More Hexachord Grids

This will be my last post for awhile on hexachord theory. Unless I
come up with a way to tie the basic C4 X C3 structure into tuning
theory, as a 7-limit structure, that ties into my hexachord theory
and with the lattices that are discussed here.

The following is hexachord grids for all eight group operations I
have been working with. They are C12, D4-X C3, S3-X C4, D12, C12 X S2
(C),
D4- X C3 X S2(C), S3- X C4 X S2(C) and D12 X SC(2). Once again,
I am only taking backwards permutations of D4 and S3, and S2(C) is
the S2 complementability group.

The columns are weights 0,1,2,3. Weights are mod6, and since it
is absolute value weighting, you obtain 0,1,2,3,2,1,0,1,2,3,2,1,0
for the mod 12 residues. The rows are the number of tritones
in a hexachord, once again 0,1,2,3. I go left to right for weights
and top to bottom for tritones.

One final note: This give mappings for all transpositions (Tn
mapping). There are two uses of transposition: The Tn that a
hexachord maps to itself under, and the transpositions of the
hexachord itself. They are NOT the same thing. I have divided
everything out by 12, since all 12 transpositions are represented
for each hexachord (Second meaning). In my spreadsheet, I have also
given the same grids based on the hexachord letters. As I have
stated previously, each letter has 1,2,4, or 8 expressions,
depending on Z, A, M5. (Z-relation, Asymmetrical, or M5 partner that
is different). It's too hard to print these here with Yahoo's bad
formating.

C12

0,0,0,6
8,16,16,0
4,10,8,8
2,0,2,0

D4- X C3

0,0,0,2
0,0,0,0
4,2,8,0
2,0,2,0

S3- X C4

0,0,0,4
2,0,0,0
2,0,0,2
2,0,0,0

D12

0,0,0,4
4,0,0,0
4,0,0,6
2,0,0,0

C12 X S2(C)

0,0,0,6
0,0,0,0
0,0,0,0
2,0,0,0

D4- X C3 X S2(C)

0,0,0,2
4,0,0,0
0,0,0,4
2,0,0,0

S3- X C4 X S2(C)

0,0,0,4
0,8,0,0
0,0,0,4
2,0,2,0

D12 X SC(2)

0,0,0,4
4,0,8,0
0,10,0,2
2,0,2,0

The reason I have not included S3 X D4 is because it can be found
from the first four (Just add up and divide by four) S3 X D4 X S2(C)
can be found the same way from the last four. And the 26 final
hexachords by adding it all up and dividing by eight.

Now, just for fun, here is just one of them. I might change my mind
and post these too. (Based on letters). Here is D4- X C3:

-,-,-,AY
-,-,-,-
GK,H,FZ,-
UP,-,I,-

I guess that's not so bad after all. If anyone wants I will post
these too, otherwise they will be in my spreadsheet along with
other things.

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:
>
> This will be my last post for awhile on hexachord theory. Unless I
> come up with a way to tie the basic C4 X C3 structure into tuning
> theory, as a 7-limit structure, that ties into my hexachord theory
> and with the lattices that are discussed here.

I can't see how that would work.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@>
> wrote:
> >
> > This will be my last post for awhile on hexachord theory. Unless I
> > come up with a way to tie the basic C4 X C3 structure into tuning
> > theory, as a 7-limit structure, that ties into my hexachord theory
> > and with the lattices that are discussed here.
>
> I can't see how that would work.

Thanks for your response. You did mention that non-cyclic group
structures don't have sensible torsion parts. (Or something like that)
so I am starting with the 80 main hexachords and seeing what I can
come up with.

My goal with hexachords, which may be misfounded, is to relate
them to your seven limit lattices. I am also reading about the
same kind of lattice in SPLAG. I use hexachords to represent
any set from 0 to 12, because 0-5 are just subsets, and 7-12 are
just supersets. (Complements of subsets). Complementation itself
is just S2-comp anyway, and then with mirror inverse I can get
this down to my 35 hexachords. Adding rows 2,3 and 6,7 into the mix
brings this down to 26 letters of my system. So the progression is
to go from 80 to 50 (or 44) to 35 to 26.

