re: Lattice of chords

Hi Gene

If you take any of the triads in the 2D five limit lattice
and multiply all the ratios together, I think you get your
indexing of the chord - is that right? So is your observation
simply that one can assign this number to every major
and minor triad and identify it in this way?

If so the 2D lattice of chords is already shown near the end of
the intro to hexany page - if you click on the gaps between the circles you hear the
triads, and I think your indexing scheme in 2D would give us those.

Can you say if I've understood correctly? If not, in what way does
your pattern of chords differ from the five limit lattice?

Same in 3D I think you get the usual 3D lattice of seven limit
major / minor tetrads.

Not sure I understand about the 3D cubic lattice yet, but
thought to ask about that first.

Thanks,

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> Same in 3D I think you get the usual 3D lattice of seven limit
> major / minor tetrads.
>
> Not sure I understand about the 3D cubic lattice yet, but
> thought to ask about that first.
>
> Thanks,
>
> Robert

robert and monz,

it's really simple.

in the octahedral-tetrahedral lattice, which we all know is the 7-
limit lattice of notes, the centers of the octahedra can be seen as
the centers of cubes in a cubic lattice. the tetrahedra nestle the
vertices of these same cubes, such that the center of each
tetrahedron is at a vertex of the cubic lattice. since each
tetrahedron represents a 7-limit tetrad, the 7-limit tetrads form a
cubic lattice! that's all there is to it. (of course, the hexanies
(which is what the octahedra represent) form a cubic lattice too!)

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> If you take any of the triads in the 2D five limit lattice
> and multiply all the ratios together, I think you get your
> indexing of the chord - is that right?

You get a 5-limit indexing, which is not the same as the 7-limit indexing.

So is your observation
> simply that one can assign this number to every major
> and minor triad and identify it in this way?

I was talking about tetrads, but it is also true of triads.

> If so the 2D lattice of chords is already shown near the end of
> the intro to hexany page - if you click on the gaps between the circles you hear the
> triads, and I think your indexing scheme in 2D would give us those.

Strictly speaking, there isn't a 2D lattice of chords, which was part of my point. There are instead two separate lattices, one for major and one for minor, which form a hexagonal tiling of chords rather than a lattice. This is typical; what is different about 3D is that the 7-limit tetrads form a true lattice.

> Can you say if I've understood correctly? If not, in what way does
> your pattern of chords differ from the five limit lattice?

Because it *is* a lattice.

> Same in 3D I think you get the usual 3D lattice of seven limit
> major / minor tetrads.
>
> Not sure I understand about the 3D cubic lattice yet, but
> thought to ask about that first.

The usual 3D lattice is the cubic lattice; my notation shows this perspicuouly.

>robert and monz,
>
>it's really simple.
>
>in the octahedral-tetrahedral lattice, which we all know is the
>7-limit lattice of notes, the centers of the octahedra can be
>seen as the centers of cubes in a cubic lattice. the tetrahedra
>nestle the vertices of these same cubes, such that the center of
>each tetrahedron is at a vertex of the cubic lattice. since each
>tetrahedron represents a 7-limit tetrad, the 7-limit tetrads form
>a cubic lattice! that's all there is to it. (of course, the
>hexanies (which is what the octahedra represent) form a cubic
>lattice too!)

Eureka. Thanks Paul, for finding such an appropriate explanation.

-Carl

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:

> Eureka. Thanks Paul, for finding such an appropriate explanation.

I was going to try geometry next. Maybe I should have tried it first, but I have the prejudices of an algebraist.

Hi Gene,

> Don't mix up the lattice of notes with the lattice of chords!

I wasn't. However, can see where I did get mixed up because
you want to put a dot in the middle of each triangle in that
alternating rhombus tiling rather than each rhombus,
i.e. two to each rhombus. Each of those dots is a tetrad.

Anyway, here is a helpful diagram to show how the octahedron and
tetrahedron tiling works:

http://www.angelfire.com/mt/marksomers/Fig12.2.html

from figure 12.2 in
http://www.angelfire.com/mt/marksomers/115.html
(about Buckminster Fuller's ideas for this tiling).

so you have tetrahedra inscribed in every cube of the regular cubic
tiling, and you have octahedra centred on alternate vertices of
all the cubes.

Incidentally here is another page with a proposal to use octahedra
and tetrahedra as building blocks for buildings:

http://www.tabletoptelephone.com/~hopspage/Index.html

Then to find one of the planes of the 3 5 7 lattice, you just look at
the plane parallel to any face of the octahedron (equivalently
tetrahedron)

Then as you said, the vertices of the 3 5 7 lattice are
the vertices of the cubic tiling - but with alternate
vertices omitted in each plane of the cubic tiling..

This makes the tiling for the notes into the face centred cubic
(the notes rather than the tetrads).

see

http://www.ucdsb.on.ca/tiss/stretton/chem2/arch19.htm

- the unit cell shown there is the same as the entire diagram
in

http://www.angelfire.com/mt/marksomers/Fig12.2.html

and you can see the octahedron in the centre, and trace
the way that it connects with the others using triangles
- while the tetrads are at the centres of all the cubes in
http://www.angelfire.com/mt/marksomers/Fig12.2.html
so those form a cubic tiling as you say.

Which I think makes it all clear for those who like to
look at these things geometrically.

Robert

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> --- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
>
> > Eureka. Thanks Paul, for finding such an appropriate
explanation.
>
> I was going to try geometry next. Maybe I should have tried it
>first, but I have the prejudices of an algebraist.

gene, i think by now it's clear that your algebraic prejudices make
communication with other tuning theorists difficult. no fault of
your own, of course. i'm still hoping i have time to play with the
Grassman Algebra stuff so that i can develop a geometric
understanding of it -- until i do that, i fear i won't understand it at
all.

p.s. i have no idea why you say the 5-limit triads don't form a
lattice. by my "geometric" explanation, they would form the same
type of lattice as 5-limit pitches. so where does the train fall off
the tracks?

i wrote,

> p.s. i have no idea why you say the 5-limit triads don't form a
> lattice. by my "geometric" explanation, they would form the
same
> type of lattice as 5-limit pitches. so where does the train fall off
> the tracks?

in my brain. sorry gene, i see it now, it's indeed not a lattice, but
rather the vertices of the hexagonal tiling, as you say.