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 3L 4s, 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). 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 (see Modal UDP Notation). 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, 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:

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

Any of the 7 modes of each of these scales can of course be used.

Tunings of 19edo, 22edo and 41edo are encouraged.