I know I work intuitively due to (big) gaps in my training. This is
due to the fact that I worked on this for years before joinging this
newsgroup (I have hundereds of pages of calculations, computer
printouts, charts, graphs etc) I didn't even know I was working with
both S4 X S3 and D4 X S3 and S2-comp for example. I still don't know
what to call my measures based on tritone counts (I use I-ching
symbols) So, actually, any responses that I get from you or anyone are
greatly appreciated. Even negative ones!

Paul
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
> >
> > This will be my last post for awhile on hexachord theory. Unless
I
> > come up with a way to tie the basic C4 X C3 structure into tuning
> > theory, as a 7-limit structure, that ties into my hexachord theory
> > and with the lattices that are discussed here.
>
> I can't see how that would work.

Hexany: J5 (0,3,4,7,9,10)

This is a symmetrical, non-M5, Z-related hexachord. It has
only 2 expressions and 12 transpositions. Other hexanies
are just transposes of this one. The complement is (0,1,3,6,8,9)
which has absolutely no value in the 7-limit lattice whatsoever.

See I am tying hexachords with hexanies, rather weakly, but it's
just a start. Scratch anything about C4 X C3...for now

Now to analyze tetrachords, and the 7-limit Tonality Diamond.
Utonal and Otonal certainly relate to mirror-image of course.

I cannot seem to reconcile what I am reading in SPLAG with
your Seven-Limit Lattices Webpage. SPLAG uses the rhombic
dodecahedron and its dual the truncated octohedron. (p.34)
The first is a fcc lattice and the second is bcc. So I will
stick with your page for now.

Sorry I don't have time to give this better attention today.
Mondays are hectic. The 13 note TD and the 14 note SH should
have hexachord theory relationships. I know, this is 12-et
Boring!!!!!!!!

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> So, actually, any responses that I get from you or anyone are
> greatly appreciated. Even negative ones!

One form of feedback you've gotten before is that no one can figure out
why you confine yourself to 12-et. As a 7-limit tuning, it isn't very
good, and if I was going to try to relate something to the lattice of 8-
limit JI, I'd use something with more of an honest 7-limit. 31 comes to
mind. Of course, that's a lot more notes, but the fact that it's a
prime number simplies the group situation.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> I cannot seem to reconcile what I am reading in SPLAG with
> your Seven-Limit Lattices Webpage. SPLAG uses the rhombic
> dodecahedron and its dual the truncated octohedron. (p.34)
> The first is a fcc lattice and the second is bcc. So I will
> stick with your page for now.

My web page follows chapter 4 of SPLAG in terms of terminology.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@>
> wrote:
>
> > So, actually, any responses that I get from you or anyone are
> > greatly appreciated. Even negative ones!
>
> One form of feedback you've gotten before is that no one can figure
out
> why you confine yourself to 12-et. As a 7-limit tuning, it isn't very
> good, and if I was going to try to relate something to the lattice of
8-
> limit JI, I'd use something with more of an honest 7-limit. 31 comes
to
> mind. Of course, that's a lot more notes, but the fact that it's a
> prime number simplies the group situation.

I guess it's because I find 12-et important, I am obsessed with
hexachord theory and with the whole 4 X 3 thing in 12-tET. I am
also interested in 22-et, and 24-et because of Steiner systems,
and I am learning some of the other ones too. I wish I had more
time. I might have to scrap 12-tET as much of a 7-limit system,
and think of it as a 5-limit system. I just intuitively believe
that the factors of twelve (2*2*3) have 7-limit properties
largely due to the tritone. (And relating C4 X C3 with tables
based on 3,5 and 7 prime numbers). I know C4 is NOT C2 X C2
of course so I am making some progress.

