Using Scala to transform just intonation: Difference between revisions
Wikispaces>genewardsmith **Imported revision 250896296 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 250916654 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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 | : 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> | : The original revision id was <tt>250916654</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> | ||
| Line 27: | Line 27: | ||
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. | ||
</pre></div> | =7-limit transformations= | ||
The native 7-limit transformations work very much the same as the 5-limit transformations. Instead of triads we have tetrads, instead of invariance under 3edo we have invariance under [[4edo]], and instead of the group of the triangle we have the [[http://groupprops.subwiki.org/wiki/Dihedral_group:D8|group of the square]]. | |||
We first note that the major-minor involution extends to the 7-limit, with the projection being "5 24/5 7 48/7". Instead of a transformation of order three, we get one of order four, by "3 14/5 5 24/5 7 32/5". Applying this twice leads to "3 8/3 5 14/3 7 20/3" and three times to "3 20/7 5 32/7 7 48/7", and applying it once again leads back to 3 5 7. Since "3 14/5 5 24/5 7 32/5" is of order four, "3 8/3 5 14/3 7 20/3" is another involution. We end up with a total of eight transformations: | |||
3->3 5->5 7->7 | |||
3->14/5 5->24/5 7->32/5 | |||
3->8/3 5->14/3 7->20/3 | |||
3->20/7 5->32/7 7->48/7 | |||
3->3 5->24/5 7->48/5 | |||
3->14/5 5->14/3 7->7 | |||
3->8/3 5->32/7 7->32/5 | |||
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> | |||
<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: | <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: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:10 --><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 /> | ||
| Line 48: | Line 62: | ||
3-&gt;10/3 5-&gt;5<br /> | 3-&gt;10/3 5-&gt;5<br /> | ||
<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.</body></html></pre></div> | 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 /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit transformations"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit transformations</h1> | |||
The native 7-limit transformations work very much the same as the 5-limit transformations. Instead of triads we have tetrads, instead of invariance under 3edo we have invariance under <a class="wiki_link" href="/4edo">4edo</a>, and instead of the group of the triangle we have the <a class="wiki_link_ext" href="http://groupprops.subwiki.org/wiki/Dihedral_group:D8" rel="nofollow">group of the square</a>.<br /> | |||
<br /> | |||
We first note that the major-minor involution extends to the 7-limit, with the projection being &quot;5 24/5 7 48/7&quot;. Instead of a transformation of order three, we get one of order four, by &quot;3 14/5 5 24/5 7 32/5&quot;. Applying this twice leads to &quot;3 8/3 5 14/3 7 20/3&quot; and three times to &quot;3 20/7 5 32/7 7 48/7&quot;, and applying it once again leads back to 3 5 7. Since &quot;3 14/5 5 24/5 7 32/5&quot; is of order four, &quot;3 8/3 5 14/3 7 20/3&quot; is another involution. We end up with a total of eight transformations:<br /> | |||
<br /> | |||
3-&gt;3 5-&gt;5 7-&gt;7<br /> | |||
3-&gt;14/5 5-&gt;24/5 7-&gt;32/5<br /> | |||
3-&gt;8/3 5-&gt;14/3 7-&gt;20/3<br /> | |||
3-&gt;20/7 5-&gt;32/7 7-&gt;48/7<br /> | |||
3-&gt;3 5-&gt;24/5 7-&gt;48/5<br /> | |||
3-&gt;14/5 5-&gt;14/3 7-&gt;7<br /> | |||
3-&gt;8/3 5-&gt;32/7 7-&gt;32/5<br /> | |||
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> | |||