Tuesday, August 10, 2010

Tables of scales

I find combinatorial categorization almost pathogically compelling, so I couldn't resist pushing a little further about scales and tetrachords. I made a table with every tetrachord I've run across, picked abbreviations for them, and included synonyms from Arabic (Maqam) and Turkish (Makam) traditions, then tried to lay out as many scales as I could in terms of those tetrachords. For the scales, I looked at the seven modes of each of several source scales (major, melodic minor, harmonic minor, harmonic major, and double harmonic), including the fancy mode names and synonyms where I could find them (mostly from Wikipedia, especially the modes article).

Finally, I made tables of what scales get produced by combining tetrachords. Here we can see the names that get attached to all the combinations, and also all the blank spots on the map, scales we can play that have no name.

I had to draw some limits. I ignore the microtonal tetrachords, limiting myself to scales that can be played on an ordinary Western instrument. I had to include several tetrachords that don't give a perfect fourth: the Lydian tetrachords, which fill a tritone, and the diminished tetrachord, but I didn't write down every permutation or rotation of intervals to invent new tetrachords, I just included the ones I found reference to. I'll leave it to someone else to figure out why Lydian #2, m3-s-T, is in the canon, but the just-as-reasonable-looking m3-T-s is not. The perfect fourth tetrachords can be combined arbitrarily into scales making an octave, separated by a tone, but we need to do something different for the non-perfect ones. Either we need to combine a Lydian (tritone) tetra with a diminished, separated by a tone; or we need to put a Lydian with a perfect and separate by a semitone. Similarly, there's a couple places where an augmented second joins the two.

I also skipped some important scales. The major and minor pentatonic scales, and the related blues scale, an elaboration of the minor pentatonic, don't fit easily into a tetrachord concept, both because they have the wrong number of notes (all the above scales are heptatonic, ie have 7 notes) and because they don't have a fourth. I'll skip the bebop scales, despite their awesome names (A Doriolian ♭5? B altered quintal?) for a similar reason: they are octatonic, basically inserting a passing note into another scale. You can of course outline four note patterns in these, but it's hard to see the scale as being constructed of them. Or maybe I'm just lazy.

At some point, of course, this kind of exercise becomes silly: mapping out and analyzing parts of harmonic space which are unused, not because they are great new territories for invention and creativity which you and you alone have realized, but rather because they suck. The world's cultures have been making music for a long time, and if a scale is undiscovered, there's likely to be a good reason for that. Still, it's kind of fun to try and get a global picture of the possibilities, patterned after the scales we are used to.

Oh, yeah, the table is here, done up as a spreadsheet so it's easy for me to fix.

Refs: Wikipedia, http://www.tonalcentre.org/, various jazz harmony books, http://docs.solfege.org:81/3.9/C/scales/modes.html

Added: Another mammoth list of modes is here, 1200 of them, part of the astonishing microtonal scale analysis and construction program Scala. It's written in Ada, and has a 9/11 Truth icon at the end, just in case you need additional evidence that the author is looney, but really, I think that list of scales should be sufficient.

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