I will look at 31-et in terms of the lattice. Probably should
take a break from 12 and form some new brain pathways:)

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@>
> > wrote:
> > >
> > > This will be my last post for awhile on hexachord theory.
Unless
> I
> > > come up with a way to tie the basic C4 X C3 structure into
tuning
> > > theory, as a 7-limit structure, that ties into my hexachord
theory
> > > and with the lattices that are discussed here.
> >
> > I can't see how that would work.
>
> Hexany: J5 (0,3,4,7,9,10)
>
> This is a symmetrical, non-M5, Z-related hexachord. It has
> only 2 expressions and 12 transpositions. Other hexanies
> are just transposes of this one. The complement is (0,1,3,6,8,9)
> which has absolutely no value in the 7-limit lattice whatsoever.

Well, actually it is of use! If you take the upper part
of a stellated hexany, (using the hexany that moves 3/1 to 1/1)
and take it's complement, you obtain (8,11,1,2). This is one
side of the only Z-related tetrachord type [0,3,5,6]. It has the
interesting interval vector <1,1,1,1,1,1>. Now taking the
bottom part of of a stellated hexany, you obtain (5,6,8,11)
as the complement, which is the inverse of the above tetrachord.
It also has the interval vector <1,1,1,1,1,1>. Putting the
two together: ((5,6,8,11,1,2) which is the Z-related complement
of J5 above: (0,3,4,7,9,10) (Also called J5 in my system).
Of course that's trivial, it's just the complement of the original
hexany.

Now taking the intersection gives (8,11), so the full stellated
hexany is a pretty uninteresting decachord. (0,1,2,3,4,5,6,7,10)

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
More stuff:

> > Hexany: J5 (0,3,4,7,9,10)
> >
> > This is a symmetrical, non-M5, Z-related hexachord. It has
> > only 2 expressions and 12 transpositions. Other hexanies
> > are just transposes of this one. The complement is (0,1,3,6,8,9)
> > which has absolutely no value in the 7-limit lattice whatsoever.
>
> Well, actually it is of use! If you take the upper part
> of a stellated hexany, (using the hexany that moves 3/1 to 1/1)
> and take it's complement, you obtain (8,11,1,2). This is one
> side of the only Z-related tetrachord type [0,3,5,6]. It has the
> interesting interval vector <1,1,1,1,1,1>. Now taking the
> bottom part of of a stellated hexany, you obtain (5,6,8,11)
> as the complement, which is the inverse of the above tetrachord.
> It also has the interval vector <1,1,1,1,1,1>. Putting the
> two together: ((5,6,8,11,1,2) which is the Z-related complement
> of J5 above: (0,3,4,7,9,10) (Also called J5 in my system).
> Of course that's trivial, it's just the complement of the original
> hexany.
>
> Now taking the intersection gives (8,11), so the full stellated
> hexany is a pretty uninteresting decachord. (0,1,2,3,4,5,6,7,9,10)

The upper part of the stellated hexany has interval vector
<5,5,5,5,5,3> and of course so does the bottom part. The full thing
has interval vector <8,8,9,8,8,4> because it's complement is
merely <0,0,1,0,0,0> (Rule: the interval vector of a complement
is just 12-2n higher than the smaller set class n for each interval
vector value) For the tritone class it is 6-n, half the amount.

Each interval vector has Binom(n,2) intervals.

I'll doublecheck my calculations at home trying other hexanies.
And of course the full 7-limit tonality diamond.

Lastly, the interval vector of the hexany J5 itself and it's
Z-related complement is <3,2,3,2,3,2> So that's a pretty pattern
to go with <1,1,1,1,1,1>

I think it is interesting that it is two diminished chords at the
fifth with one flipped backwards in terms of JI (reciprocals).

The interval vector of a tetrachord is just <0,1,2,1,1,1>

Paul Hj