Inversionally Symmetrical Hexatonics and Heptatonics

As a bit of a supplement to my previous post, here's a set of clockface diagrams for all the six- and seven-note scales in 12EDO that are equal to their own inversions. As I observed there, these might be interesting resources because every descending melody such a scale contains also exists as an identical descending version. I'll say more about that after the first pic.

So here are all the inversionally symmetrical hexatonics:

At the bottom right is the whole tone scale and next to it the augmented hexatonic. We also have 6-z6 and 6-z38, which I wrote about in relation to harmony with contrary motion and also 6-z29 and 6-z50, which are Moonlight and Bridge. So lots of interesting stuff that's proved to be fruitful (at least to me) alongside lots of things I haven't really looked at before, which is a good mixture.

Each hexatonic is exactly half the set of all available notes so the complement of a hexatonic is another hexatonic, which makes them handy for atonal composition. What's more, the complement of an inversionally symmetrical scale is always inversionally symmetrical. In retrospect I should have paired the diagrams above so that's easy to see but I made them all by hand before I thought of it -- at least it's not too hard to eyeball the shape of the complement (the white notes) and then find the matching shape elsewhere in the diagram. This is made easier by the fact that the mirror symmetry line will be at the same angle for both.

I did better when it came to the heptatonics. There are only 10 of these but of course there are also 10 pentatonics that are their complements. This time I've paired each heptatonic next to its complement. (I don't think it's any more than a coincidence that we again get 20 scales.)

The top row is just chromatic soup. The next three rows contain melakatas and their complements: 7-15 is Jalarnavam; 7-z17 is Yagapriya; 7-z12 is Tanarupi; 7-22 is Rasikapriya, whose complement is MiP; 7-33 is Neapolitan. Then the bottom row contains 7-35, the Major scale and 7-34, which is Melodic Minor.

There are a few things I want to take away from this rather silly exercise. I'd like to take a closer look at Jalarnavam, Tanarupi and Rasikapriya and their complementary pentatonics. I'd also like to look at complementary pairs of different hexatonics from the list above. In both cases, one angle is the 12-tone one where we look to split the chromatic into two halves and exploit the extra intervallic consistency that these structures have. It could be that for 12-tone improvisation, these give you an advantage because in that world groups of intervals are the repeating building-blocks that create a sense of structure and internal identity.