Using Scala to transform just intonation - Revision history

Xenwolf: recat

2020-12-05T14:48:24Z

recat

← Older revision		Revision as of 14:48, 5 December 2020
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	=Other transformations=		=Other transformations=
	Many other interesting transformations can be performed using Scala. For one example, "5 56/11 7 80/11 11 128/11" is another involution, this time of the 11-limit. "5 44/9 7 22/3" on the other hand projects the 11-limit down to its 2.3.11 subgroup, and "5 24/5 7 36/5 11 52/5 13 64/5" projects the 13-limit down to the 2.3.5.13 subgroup.		Many other interesting transformations can be performed using Scala. For one example, "5 56/11 7 80/11 11 128/11" is another involution, this time of the 11-limit. "5 44/9 7 22/3" on the other hand projects the 11-limit down to its 2.3.11 subgroup, and "5 24/5 7 36/5 11 52/5 13 64/5" projects the 13-limit down to the 2.3.5.13 subgroup.
	~~[[Category:ji]]~~
	[[Category:~~just~~]]		[[Category:Just intonation]]
	[[Category:Scala]]		[[Category:Scala]]
	[[Category:~~transformation~~]]		[[Category:Transformation]]

Keenan Pepper: Category:Scala instead of Category:Pages with Scala files

2020-12-04T22:19:08Z

Category:Scala instead of Category:Pages with Scala files

← Older revision		Revision as of 22:19, 4 December 2020
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	[[Category:ji]]		[[Category:ji]]
	[[Category:just]]		[[Category:just]]
	[[Category:~~Pages with~~ Scala ~~files~~]]		[[Category:Scala]]
	[[Category:transformation]]		[[Category:transformation]]

Keenan Pepper: Moving from Category:Scala to Category:Pages with Scala files using Cat-a-lot

2020-12-04T22:16:55Z

Moving from Category:Scala to Category:Pages with Scala files using Cat-a-lot

← Older revision		Revision as of 22:16, 4 December 2020
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	[[Category:ji]]		[[Category:ji]]
	[[Category:just]]		[[Category:just]]
	[[Category:~~scala~~]]		[[Category:Pages with Scala files]]
	[[Category:transformation]]		[[Category:transformation]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>genewardsmith: Imported revision 251327470 - Original comment:

2011-09-06T17:28:37Z

**Imported revision 251327470 - Original comment: **

← Older revision		Revision as of 17:28, 6 September 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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>2011-09-06 14:43:52 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-09-06 17:28:37 UTC</tt>.<br>
	: The original revision id was <tt>~~251270206~~</tt>.<br>		: The original revision id was <tt>251327470</tt>.<br>
	: The revision comment was: <tt></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>		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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	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]].		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>		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.

