Magic Tetrachords: Difference between revisions
Wikispaces>hearneg **Imported revision 507840372 - Original comment: ** |
Wikispaces>hearneg **Imported revision 507843428 - 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:hearneg|hearneg]] and made on <tt>2014-05-09 | : This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-05-09 16:05:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>507843428</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 20: | Line 20: | ||
This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3. | This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3. | ||
It follows as before that from 0|6 | It follows as before that from 0|6 we obtain the inverse, 0|6 #4, ssAdLsL: 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1, with an enharmonic lower tetrachord of 1/1 28/27 16/15 4/3 and an upper chromatic tetrachord of 1/1 16/15 10/9 4/3. | ||
Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest. | Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest. | ||
| Line 26: | Line 26: | ||
From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other. | From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other. | ||
From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, but that is outside the scope of this article, and is left as an exercise for the reader. | From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, and from other 1-MODMOS, other, non-diatonic tetrachordal scales can be constructed, but that is outside the scope of this article, and is left as an exercise for the reader. | ||
==Table of Results:== | ==Table of Results:== | ||
| Line 34: | Line 34: | ||
|| 6|0 b5 || LsLdAss || 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1 || Chromatic: 1/1 6/5 5/4 4/3 || Enharmonic: 1/1 5/4 9/7 4/3 || | || 6|0 b5 || LsLdAss || 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1 || Chromatic: 1/1 6/5 5/4 4/3 || Enharmonic: 1/1 5/4 9/7 4/3 || | ||
|| 0|6 #4 || ssAdLsL || 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1 || Enharmonic: 1/1 28/27 16/15 4/3 || Chromatic: 1/1 16/15 10/9 4/3 || | || 0|6 #4 || ssAdLsL || 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1 || Enharmonic: 1/1 28/27 16/15 4/3 || Chromatic: 1/1 16/15 10/9 4/3 || | ||
|| 3|3 #4 b5 || sLLdLLs || 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1 || Chromatic: 1/1 28/27 5/4 4/3 || Chromatic: 1/1 16/15 9/7 2/1 ||</pre></div> | || 3|3 #4 b5 || sLLdLLs || 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1 || Chromatic: 1/1 28/27 5/4 4/3 || Chromatic: 1/1 16/15 9/7 2/1 || | ||
Any of the 7 modes of each of these scales can of course be used. | |||
Tunings of 19edo, 22edo and 41edo are encouraged.</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>Magic Tetrachords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Magic Tetrachords!"></a><!-- ws:end:WikiTextHeadingRule:0 -->Magic Tetrachords!</h1> | <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>Magic Tetrachords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Magic Tetrachords!"></a><!-- ws:end:WikiTextHeadingRule:0 -->Magic Tetrachords!</h1> | ||
| Line 50: | Line 52: | ||
This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3.<br /> | This time lowering the minor fifth, 14/9, by a chroma, to obtain a diminished fifth of 4/3, we now have 6|0 b5, LsLdAss (where A is 7/5~45/32): 1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1. Our out of order, mixed tetrachords are now of different genus, the upper, 1/1 5/4 9/7 4/3 of enharmonic genus and the lower our now familiar chromatic: 1/1 6/5 5/4 4/3.<br /> | ||
<br /> | <br /> | ||
It follows as before that from 0|6 | It follows as before that from 0|6 we obtain the inverse, 0|6 #4, ssAdLsL: 1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1, with an enharmonic lower tetrachord of 1/1 28/27 16/15 4/3 and an upper chromatic tetrachord of 1/1 16/15 10/9 4/3.<br /> | ||
<br /> | <br /> | ||
Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest.<br /> | Looking now at modes of Magic[7] that do not contain 4/3 or 3/2: From 4|2 and 2|4 2-MODMOS lead us to the same scales we obtained from 5|1 and 1|5 respectively, so that is of little interest.<br /> | ||
| Line 56: | Line 58: | ||
From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other.<br /> | From 3|3 however, both raising the major fourth to 3/2 and lowering the minor fifth to 4/3 leads us to 3|3 #4 b5, sLLdLLs: 1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1, a symmetrical scale with chromatic tetrachords of 1/1 28/27 5/4 4/3 and 1/1 16/15 9/7 2/1, inversions of each other.<br /> | ||
<br /> | <br /> | ||
From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, but that is outside the scope of this article, and is left as an exercise for the reader.<br /> | From the 1-MODMOS above, 2-MODMOS can be obtained with two diminished seconds of 8/9, and one diminished fourth of 10/9, of one diatonic and one chromatic tetrachord, moving us another step closer to the Zarlino-Ptolemy diatonic scale, and from other 1-MODMOS, other, non-diatonic tetrachordal scales can be constructed, but that is outside the scope of this article, and is left as an exercise for the reader.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Magic Tetrachords!-Table of Results:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Table of Results:</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Magic Tetrachords!-Table of Results:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Table of Results:</h2> | ||
| Line 136: | Line 138: | ||
</table> | </table> | ||
</body></html></pre></div> | Any of the 7 modes of each of these scales can of course be used.<br /> | ||
Tunings of 19edo, 22edo and 41edo are encouraged.</body></html></pre></div> | |||