Using Scala to transform just intonation: 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>2011-09-~~05 20~~:08:08 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-06 14:43:52 UTC</tt>.<br>

: The original revision id was <tt>~~250987280~~</tt>.<br>

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3->20/7 5->5 7->20/3

Just as the 5-limit transformations are left invariant by <3 5 7|, these native 7-limit transformations are left invariant by <4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that <4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend "3 16/5 5 24/5" to the 7-limit as "3 16/5 5 24/5 7 28/5", then applying it twice leads to "3 10/3 5 16/3 7 14/3" and three times to "7 7/2"; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, "3 10/3 5 16/3 7 28/3". If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from "3 14/5 5 24/5 7 32/5", for a group of order 24, the [[http://mathworld.wolfram.com/TetrahedralGroup.html|group of the tetrahedron]]. If we add to our transformations the inversion, "2 1/2 3 1/3 5 1/5 7 1/7", we end up with a group of order 48, the [[http://mathworld.wolfram.com/OctahedralGroup.html|group of the octahedron]], the full set of symmetries of a hexany. This is illustrated by the piece [[http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations|hexany phrase]].</pre></div>

Just as the 5-limit transformations are left invariant by <3 5 7|, these native 7-limit transformations are left invariant by <4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that <4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend "3 16/5 5 24/5" to the 7-limit as "3 16/5 5 24/5 7 28/5", then applying it twice leads to "3 10/3 5 16/3 7 14/3" and three times to "7 7/2"; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, "3 10/3 5 16/3 7 28/3". If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from "3 14/5 5 24/5 7 32/5", for a group of order 24, the [[http://mathworld.wolfram.com/TetrahedralGroup.html|group of the tetrahedron]]. If we add to our transformations the inversion, "2 1/2 3 1/3 5 1/5 7 1/7", we end up with a group of order 48, the [[http://mathworld.wolfram.com/OctahedralGroup.html|group of the octahedron]], the full set of symmetries of a hexany. This is illustrated by the piece [[http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations|hexany phrase]].

The transformations listed above do not exhaust the interesting 7-limit transformations. If we put "5 36/7 7 36/5" into the Factor pairs box, we interchange major with supermajor tetrads, and minor with subminor tetrads. Doing it twice restores the orginal scale; this is another involution. Doing major-minor first, followed by major-supermajor, leads to "5 14/3 7 20/3", which sends major tetrads to subminor tetrads and minor tetrads to supermajor tetrads. It isn't an involution, but it is invertible, with inverse transformation "5 21/4 7 15/2" which is what you get by doing major-supermajor first, then major-minor. </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>Using Scala to transform just intonation</title></head><body><a href="#Scala seq files">Scala seq files</a> | <a href="#x5-limit transformations">5-limit transformations</a> | <a href="#x7-limit transformations">7-limit transformations</a>

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3-&gt;20/7 5-&gt;5 7-&gt;20/3<br />

<br />

Just as the 5-limit transformations are left invariant by &lt;3 5 7|, these native 7-limit transformations are left invariant by &lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend &quot;3 16/5 5 24/5&quot; to the 7-limit as &quot;3 16/5 5 24/5 7 28/5&quot;, then applying it twice leads to &quot;3 10/3 5 16/3 7 14/3&quot; and three times to &quot;7 7/2&quot;; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, &quot;3 10/3 5 16/3 7 28/3&quot;. If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from &quot;3 14/5 5 24/5 7 32/5&quot;, for a group of order 24, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/TetrahedralGroup.html" rel="nofollow">group of the tetrahedron</a>. If we add to our transformations the inversion, &quot;2 1/2 3 1/3 5 1/5 7 1/7&quot;, we end up with a group of order 48, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow">group of the octahedron</a>, the full set of symmetries of a hexany. This is illustrated by the piece <a class="wiki_link_ext" href="http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations" rel="nofollow">hexany phrase</a>.</body></html></pre></div>