			=Other transformations=
			Many other interesting transformations can be performed using Scala. For one example, "5 56/11 7 80/11 11 128/11" is another involution, this time of the 11-limit. "5 44/9 7 22/3" on the other hand projects the 11-limit down to its 2.3.11 subgroup, and "5 24/5 7 36/5 11 52/5 13 64/5" projects the 13-limit down to the 2.3.5.13 subgroup.</pre></div>
	<h4>Original HTML content:</h4>		<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><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Scala seq files">Scala seq files</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> \| <a href="#x5-limit transformations">5-limit transformations</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> \| <a href="#x7-limit transformations">7-limit transformations</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->		<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><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Scala seq files">Scala seq files</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> \| <a href="#x5-limit transformations">5-limit transformations</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> \| <a href="#x7-limit transformations">7-limit transformations</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> \| <a href="#Other transformations">Other transformations</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: -->
	<!-- ws:end:WikiTextTocRule:10 --><br />		<!-- ws:end:WikiTextTocRule:13 --><br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scala seq files"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scala seq files</h1>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scala seq files"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scala seq files</h1>
	Suppose we have a piece in <a class="wiki_link" href="/Just%20intonation">Just intonation</a> which we want to put into the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/seq_format.html" rel="nofollow">Scala seq file</a> format, which is Scala's musical score format. We can enclose the pitch values in parenthesis, so that they have the form (5/4) for a major third or (7/6) for a subminor third. Another form is monzo format, where (\|-2 0 1&gt;) can be used in place of (5/4), and (\|-1 -1 0 1&gt;) in place of (7/6). An alternative is to use Scala degree numbers. This involves using lines like &quot;4564 note 61 47&quot; in the seq file, where the number right after &quot;note&quot; is the degree or note number. Scala needs additional information in the form of a scale or a declaration of what equal division is being used to interpret the note number, but one of the advantages is that by changing the scale, you can change the music Scala outputs in the form of a midi file.<br />		Suppose we have a piece in <a class="wiki_link" href="/Just%20intonation">Just intonation</a> which we want to put into the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/seq_format.html" rel="nofollow">Scala seq file</a> format, which is Scala's musical score format. We can enclose the pitch values in parenthesis, so that they have the form (5/4) for a major third or (7/6) for a subminor third. Another form is monzo format, where (\|-2 0 1&gt;) can be used in place of (5/4), and (\|-1 -1 0 1&gt;) in place of (7/6). An alternative is to use Scala degree numbers. This involves using lines like &quot;4564 note 61 47&quot; in the seq file, where the number right after &quot;note&quot; is the degree or note number. Scala needs additional information in the form of a scale or a declaration of what equal division is being used to interpret the note number, but one of the advantages is that by changing the scale, you can change the music Scala outputs in the form of a midi file.<br />
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	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 />		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 />		<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>		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. <br />
			<br />
			<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Other transformations"></a><!-- ws:end:WikiTextHeadingRule:6 -->Other transformations</h1>
			Many other interesting transformations can be performed using Scala. For one example, &quot;5 56/11 7 80/11 11 128/11&quot; is another involution, this time of the 11-limit. &quot;5 44/9 7 22/3&quot; on the other hand projects the 11-limit down to its 2.3.11 subgroup, and &quot;5 24/5 7 36/5 11 52/5 13 64/5&quot; projects the 13-limit down to the 2.3.5.13 subgroup.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 251270206 - Original comment:

2011-09-06T14:43:52Z

**Imported revision 251270206 - Original comment: **

← Older revision		Revision as of 14:43, 6 September 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 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>		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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	3->20/7 5->5 7->20/3		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>		<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><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Scala seq files">Scala seq files</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> \| <a href="#x5-limit transformations">5-limit transformations</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> \| <a href="#x7-limit transformations">7-limit transformations</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->		<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><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Scala seq files">Scala seq files</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> \| <a href="#x5-limit transformations">5-limit transformations</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> \| <a href="#x7-limit transformations">7-limit transformations</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
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	3-&gt;20/7 5-&gt;5 7-&gt;20/3<br />		3-&gt;20/7 5-&gt;5 7-&gt;20/3<br />
	<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>

Wikispaces>genewardsmith: Imported revision 250987280 - Original comment:

2011-09-05T20:08:08Z

**Imported revision 250987280 - Original comment: **

← Older revision		Revision as of 20:08, 5 September 2011
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 15:37:00 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-09-05 20:08:08 UTC</tt>.<br>
	: The original revision id was <tt>~~250916654~~</tt>.<br>		: The original revision id was <tt>250987280</tt>.<br>
	: The revision comment was: <tt></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>		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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	If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the [[3edo]] versions of all of these, which turn out to be the same: the 3edo "skeleton", as we might call it, of 5-limit just intonation is left invariant by these transformations.		If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the [[3edo]] versions of all of these, which turn out to be the same: the 3edo "skeleton", as we might call it, of 5-limit just intonation is left invariant by these transformations.

			[[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah0.mp3\|Do Wah Diddy Diddy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah1.mp3\|3 16/5 5 24/5]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah2.mp3\|3 10/3 5 16/3]]

	=7-limit transformations=		=7-limit transformations=
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	<br />		<br />
	If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the <a class="wiki_link" href="/3edo">3edo</a> versions of all of these, which turn out to be the same: the 3edo &quot;skeleton&quot;, as we might call it, of 5-limit just intonation is left invariant by these transformations. <br />		If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the <a class="wiki_link" href="/3edo">3edo</a> versions of all of these, which turn out to be the same: the 3edo &quot;skeleton&quot;, as we might call it, of 5-limit just intonation is left invariant by these transformations. <br />
			<br />
			<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah0.mp3" rel="nofollow">Do Wah Diddy Diddy</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah1.mp3" rel="nofollow">3 16/5 5 24/5</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/dowah2.mp3" rel="nofollow">3 10/3 5 16/3</a><br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit transformations"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit transformations</h1>		<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit transformations"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit transformations</h1>

