7-limit lattice

Here is a depiction of the oct-tet lattice we've been using to map 7-limit
JI, from Buckminster Fuller (who was using it to illustrate his philosophy,
not music):

http://www.rwgrayprojects.com/synergetics/s05/figs/f3710.html

Hi Joseph

I wonder if it may help to give a better picture of lattice diagrams
to know that the 7-limit lattice is none other than the lattice green
grocers use to stack oranges. (Also known as the fruit stacking
lattice).

I just realised that recently, and it became much clearer.

Suppose you've just completed a layer, and want to add another orange.

You do that by placing it on top of three others, so it will complete a tetrahedron.
Also once you've added three oranges next to each other, they
complete a tetrahedron with one of the previous oranges, this time
with the point downwards.

So each new layer of oranges makes a series of tetrahedra.
with the previous one, alternating between ones pointing
upwards and ones pointing downwards:

Showing two layers (first layer shown with dashes):
1---------3---------9---------27
/ \ / \ / \ /
/ \ 7 / \ 21 / \ 63 /
/ \ / \ / \ /
/ \ / \ / \ /
5/3-------5--------15--------45
\ / \ / \ /
\35/3 / \ 35 / \ 105 /
\ / \ / \ /
\ / \ / \ /
25/3-------25-------125

(multiply by 3 horizontally, 5 diagonally down to right, and
by 7 in the 1->7 direction, which is upwards out of the page)

Here, 1 3 5 7 is a tetrahedron pointing upwardds, and 7 35 35/3 5
is one pointing downwards

However tetrahedra by themselves can't fill space, they need to alternate
with octahedra, which fill the gaps between them. I.e. with hexanies.

We can see them here, for instance, 3 5 15 7 21 35 is our familiar
3, 5, 7 hexany.

One can place each new layer of oranges in two ways.

However, one natural method to use is to place each new orange
to fill the gap in the layer two layers below.

If you use that method, you end up with the 7-limit lattice.

Showing three layers:
1---------3---------9---------27
/ \ / \ / \ /
/ \ 7 / \ 21 / \ 63 /
/49/3 \ / 49 \ / 147 \ /
/ \ / \ / \ /
5/3-------5--------15--------45
\ / \ / \ /
\35/3 / \ 35 / \ 105 /
\ /245/3\ / 245 \ /
\ / \ / \ /
25/3-------25-------125

Here, 1, 7 and 49 lie on a straight line, so the idea is that the
third layer is above the other two.

This way of arranging the oranges is known as the face centred
cubic packing.

http://cst-www.nrl.navy.mil/lattice/struk/a1.html

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_16696.html#18885

> Hi Joseph
>
> I wonder if it may help to give a better picture of lattice diagrams
> to know that the 7-limit lattice is none other than the lattice
green grocers use to stack oranges. (Also known as the fruit stacking
> lattice).
>

Thanks so much, Robert, for your commentary. Yes, actually I
understood this much of it... I was just getting momentarily confused
about the 7-limit dimension with the spellings used...

It seems as though the VRML lattices are really the way to go! I've
very much enjoyed your work with this so far! The idea of rotating
the lattices in VRML is really spectacular.

Maybe someday there should be a "library" of some "common" tuning
lattices in VRML so that beginners and others trying to understand
tuning could grasp them better. I think it would be a valuable
addition, and perhaps could link to Joe Monzo's or John Starrett's
microtonal site.

Thanks again as ever!

_______ __ ______ __
Joseph Pehrson

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
Thanks Robert,

This is all true, but the convention for 7-limit tuning lattices is
for a multiply by 5 to go _up_ to the right and a multiply by 7 to go
_up_ to the right and out of the page.

Relative to this convention, you had up and down swapped in your
examples.

Regards,
-- Dave Keenan

Hi Joseph,

Thanks!

Glad you've enjoyed my VRML.

I'm sure there will be more to come, and I'm doing a few
VRML experiments at the moment with the hope of eventually doing
Christopher Bailey's trees, or something like them, in
VRML.

This 7-limit fruit stacking lattice would be a nice one to do in VRML
some time actually.

I'm not sure how comprehensive a library I'd manage to do, but
am interested to put a few of the models to date plus some new
ones onto a web page some day, of course linking to the microtonal
sites about lattices.

Probably a mixture of 2d gifs and image maps (like my
2D octony projection) for the ones that can be done in 2D,
then VRML for the ones that need to be done in 3D.

Meanwhile, main thing right now is to finish FTS 1.09,
which is going pretty well.

Apologies for using up instead of down for those who are
used to seeing the 7-limit lattice the other way up.

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:

/tuning/topicId_16696.html#18914

>>
> I'm not sure how comprehensive a library I'd manage to do, but
> am interested to put a few of the models to date plus some new
> ones onto a web page some day, of course linking to the microtonal
> sites about lattices.

This would be a great asset for many of us trying to slowly learn
this stuff! I really hope you get to do this Robert. If you could
start with the simple ones and work toward complexity, this would be
a powerful tool, I believe to understanding this subject.

Thanks again!

_______ _____ ___ _
Joseph Pehrson