Just as the 5-limit transformations are left invariant by &lt;3 5 7|, these native 7-limit transformations are left invariant by &lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend &quot;3 16/5 5 24/5&quot; to the 7-limit as &quot;3 16/5 5 24/5 7 28/5&quot;, then applying it twice leads to &quot;3 10/3 5 16/3 7 14/3&quot; and three times to &quot;7 7/2&quot;; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, &quot;3 10/3 5 16/3 7 28/3&quot;. If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from &quot;3 14/5 5 24/5 7 32/5&quot;, for a group of order 24, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/TetrahedralGroup.html" rel="nofollow">group of the tetrahedron</a>. If we add to our transformations the inversion, &quot;2 1/2 3 1/3 5 1/5 7 1/7&quot;, we end up with a group of order 48, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow">group of the octahedron</a>, the full set of symmetries of a hexany. This is illustrated by the piece <a class="wiki_link_ext" href="http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations" rel="nofollow">hexany phrase</a>.<br />

<br />

The transformations listed above do not exhaust the interesting 7-limit transformations. If we put &quot;5 36/7 7 36/5&quot; into the Factor pairs box, we interchange major with supermajor tetrads, and minor with subminor tetrads. Doing it twice restores the orginal scale; this is another involution. Doing major-minor first, followed by major-supermajor, leads to &quot;5 14/3 7 20/3&quot;, which sends major tetrads to subminor tetrads and minor tetrads to supermajor tetrads. It isn't an involution, but it is invertible, with inverse transformation &quot;5 21/4 7 15/2&quot; which is what you get by doing major-supermajor first, then major-minor.</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>2011-09-05 20:08:08 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-06 14:43:52 UTC</tt>.<br>
-: The original revision id was <tt>250987280</tt>.<br>
+: The original revision id was <tt>251270206</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 43: / Line 43: @@
 -&gt;20/7 5-&gt;5 7-&gt;20/3
-Just as the 5-limit transformations are left invariant by &lt;3 5 7|, these native 7-limit transformations are left invariant by &lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend "3 16/5 5 24/5" to the 7-limit as "3 16/5 5 24/5 7 28/5", then applying it twice leads to "3 10/3 5 16/3 7 14/3" and three times to "7 7/2"; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, "3 10/3 5 16/3 7 28/3". If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from "3 14/5 5 24/5 7 32/5", for a group of order 24, the [[http://mathworld.wolfram.com/TetrahedralGroup.html|group of the tetrahedron]]. If we add to our transformations the inversion, "2 1/2 3 1/3 5 1/5 7 1/7", we end up with a group of order 48, the [[http://mathworld.wolfram.com/OctahedralGroup.html|group of the octahedron]], the full set of symmetries of a hexany. This is illustrated by the piece [[http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations|hexany phrase]].</pre></div>
+Just as the 5-limit transformations are left invariant by &lt;3 5 7|, these native 7-limit transformations are left invariant by &lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend "3 16/5 5 24/5" to the 7-limit as "3 16/5 5 24/5 7 28/5", then applying it twice leads to "3 10/3 5 16/3 7 14/3" and three times to "7 7/2"; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, "3 10/3 5 16/3 7 28/3". If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from "3 14/5 5 24/5 7 32/5", for a group of order 24, the [[http://mathworld.wolfram.com/TetrahedralGroup.html|group of the tetrahedron]]. If we add to our transformations the inversion, "2 1/2 3 1/3 5 1/5 7 1/7", we end up with a group of order 48, the [[http://mathworld.wolfram.com/OctahedralGroup.html|group of the octahedron]], the full set of symmetries of a hexany. This is illustrated by the piece [[http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations|hexany phrase]].
