Hexachord Theory in a Nutshell

THEORY IN A NUTSHELL

1. The 132 Steiner hexads comprise 1/7 of the 924 total hexads (hexachords), in fact they comprise 1/7 in every BW (Black and White) category:

0B/6W 1B/5W 2B/4W 3B/3W 4B/2W 5B/1W

7 105 350 350 105 7 = 924 (Totals)
1 15 50 50 15 1 = 132 (Steiner Hexads)

2. A Steiner System is a Block Design such that each of the 132 Hexads contains 6 Pentads. In fact, every 792 Pentad is contained
exactly once, by some Hexad. No two Steiner Hexads contain the same Pentad.

3. (Technical) M12 is constructed from the Exceptional Outer Automorphism of the Symmetric(6) Group, using duads and triads. This is modelled by a ProjectiveGeometry(3,2). M12 is the Symmetry Group (Automorphism Group) of Steiner(5,6,12), a Block design as mentioned previously.

4. By Diatonic Transposition of the Whites, we can span all 924 hexachords, together and in each BW category. Open: Can Seven independent Steiner Systems be constructed? The remaining hexads
from 924-132=792 comprise a Sextuple Steiner System. The total is a trivial Septuple Steiner System.

5. The Black keys used are passively determined from this construction, they are merely the remaining combinations in each BW category. However, there might be a pattern there also.

6. Definition of Diatonic Transposition: 024->245->457, ie CDE->DEF->EFG etc. so with the 350 3B3W, 50 SS hexads -> 350 hexads, each 35 3W is used 10 times, and there are exactly 10 3B to use with each of these. 3W is C(7,3) 3B is C(5,3)

7. Ultimately, I will show that 35 hexachord types of D4 X S3 X S2 also have this pattern. Every hexachord type can be found
from 3W and 3B keys. (Known) Basically, I will show that the 35 hexachord types are merely 5 x 7, generated by the Steiner System above.

Note:

D4 X S3 X S2 is a Fourfold Symmetry whose hexachords can be found
from 80 + 20 + 8 + 32 = 140, 140/4 = 35. The separate parts are
found from Polya Necklace Theory.