Enumerative Combinatorics

What I am trying to achieve:

Since I have exhausted everything I wanted to know, with respect
to Enumerative Combinatorics on Necklaces, my goal now is
to apply the same math to Lattices. Starting simply, in 12-tET
a 5-limit Lattice based on 3 and 5 in two dimensions.

What can be determined about different set-classes? For example,
80 hexachords, 66 pentachords and so forth.

I am awaiting a book on the subject by Stanley, and I am sure
that Polya theory can be applied to Lattices as well as Necklaces.

So that's it in a nutshell. Of course, I would like to carry over
everything I have done with interval vectors, group theory,
affine group, FLIDs (a la Dr. Jon Wild) difference sets, projective
spaces and Steiner systems.

Thanks

PGH

Hi Paul,

I regret to say that i haven't followed any of your
Necklace posts (for lack of time to get really involved),
and am especially sorry about this since you contacted
me directly about it too. I hope to catch up some day.

But what you want to do with Lattices sounds exactly
like what i had in mind way back when i first developed
the lattice concept independently, around 1995, before
i knew about the tuning list or Fokker, Wilson, et al.
So hopefully i'll be able to read and comprehend some
of what you will write about this.

You might find this helpful: the fourth graphic from
the bottom on my 12-edo page, which shows a 3,5-space
bingo-card lattice for 12-edo, with several of the
possible periodicity-blocks outlined in green.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> What I am trying to achieve:
>
> Since I have exhausted everything I wanted to know, with respect
> to Enumerative Combinatorics on Necklaces, my goal now is
> to apply the same math to Lattices. Starting simply, in 12-tET
> a 5-limit Lattice based on 3 and 5 in two dimensions.
>
> What can be determined about different set-classes? For example,
> 80 hexachords, 66 pentachords and so forth.
>
> I am awaiting a book on the subject by Stanley, and I am sure
> that Polya theory can be applied to Lattices as well as Necklaces.
>
> So that's it in a nutshell. Of course, I would like to carry over
> everything I have done with interval vectors, group theory,
> affine group, FLIDs (a la Dr. Jon Wild) difference sets, projective
> spaces and Steiner systems.
>
> Thanks
>
> PGH

--- In tuning-math@yahoogroups.com, "monz" <joemonz@...> wrote:
Hi Monz,

Thanks. I will look at this. I don't know why I am so obssessed
with applying musical set theory to lattices, but then again,
why not. So far branching out into two dimensions for this
hasn't added any information, but who knows.

>
> Hi Paul,
>
>
> I regret to say that i haven't followed any of your
> Necklace posts (for lack of time to get really involved),
> and am especially sorry about this since you contacted
> me directly about it too. I hope to catch up some day.
>
> But what you want to do with Lattices sounds exactly
> like what i had in mind way back when i first developed
> the lattice concept independently, around 1995, before
> i knew about the tuning list or Fokker, Wilson, et al.
> So hopefully i'll be able to read and comprehend some
> of what you will write about this.
>
> You might find this helpful: the fourth graphic from
> the bottom on my 12-edo page, which shows a 3,5-space
> bingo-card lattice for 12-edo, with several of the
> possible periodicity-blocks outlined in green.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > What I am trying to achieve:
> >
> > Since I have exhausted everything I wanted to know, with respect
> > to Enumerative Combinatorics on Necklaces, my goal now is
> > to apply the same math to Lattices. Starting simply, in 12-tET
> > a 5-limit Lattice based on 3 and 5 in two dimensions.
> >
> > What can be determined about different set-classes? For example,
> > 80 hexachords, 66 pentachords and so forth.
> >
> > I am awaiting a book on the subject by Stanley, and I am sure
> > that Polya theory can be applied to Lattices as well as Necklaces.
> >
> > So that's it in a nutshell. Of course, I would like to carry over
> > everything I have done with interval vectors, group theory,
> > affine group, FLIDs (a la Dr. Jon Wild) difference sets,
projective
> > spaces and Steiner systems.
> >
> > Thanks
> >
> > PGH
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> What I am trying to achieve:
>
> Since I have exhausted everything I wanted to know, with respect
> to Enumerative Combinatorics on Necklaces, my goal now is
> to apply the same math to Lattices. Starting simply, in 12-tET
> a 5-limit Lattice based on 3 and 5 in two dimensions.
>
> What can be determined about different set-classes? For example,
> 80 hexachords, 66 pentachords and so forth.
>
> I am awaiting a book on the subject by Stanley, and I am sure
> that Polya theory can be applied to Lattices as well as Necklaces.
>

Polya theory can be applied to everything that is finite. So at least
the original theory won't work for infinite lattices - maybe there is
some generalized theory that does.
There might, however, be some aplications of Polya theory for finite
periodicity blocks - taking into account only symmetries within one
periodicity block.
--
Hans Straub

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > What I am trying to achieve:
> >
> > Since I have exhausted everything I wanted to know, with respect
> > to Enumerative Combinatorics on Necklaces, my goal now is
> > to apply the same math to Lattices. Starting simply, in 12-tET
> > a 5-limit Lattice based on 3 and 5 in two dimensions.
> >
> > What can be determined about different set-classes? For example,
> > 80 hexachords, 66 pentachords and so forth.
> >
> > I am awaiting a book on the subject by Stanley, and I am sure
> > that Polya theory can be applied to Lattices as well as Necklaces.
> >
>
> Polya theory can be applied to everything that is finite. So at
least
> the original theory won't work for infinite lattices - maybe there
is
> some generalized theory that does.
> There might, however, be some aplications of Polya theory for
finite
> periodicity blocks - taking into account only symmetries within one
> periodicity block.
> --
> Hans Straub
>
Yes, that is the sort of thing I am looking for. My book still hasn't
arrived, so I cannot look any of this up yet. Of course, group theory
itself can be applied to the infinite, if not Polya's enumerative
combinatorics....

PGH