+The transformations listed above do not exhaust the interesting 7-limit transformations. If we put "5 36/7 7 36/5" into the Factor pairs box, we interchange major with supermajor tetrads, and minor with subminor tetrads. Doing it twice restores the orginal scale; this is another involution. Doing major-minor first, followed by major-supermajor, leads to "5 14/3 7 20/3", which sends major tetrads to subminor tetrads and minor tetrads to supermajor tetrads. It isn't an involution, but it is invertible, with inverse transformation "5 21/4 7 15/2" which is what you get by doing major-supermajor first, then major-minor. </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;Using Scala to transform just intonation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Scala seq files"&gt;Scala seq files&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#x5-limit transformations"&gt;5-limit transformations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#x7-limit transformations"&gt;7-limit transformations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
@@ Line 82: / Line 84: @@
 -&amp;gt;20/7 5-&amp;gt;5 7-&amp;gt;20/3&lt;br /&gt;
 &lt;br /&gt;
-Just as the 5-limit transformations are left invariant by &amp;lt;3 5 7|, these native 7-limit transformations are left invariant by &amp;lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &amp;lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend &amp;quot;3 16/5 5 24/5&amp;quot; to the 7-limit as &amp;quot;3 16/5 5 24/5 7 28/5&amp;quot;, then applying it twice leads to &amp;quot;3 10/3 5 16/3 7 14/3&amp;quot; and three times to &amp;quot;7 7/2&amp;quot;; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, &amp;quot;3 10/3 5 16/3 7 28/3&amp;quot;. If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from &amp;quot;3 14/5 5 24/5 7 32/5&amp;quot;, for a group of order 24, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/TetrahedralGroup.html" rel="nofollow"&gt;group of the tetrahedron&lt;/a&gt;. If we add to our transformations the inversion, &amp;quot;2 1/2 3 1/3 5 1/5 7 1/7&amp;quot;, we end up with a group of order 48, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow"&gt;group of the octahedron&lt;/a&gt;, the full set of symmetries of a hexany. This is illustrated by the piece &lt;a class="wiki_link_ext" href="http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations" rel="nofollow"&gt;hexany phrase&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Just as the 5-limit transformations are left invariant by &amp;lt;3 5 7|, these native 7-limit transformations are left invariant by &amp;lt;4 6 9 11|. However, we can increase the number of 7-limit transformations by no longer requiring that &amp;lt;4 6 9 11| be left invariant, and extending the 5-limit transformations to the 7-limit. If we extend &amp;quot;3 16/5 5 24/5&amp;quot; to the 7-limit as &amp;quot;3 16/5 5 24/5 7 28/5&amp;quot;, then applying it twice leads to &amp;quot;3 10/3 5 16/3 7 14/3&amp;quot; and three times to &amp;quot;7 7/2&amp;quot;; the transformation is of finite order modulo octave equivalence, but not absolutely. We can cure this by using it together with its inverse transformation, &amp;quot;3 10/3 5 16/3 7 28/3&amp;quot;. If we do this, we get a cycle of order two from major-minor, of order three from transformations lifted from the 5-limit, and of order four from &amp;quot;3 14/5 5 24/5 7 32/5&amp;quot;, for a group of order 24, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/TetrahedralGroup.html" rel="nofollow"&gt;group of the tetrahedron&lt;/a&gt;. If we add to our transformations the inversion, &amp;quot;2 1/2 3 1/3 5 1/5 7 1/7&amp;quot;, we end up with a group of order 48, the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/OctahedralGroup.html" rel="nofollow"&gt;group of the octahedron&lt;/a&gt;, the full set of symmetries of a hexany. This is illustrated by the piece &lt;a class="wiki_link_ext" href="http://robertinventor.tripod.com/tunes/tunes.htm#hexany_phrase_transformations" rel="nofollow"&gt;hexany phrase&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+The transformations listed above do not exhaust the interesting 7-limit transformations. If we put &amp;quot;5 36/7 7 36/5&amp;quot; into the Factor pairs box, we interchange major with supermajor tetrads, and minor with subminor tetrads. Doing it twice restores the orginal scale; this is another involution. Doing major-minor first, followed by major-supermajor, leads to &amp;quot;5 14/3 7 20/3&amp;quot;, which sends major tetrads to subminor tetrads and minor tetrads to supermajor tetrads. It isn't an involution, but it is invertible, with inverse transformation &amp;quot;5 21/4 7 15/2&amp;quot; which is what you get by doing major-supermajor first, then major-minor.&lt;/body&gt;&lt;/html&gt;</pre></div>