">Star transversal</a> | <a href="#Notes of star">Notes of star</a> | <a href="#Chords of star">Chords of star</a> | <a href="#Transformations">Transformations</a>

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<br />

<h1 id="toc3"><a name="Transformations"></a>Transformations</h1>

Star has many permutations of its notes which send dyadic chords to other dyadic chords. ~~If we~~ use the standard <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cycle_notation" rel="nofollow">cycle notation</a> used with permutation groups to mean</body></html></pre></div>

Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cycle_notation" rel="nofollow">cycle notation</a> used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Elementary_abelian_group" rel="nofollow">elementary abelian 2-group</a> isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory <a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales#Examples-Star">here</a>.</body></html></pre></div>

@@ 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>2012-09-04 11:39:19 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-04 13:15:00 UTC</tt>.<br>
-: The original revision id was <tt>361929750</tt>.<br>
+: The original revision id was <tt>361963034</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 20: / Line 20: @@
 =Transformations=
-Star has many permutations of its notes which send dyadic chords to other dyadic chords. If we use the standard [[http://en.wikipedia.org/wiki/Cycle_notation|cycle notation]] used with permutation groups to mean </pre></div>
+Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard [[http://en.wikipedia.org/wiki/Cycle_notation|cycle notation]] used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an [[http://en.wikipedia.org/wiki/Elementary_abelian_group|elementary abelian 2-group]] isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory [[Graph-theoretic properties of scales#Examples-Star|here]].</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;Star and Nova&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Star transversal"&gt;Star transversal&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Notes of star"&gt;Notes of star&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Chords of star"&gt;Chords of star&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Transformations"&gt;Transformations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Transformations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Transformations&lt;/h1&gt;
-Star has many permutations of its notes which send dyadic chords to other dyadic chords. If we use the standard &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cycle_notation" rel="nofollow"&gt;cycle notation&lt;/a&gt; used with permutation groups to mean&lt;/body&gt;&lt;/html&gt;</pre></div>
+Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cycle_notation" rel="nofollow"&gt;cycle notation&lt;/a&gt; used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Elementary_abelian_group" rel="nofollow"&gt;elementary abelian 2-group&lt;/a&gt; isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory &lt;a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales#Examples-Star"&gt;here&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>