A 9-limit diamond packing

If we start by considering 5,7, and 9 to be equal, we can put note-
classes of the form 9^a 5^b 7^c into a fcc lattice like the 7-limit
lattice. If we take the centroid of 1,9,5,7, where 9 is represented by
[0 1 1], 5 by [1 0 1] and 7 by [1 1 0], we get ([0 0 0]+[0 1 1]+[1 0
1]+[1 1 0])/4 = [1/2 1/2 1/2]. I propose placing 3 there. This would
gives us a diamond packing, not a lattice--that is, 9-limit notes are
arranged like the carbon atom of a diamond.

However, we have both otonalities and utonalities to deal with. If we
scale everything up by two, we have 3 at [1 1 1] surrounded by 9, 5,
and 7, and inverting this gives 1/3 at [-1 -1 -1] surrounded by 1/9
1/5 and 1/7. Transposing both of these to the unison, we have the
unison surrounded by two tetrahedra, which represent the 1:5:7:9 and
1:1/5:1/7:1/9 tetrads respectively, and are represented by [-1 1 1],
[1 -1 1], [1 1 -1] and [-1 -1 -1] for the major tetrad, with sum of
coordinates congruent to 1 mod 4, and [1 -1 -1], [-1 1 -1], [-1 -1 1]
and [1 1 1] a minor tetrad, with sum of coordinates congruent to -1
mod 4. The fcc lattice we started with is now everything with even
coordinates which sum to 0 mod 4, if we add everything with even
coordinates which sum to 2 mod 4, we get the body-centered cubic
lattice. We can do this by adding the minor tetrad around [1 1 1] to
the major one. Now I've got a bcc lattice of notes, consisting of
every triple with all even or all odd coodinates, but since 3*3=9 I
have the same note represented more than once.

Can you, uh, draw it?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If we start by considering 5,7, and 9 to be equal, we can put note-
> classes of the form 9^a 5^b 7^c into a fcc lattice like the 7-limit
> lattice. If we take the centroid of 1,9,5,7, where 9 is represented
by
> [0 1 1], 5 by [1 0 1] and 7 by [1 1 0], we get ([0 0 0]+[0 1 1]+[1
0
> 1]+[1 1 0])/4 = [1/2 1/2 1/2]. I propose placing 3 there. This
would
> gives us a diamond packing, not a lattice--that is, 9-limit notes
are
> arranged like the carbon atom of a diamond.
>
> However, we have both otonalities and utonalities to deal with. If
we
> scale everything up by two, we have 3 at [1 1 1] surrounded by 9,
5,
> and 7, and inverting this gives 1/3 at [-1 -1 -1] surrounded by 1/9
> 1/5 and 1/7. Transposing both of these to the unison, we have the
> unison surrounded by two tetrahedra, which represent the 1:5:7:9
and
> 1:1/5:1/7:1/9 tetrads respectively, and are represented by [-1 1
1],
> [1 -1 1], [1 1 -1] and [-1 -1 -1] for the major tetrad, with sum of
> coordinates congruent to 1 mod 4, and [1 -1 -1], [-1 1 -1], [-1 -1
1]
> and [1 1 1] a minor tetrad, with sum of coordinates congruent to -1
> mod 4.

This is where you lose me. What has even coordinates? Thanks

The fcc lattice we started with is now everything with even
> coordinates which sum to 0 mod 4, if we add everything with even
> coordinates which sum to 2 mod 4, we get the body-centered cubic
> lattice. We can do this by adding the minor tetrad around [1 1 1]
to
> the major one.

I get 0 mod 4 adding the minor tetrad to the major one...

Now I've got a bcc lattice of notes, consisting of
> every triple with all even or all odd coodinates, but since 3*3=9 I
> have the same note represented more than once.

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

> The fcc lattice we started with is now everything with even
> > coordinates which sum to 0 mod 4, if we add everything with even
> > coordinates which sum to 2 mod 4, we get the body-centered cubic
> > lattice. We can do this by adding the minor tetrad around [1 1 1]
> to
> > the major one.

> I get 0 mod 4 adding the minor tetrad to the major one...

I meant we have the four notes 1-5-7-9 of a major tetrad surrounding
3, and we also can add 9-9/5-9/7-9/9 around it.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > The fcc lattice we started with is now everything with even
> > > coordinates which sum to 0 mod 4, if we add everything with
even
> > > coordinates which sum to 2 mod 4, we get the body-centered
cubic
> > > lattice. We can do this by adding the minor tetrad around [1 1
1]
> > to
> > > the major one.
>
> > I get 0 mod 4 adding the minor tetrad to the major one...
>
> I meant we have the four notes 1-5-7-9 of a major tetrad surrounding
> 3, and we also can add 9-9/5-9/7-9/9 around it.

Okay, and both are approximated by Bb-C-D-E in absolute pitches...I
think if you gave me (new) the coordinates for 9-9/5-9/7-9/9 I could
figure out what you mean.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > The fcc lattice we started with is now everything with even
> > > > coordinates which sum to 0 mod 4, if we add everything with
> even
> > > > coordinates which sum to 2 mod 4, we get the body-centered
> cubic
> > > > lattice. We can do this by adding the minor tetrad around [1
1
> 1]
> > > to
> > > > the major one.
> >
> > > I get 0 mod 4 adding the minor tetrad to the major one...
> >
> > I meant we have the four notes 1-5-7-9 of a major tetrad
surrounding
> > 3, and we also can add 9-9/5-9/7-9/9 around it.
>
> Okay, and both are approximated by Bb-C-D-E in absolute pitches...I
> think if you gave me (new) the coordinates for 9-9/5-9/7-9/9 I
could
> figure out what you mean.

I figured it out. I really like all this geometric stuff, and
Wikipedia + Mathworld has a ton of stuff between them

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > >
> > > > The fcc lattice we started with is now everything with even
> > > > > coordinates which sum to 0 mod 4, if we add everything with
> > even
> > > > > coordinates which sum to 2 mod 4, we get the body-centered
> > cubic
> > > > > lattice. We can do this by adding the minor tetrad around
[1 1
> > 1]
> > > > to
> > > > > the major one.
> > >
> > > > I get 0 mod 4 adding the minor tetrad to the major one...
> > >
> > > I meant we have the four notes 1-5-7-9 of a major tetrad
> surrounding
> > > 3, and we also can add 9-9/5-9/7-9/9 around it.
> >
> > Okay, and both are approximated by Bb-C-D-E in absolute
pitches...I
> > think if you gave me (new) the coordinates for 9-9/5-9/7-9/9 I
> could
> > figure out what you mean.
>
> I figured it out. I really like all this geometric stuff, and
> Wikipedia + Mathworld has a ton of stuff between them

(Even though I only get 2 mod 4 with 9*9,9*5,9*7,9*1 as opposed to
9-9/5-9/7-9/9 where I obtain 0 mod 4) I must be missing a factor
somewhere