Hexachord Theorem, from Lewin and Maximally Even Sets

Here is an interesting find, a paper on "David Lewin and Maximally Even Sets". As a "pleasant by-product" of Lewin's work with the FFT (Fast Fourier Transform), Amiot also presents a short proof of the Hexachord Theorem. (Theorem 1.9). It's only two lines. (p. 4 bottom p. 5 top)

http://canonsrythmiques.free.fr/4journalOfM&M/lewinMESets.pdf

Here's a one-page description of the Hexachord Theorem:

http://myweb.lsbu.ac.uk/~whittyr/MathSci/TheoremOfTheDay/MusicAndArt/Hexachord/TotDHexachord.pdf

Of course, Maximally-Even Sets, MOS, Myhill's Propriety (and maybe
Rothenberg Propriety?) are all related. It's fun to see this tie-in
with Hexachord Theory, which is my baby I guess, the Hexachord Theorem
leads to the Z-relation (Isomeric Sets) and the main issue of
"complementability" or the S2-symmetry. (Symmetry Types of Periodic
Sequences, Gilbert and Riordan, Ill. Journ. of Mus. 1961)

More tie-ins:

1. Z-relation, Flat-Line Interval Distribution, Difference Sets, Projective Spaces, etc.

2. M5-relation, Affine Group, Necklace Theory, D4 X S3 for example.

The jump between Z-relation and FLIDs is a bit tenous but if I review
Jon Wild's paper I think it is in there.

Another interesting tangent would go between Projective Spaces,
Steiner Systems, M12, and ultimately, the Monster group.

The musical aspect of M12 and S(5,6,12) will be covered in my
paper on M12 and the Piano Keyboard Layout, where Diatonic Transposition spans all 924 hexachords, seeded from 132 Steiner Hexads. (Seven
fold expansion.)

And that leads to Diatonic Set Theory, I guess.

PGH