Wikispaces>genewardsmith: Imported revision 250916654 - Original comment:

2011-09-05T15:37:00Z

**Imported revision 250916654 - Original comment: **

@@ Line 1: / Line 1: @@
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 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 14:17:30 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-05 15:37:00 UTC</tt>.<br>
-: The original revision id was <tt>250896296</tt>.<br>
+: The original revision id was <tt>250916654</tt>.<br>
 : The revision comment was: <tt></tt><br>
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 If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the [[3edo]] versions of all of these, which turn out to be the same: the 3edo "skeleton", as we might call it, of 5-limit just intonation is left invariant by these transformations.
-</pre></div>
+=7-limit transformations=
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-<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:4:&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:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#Scala seq files"&gt;Scala seq files&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt; | &lt;a href="#x5-limit transformations"&gt;5-limit transformations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&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;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;
-&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scala seq files"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scala seq files&lt;/h1&gt;
 Suppose we have a piece in &lt;a class="wiki_link" href="/Just%20intonation"&gt;Just intonation&lt;/a&gt; which we want to put into the &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/seq_format.html" rel="nofollow"&gt;Scala seq file&lt;/a&gt; format, which is Scala's musical score format. We can enclose the pitch values in parenthesis, so that they have the form (5/4) for a major third or (7/6) for a subminor third. Another form is monzo format, where (|-2 0 1&amp;gt;) can be used in place of (5/4), and (|-1 -1 0 1&amp;gt;) in place of (7/6). An alternative is to use Scala degree numbers. This involves using lines like &amp;quot;4564 note 61 47&amp;quot; in the seq file, where the number right after &amp;quot;note&amp;quot; is the degree or note number. Scala needs additional information in the form of a scale or a declaration of what equal division is being used to interpret the note number, but one of the advantages is that by changing the scale, you can change the music Scala outputs in the form of a midi file.&lt;br /&gt;
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 -&amp;gt;10/3 5-&amp;gt;5&lt;br /&gt;
 &lt;br /&gt;
-If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; versions of all of these, which turn out to be the same: the 3edo &amp;quot;skeleton&amp;quot;, as we might call it, of 5-limit just intonation is left invariant by these transformations.&lt;/body&gt;&lt;/html&gt;</pre></div>
+If we apply the Quantize command under the Modify pull-down menu, putting 3 in the box for resolution, we get the &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; versions of all of these, which turn out to be the same: the 3edo &amp;quot;skeleton&amp;quot;, as we might call it, of 5-limit just intonation is left invariant by these transformations. &lt;br /&gt;

Wikispaces>genewardsmith: Imported revision 250896296 - Original comment:

2011-09-05T14:17:30Z

**Imported revision 250896296 - Original comment: **

Show changes

Wikispaces>genewardsmith: Imported revision 250883122 - Original comment:

2011-09-05T13:17:20Z

**Imported revision 250883122 - Original comment: **

New page

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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 13:17:20 UTC</tt>.<br>
: The original revision id was <tt>250883122</tt>.<br>
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<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">[[toc|flat]]

=Scala seq files=
Suppose we have a piece in [[Just intonation]] which we want to put into the [[http://www.huygens-fokker.org/scala/seq_format.html|Scala seq file]] format, which is Scala's musical score format. We can enclose the pitch values in parenthesis, so that they have the form (5/4) for a major third or (7/6) for a subminor third. Another form is monzo format, where (|-2 0 1>) can be used in place of (5/4), and (|-1 -1 0 1>) in place of (7/6). An alternative is to use Scala degree numbers. This involves using lines like "4564 note 61 47" in the seq file, where the number right after "note" is the degree or note number. Scala needs additional information in the form of a scale or a declaration of what equal division is being used to interpret the note number, but one of the advantages is that by changing the scale, you can change the music Scala outputs in the form of a midi file.

