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Author Topic: "antimajor scale" in 31-EDO  (Read 4144 times)


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"antimajor scale" in 31-EDO
« on: September 15, 2012, 12:33:18 PM »

Some time ago, I discovered an interesting scale in 31-EDO, which compares to the 5-limit major scale in a similar way as the antidiatonic scale (e.g. Mavila[7]) compares to the diatonic scale.

The diatonic scale has the form LLsLLLs (5 large steps, 2 small steps), while the antidiatonic scale has the inverted form ssLsssL (2L5s).
The 5-limit major scale has 5 whole tones and 2 semitones. However, unlike the diatonic scale, this scale has two different whole tones: A major tone 9:8 and a minor tone 10:9, and its form is LmsLmLs (L = 9:8, m = 10:9, s = 16:15). I wondered how an inversion of this scale form may sound, and tried to build scales with the inverted form smLsmsL in 31-EDO; this one caught my interest:

Code: [Select]
0     3    8   14   17    22  25   31
 0   116  310  542  658   852 968 1200 cents
1:1 16:15 6:5 11:8 16:11 13:8 7:4  2:1

With L = 6 (8:7), m = 5 (the "meantone") and s = 3 (16:15), I find it melodically very pleasant. The first three notes (0 3 8) are part of the diatonic scale (meantone[7]), and the other four notes (14 17 22 25) are part of the harmonic / melodic minor scale, so there should be some familiarity for listeners of western classical music. However, the scale as a whole contains many higher limit harmonies, so it feels pretty xenharmonic nonetheless.

Apart from having the inverted form of the major scale, LmsLmLs, there's another property that makes it kind of an opposite to the major scale: With 11:8, 13:8 and 7:4, it contains all (higher limit) overtones between the 8th and 16th overtone that the major scale doesn't contain, while it contains none of the (5-limit) overtones that the major scale contains (except for the octave, of course!). So there are two properties that make it some kind of an "antimajor scale" (though the latter one alone wouldn't be enough to justify that name). That's why I'm calling it an "antimajor scale", unless I find out there's already a name for it. Hope there's not too much confusion with the name "antidiatonic scale", though...

A scale diagram for standard tuning E A D G B E (13 13 13 10 13):

This looks pretty chaotic and hard to memorize, something like E A^ DL G Bv E (14 11 14 9 14) might be suited better for this scale:
(^ = +1 step in 31-EDO, v/L = -1 step)

In case you're wondering:
When both 6-step intervals are divided into two 3-step intervals each, the result is Casablanca[9] - that's why I added the dashed circles. Casablanca[9] is the MOS-scale with a 14\31 generator (11:8), i. e. the scale that is generated by stacking 14-step intervals on top of themselves until there are 9 tones (using octave equivalency).
MOSes (= Moments of Symmetry) allow for simple modulation - just slightly modify one tone (like adding a '#' or 'b')! If you know how to reach a nearby MOS, and how modulation works for that MOS, you can use this to move around. ;)
« Last Edit: September 15, 2012, 12:46:12 PM by Gedankenwelt »