Use of I-Ching symbols

Many years ago I developed a way of representing all the subsets of
12-tet with I-Ching symbols (primarily to study hexachords). Since
there are 2^12 possible subsets (including the empty set), these
are represented by 4^6 combinations of symbols. Of course this also
equals 2^12 or 4096 and it is one-to-one and onto.

First use the following as a starting template (even though a circle
of fifths or fourths would work as well):

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

The sets are A and B, complementary sets. (For example, a hexachord
and its complement, or a septachord and its pentachord complement,
etc) The symbols used are as follows: - - maps (a, b), --- maps (a, a)
-o- maps (b,a) and -e- maps (b,b) where a is a member of A and b is a
member of B. (-e- is actually o with --- going through it).

The nice thing about this system is that it represents both a chord
and its complement together, so in the case of hexachords, it
represents a hexachord and its z-relation, since all z-related
hexachords are z-related to their complements. Its also easy to
study the "M5" relationship with these symbols.

When I get time I will post all 35 hexachord types to the my Files
directory using the I-Ching symbols.

Now, for hexachords, you can classify them based on (0,1,2 or 3)
tritones. For (0,1) tritone sets, you can hold the tritone constant
in both A and B and vary the other notes. For (2,3) tritones, do
just the opposite. Hold the non-tritones constant and mix up the
tritones. I will post these results to Files menu soon. This will
use I-Ching symbols.

After reducing for the Z-relation you obtain 35 hexachord types.
Reducing further for the M5 relationship you get 26 hexachord types.
I have a system that uses the 26 letters of the alphabet to represent
these chord types. (If the hexachord has an "M5" partner I use
B1 and B5 for example) I will post this also to Files soon.

Going back a paragraph: Create an illustration which maps the results
indicated as vertical relationships. For the horizontal relationships
use (what I believe is) C3 X S4 scrambling. This creates a tight
little grid 8 columns across and 12 rows going down. I will post
this tomorrow. Here's how the rows go:

AU
BDV
CEW
FT
G
HI
JK
LNO
MQX
PY
RZ
S

And the columns: (I've made them rows here)

AEMY
BNT
CSX
DLR
F/Z G/K
HQW
IPU
JOV

I realize this doesn't mean anything until I post my files, and
give you all the key for (A - Z) (Also, I need to explain
F/Z G/K line.) I'm also going to figure out how to arrange chords
for other group theoretical symmetries, (for example S3 X C4) and the
like. The thing that I think is neat about the above is that
the vertical and horizontal are independent (no two chord types
appear in a column AND a row, or in more than one column or row). I
have already reduced for the M5 relationships because C3 X S4 absorbs
them -- but I can expand them out easily. I could probably use
some help to make this more rigorous. After I post all my files I
will post another message to "tie up loose ends". I also use these
illustrations to study the different types of z-relations. More
on that later.

I have also printed out 4 X 3 grids for all the chords in 12-tet
but I don't think I will post these, that would be overkill. The
4 X 3 grids are based on minor third as the x-axis and major third
as the y-axis. I also have classified hexachords, pentachords, etc.
in staircase grids based on counts per row and column etc. I think I
might post those to the Files menu. Anyway, the 4 X 3 grids are good
for studying certain relationships and the I-Ching symbols are good
for others.

Last, you can arrange all the hexachords based on "families" which
are based on hexachords with no tritones and hexachords with 3
tritones. The 0-tritone parents have children in (0,1) and the 3-
tritone parents have children in (2,3).

Well anyway I'd better quit while I'm ahead now. Am I making any
sense?

Paul