Magic Tetrachords
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=Magic Tetrachords!= Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], Magic Tetrachords! Magic[7] has structure 3L4s, with a large step of 6/5 and a small step of 25/24~28/27~36/35. The chroma is therefore 7/6, so the diminished second is 8/9 (so like, a tone in the wrong direction. Weird hey) and the augmented seventh 9/4. From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-MODMOS with 4/3 and 3/2. Eg. Take 5|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1. We obtain 3/2 as the fourth of the scale, rather than the fifth, and we want 4/3 before it. If we lower the minor fifth, 14/9, by a chroma, we obtain a diminished fifth of 4/3, where the interval in between the major fourth and the diminished fifth is a diminished second of 8/9! We now have 5|1 b5, LsLdLLs: 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1. So we have an out of order tetrachordal scale, of chromatic genus. Why not just play the notes in order of pitch? Then, as if by 'magic', we have a tetrachordal scale! The two tetrachords are 1/1 6/5 5/4 4/3 and 1/1 16/15 9/7 4/3. Since the tetrachords are not the same our tetrachordal scale is classified as ‘mixed.’ Noting that 1|5 – sLssLsL – is the inversion of 5|1, we deduce that by raising the major fourth, 9/7, by the chroma, resulting in an augmented fourth of 3/2, we obtain the inversion of our first tetrachordal scale, 1|5 #4, sLLdLsL: 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1, with out of order tetrachords 1/1 28/27 5/4 4/3 and 1/1 16/15 10/9 4/3. Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1. 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, 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. 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. ==Table of Results:== ||~ UDP Notation ||~ Steps ||~ Ratios ||~ Lower Tetrachord ||~ Upper Tetrachord || || 5|1 b5 || LsLdLLs || 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1 || Chromatic: 1/1 6/5 5/4 4/3 || Chromatic: 1/1 16/15 9/7 4/3 || || 1|5 #4 || sLLdLsL || 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1 || Chromatic: 1/1 28/27 5/4 4/3 || Chromatic: 1/1 16/15 10/9 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 || || 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 ||
Original HTML content:
<html><head><title>Magic Tetrachords</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Magic Tetrachords!"></a><!-- ws:end:WikiTextHeadingRule:0 -->Magic Tetrachords!</h1>
<br />
Dissatisfied with Magic[7]? A fan of tetrachordal scales? Try tetrachordal MODMOS of Magic[7], Magic Tetrachords!<br />
<br />
Magic[7] has structure 3L4s, with a large step of 6/5 and a small step of 25/24~28/27~36/35. The chroma is therefore 7/6, so the diminished second is 8/9 (so like, a tone in the wrong direction. Weird hey) and the augmented seventh 9/4. From any mode of Magic[7] with 4/3 or 3/2 can be constructed a 1-MODMOS with 4/3 and 3/2.<br />
<br />
Eg. Take 5|1. We have LsLssLs: 1/1 6/5 5/4 3/2 14/9 8/5 27/14 2/1.<br />
We obtain 3/2 as the fourth of the scale, rather than the fifth, and we want 4/3 before it. If we lower the minor fifth, 14/9, by a chroma, we obtain a diminished fifth of 4/3, where the interval in between the major fourth and the diminished fifth is a diminished second of 8/9! We now have 5|1 b5, LsLdLLs: 1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1. So we have an out of order tetrachordal scale, of chromatic genus. Why not just play the notes in order of pitch? Then, as if by 'magic', we have a tetrachordal scale! The two tetrachords are 1/1 6/5 5/4 4/3 and 1/1 16/15 9/7 4/3. Since the tetrachords are not the same our tetrachordal scale is classified as ‘mixed.’<br />
<br />
Noting that 1|5 – sLssLsL – is the inversion of 5|1, we deduce that by raising the major fourth, 9/7, by the chroma, resulting in an augmented fourth of 3/2, we obtain the inversion of our first tetrachordal scale, 1|5 #4, sLLdLsL: 1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1, with out of order tetrachords 1/1 28/27 5/4 4/3 and 1/1 16/15 10/9 4/3.<br />
<br />
Taking now 6|0, we have LsLsLss: 1/1 6/5 5/4 3/2 14/9 15/8 27/14 2/1.<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 />
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 />
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 />
<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 />
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 />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Magic Tetrachords!-Table of Results:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Table of Results:</h2>
<table class="wiki_table">
<tr>
<th>UDP Notation<br />
</th>
<th>Steps<br />
</th>
<th>Ratios<br />
</th>
<th>Lower Tetrachord<br />
</th>
<th>Upper Tetrachord<br />
</th>
</tr>
<tr>
<td>5|1 b5<br />
</td>
<td>LsLdLLs<br />
</td>
<td>1/1 6/5 5/4 3/2 4/3 8/5 27/14 2/1<br />
</td>
<td>Chromatic: 1/1 6/5 5/4 4/3<br />
</td>
<td>Chromatic: 1/1 16/15 9/7 4/3<br />
</td>
</tr>
<tr>
<td>1|5 #4<br />
</td>
<td>sLLdLsL<br />
</td>
<td>1/1 28/27 5/4 3/2 4/3 8/5 5/3 2/1<br />
</td>
<td>Chromatic: 1/1 28/27 5/4 4/3<br />
</td>
<td>Chromatic: 1/1 16/15 10/9 4/3<br />
</td>
</tr>
<tr>
<td>6|0 b5<br />
</td>
<td>LsLdAss<br />
</td>
<td>1/1 6/5 5/4 3/2 4/3 15/8 27/14 2/1<br />
</td>
<td>Chromatic: 1/1 6/5 5/4 4/3<br />
</td>
<td>Enharmonic: 1/1 5/4 9/7 4/3<br />
</td>
</tr>
<tr>
<td>0|6 #4<br />
</td>
<td>ssAdLsL<br />
</td>
<td>1/1 28/27 16/15 3/2 4/3 8/5 5/3 2/1<br />
</td>
<td>Enharmonic: 1/1 28/27 16/15 4/3<br />
</td>
<td>Chromatic: 1/1 16/15 10/9 4/3<br />
</td>
</tr>
<tr>
<td>3|3 #4 b5<br />
</td>
<td>sLLdLLs<br />
</td>
<td>1/1 28/27 5/4 3/2 4/3 8/5 27/14 2/1<br />
</td>
<td>Chromatic: 1/1 28/27 5/4 4/3<br />
</td>
<td>Chromatic: 1/1 16/15 9/7 2/1<br />
</td>
</tr>
</table>
</body></html>