I-Ching SYstem for Hexachords

From "The Little Book of Hexachord Theory"

In my I-Ching system, I use the four symbols and six lines to
represent musical sets.

0,6
1,7
2,8
3,9
4,10
5,11

- - is 0 in A and 6 in B
-o- is 0 in B and 6 in A
---- is 0,6 in A
-e- is 0,6 in B

where B is the complement of A

Since there are 4^6 possibilities this represents all 2^12 sets in
12t-ET

Using just hexachords, I found that one can arrange them in families
based on:

1) Holding tritones fixed in A, B for sets with 0,1 tritones and
scrambling non-tritones

2) Holding non-tritones fixed in A, B for sets with 2,3 tritones and
scrambling tritones

Columns are my "valiance families" using U,P,I, A,M, Y as parents
Rows are based on distance of non-tritones, or tritones based on above

U P I : AE M Y
V O J : B N T
W Q H : C S X
* F/Z G/K : D R L

1. No tritone: 2^6, which is 6C0+6C1+6C2+6C3+6C4+6C5+6C6

2. One tritone: 6C1*5C1*2^4, fixing tritones is (12 + 12 + 6) *
2^4, which is (2C1 * 6 + 2C1 * 6 + 2C1 * 3) * 2^4

3. Two tritones: 6C2*4C2*2^2, fixing non-tritones is (36 + 36 + 18) *
2^2 which is (4C2 * 6 + 4C2 * 6 + 4C2 * 3) * 2^2

4. Three tritones: 6C3. in terms of types is 6 + 6 + 6 + 2

This is 64 + 480 + 360 + 20 = 924 (12C6)

2^4 is 4C0 + 4C1 + 4C2 + 4C3 + 4C4
2^2 is 2C0 + 2C1 + 2C2

4^6 = 2^12 = 4096 (Grand Total)

6 transposes, 3 in the case of 0,3 symmetry

0TT: (6C0+6C1+6C2+6C3+6C4+6C5+6C6)
1TT: (2C1 * 6 + 2C1 * 6 + 2C1 * 3)(4C0 + 4C1 + 4C2 + 4C3 + 4C4)
2TT: (4C2 * 6 + 4C2 * 6 + 4C2 * 3)(2C0 + 2C1 + 2C2)
3TT: (6C3)

PGH

On 6/29/07, Paul G Hjelmstad <phjelmstad@msn.com> wrote:
> From "The Little Book of Hexachord Theory"
>
> In my I-Ching system, I use the four symbols and six lines to
> represent musical sets.

Paul, have you ever read The Glass Bead Game (a.k.a. Das
Glasperlenspiel or Magister Ludi) by Hermann Hesse? This idea
remindeds me of it strongly, and I think you'd really enjoy it.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper"
<keenanpepper@...> wrote:
>
> On 6/29/07, Paul G Hjelmstad <phjelmstad@...> wrote:
> > From "The Little Book of Hexachord Theory"
> >
> > In my I-Ching system, I use the four symbols and six lines to
> > represent musical sets.
>
> Paul, have you ever read The Glass Bead Game (a.k.a. Das
> Glasperlenspiel or Magister Ludi) by Hermann Hesse? This idea
> remindeds me of it strongly, and I think you'd really enjoy it.
>
> Keenan
>

Yes I read it, many years ago! I was working on this stuff back then
too. (I used to really like Hesse, I think you have to be at the
right age for it, now it doesn't seem that great. But I enjoyed that
book)

PGH