=Transformations=
Suppose we have a scale in 5-limit JI. One of the most basic transformations we can apply to such a scale is the major-minor involution. This changes major triads such as 1-5/4-3/2 to minor triads such as 1-6/5-3/2, and vice-versa. Suppose, for example, our scale is the Ellis duodene: 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2. If we go to "Project" under the "Modify" pull-down menu, we may put "5 24/5" into the box called "Factor pair(s)", then the resulting scale will be 10/9 9/8 5/4 6/5 4/3 27/20 3/2 5/3 8/5 15/8 9/5 2. If we use this scale in place of the duodene (be sure **not** to change the ordering!) using the "Tools" pull-down menu at "Transform sequence to midi file", we will get our original 5-limit duodene piece, only with major and minor switched about. If we put "5 24/5" in the Project command again, we get the duodene back again, which is why this transformation is called an involution.

An alternative method of accomplishing basically the same thing is to use monzo notation for the notes, only //without// parenthesis. That is, a major third is represented, not by (|-2 0 1>), but by |-2 0 1> with no parentheses. To get this seq file to produce a midi, you need to put both a val command and a corresponding equal command at the top of the file, such as "0 val <612 970 1421|" together with "0 equal 612". This has the advantage of not requiring a scale file. To produce a midi file with the majpr-minor involution, replace the mapping for 5 with one for 24/5 in [[612edo]] or whatever equal division you choose to use; in this case "0 val <612 970 1385|".</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="#Transformations">Transformations</a>
<br />
<h1 id="toc0"><a name="Scala seq files"></a>Scala seq files</h1>
Suppose we have a piece in <a class="wiki_link" href="/Just%20intonation">Just intonation</a> which we want to put into the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/seq_format.html" rel="nofollow">Scala seq file</a> format, which is Scala's musical score format. We can enclose the pitch values in parenthesis, so that they have the form (5/4) for a major third or (7/6) for a subminor third. Another form is monzo format, where (|-2 0 1&gt;) can be used in place of (5/4), and (|-1 -1 0 1&gt;) in place of (7/6). An alternative is to use Scala degree numbers. This involves using lines like &quot;4564 note 61 47&quot; in the seq file, where the number right after &quot;note&quot; is the degree or note number. Scala needs additional information in the form of a scale or a declaration of what equal division is being used to interpret the note number, but one of the advantages is that by changing the scale, you can change the music Scala outputs in the form of a midi file.<br />
<br />
<h1 id="toc1"><a name="Transformations"></a>Transformations</h1>
Suppose we have a scale in 5-limit JI. One of the most basic transformations we can apply to such a scale is the major-minor involution. This changes major triads such as 1-5/4-3/2 to minor triads such as 1-6/5-3/2, and vice-versa. Suppose, for example, our scale is the Ellis duodene: 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2. If we go to &quot;Project&quot; under the &quot;Modify&quot; pull-down menu, we may put &quot;5 24/5&quot; into the box called &quot;Factor pair(s)&quot;, then the resulting scale will be 10/9 9/8 5/4 6/5 4/3 27/20 3/2 5/3 8/5 15/8 9/5 2. If we use this scale in place of the duodene (be sure <strong>not</strong> to change the ordering!) using the &quot;Tools&quot; pull-down menu at &quot;Transform sequence to midi file&quot;, we will get our original 5-limit duodene piece, only with major and minor switched about. If we put &quot;5 24/5&quot; in the Project command again, we get the duodene back again, which is why this transformation is called an involution.<br />
<br />
An alternative method of accomplishing basically the same thing is to use monzo notation for the notes, only <em>without</em> parenthesis. That is, a major third is represented, not by (|-2 0 1&gt;), but by |-2 0 1&gt; with no parentheses. To get this seq file to produce a midi, you need to put both a val command and a corresponding equal command at the top of the file, such as &quot;0 val &lt;612 970 1421|&quot; together with &quot;0 equal 612&quot;. This has the advantage of not requiring a scale file. To produce a midi file with the majpr-minor involution, replace the mapping for 5 with one for 24/5 in <a class="wiki_link" href="/612edo">612edo</a> or whatever equal division you choose to use; in this case &quot;0 val &lt;612 970 1385|&quot;.</body></html></pre></div>

Using Scala to transform just intonation - Revision history

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