Lattices (was Intro to Hexany page)

Hi Gene, and Paul,

First of all congratulations on independently discovering /inventing the idea
of a 3D lattice of musical notes. Must have been quite a thing :-).

If one approaches it algebraicly I expect it would be quite complex, however
the musical lattices are quite simple mathematical structurs if you look at them number theoretically
and geometrically.

Start with the five limit lattice. You can think of that just as the numbers
3^a*5^b. Then a triad is 3^a5^b, 3^(a+1)*5^b, 3^a*5^(b+1), or
3^a*5^b, 3^(a-1)5^b, 3^a*5^(b-1),

Then, once you've got the idea of representing this geometrically, maybe
ones first thought is to use a square lattice with 3 horizontally and 5
vertically. But then you draw it and see that you have got 45 degree
icoseles triangles for the triads. Nicer to use a sixty degree rhombus
for the unit cell instead of a square, and then you have a tiling
by regular triangles instead.

Now to go into three dimensions, just add an extra term to the number
theoretic expression:
3^a*5^b*7^c

Triads obtained by taking numbers in pairs and incrementing one of them
at a time:
3^a5^b*7^c, 3^(a+1)*5^b*7^c, 3^a*5^(b+1)*7^c,
3^a5^b*7^c, 3^(a+1)*5^b*7^c, 3^a*5^b*7^(c+1),
3^a5^b*7^c, 3a*5^(b+1)*7^c, 3^a*5^b*7^(c+1),

or the same with -1 instead of +1.

Then again one might think of drawing the three axis at right angles,
and that is perfectly valid of course, but it looks nicer if you
make the tetrads into tetrahedra.

Then to go into four dimensions, do the same procedure, but usse
simplexes for the pentads.

Since the simplex generalises to all higher dimensions, then
that immediately gives you the n-dimensional generalisation.

So geometrically, that is all there is to it.

One can then study the algebraic properties of it as an
emergent structure.

Of course algebraic studies would be expected to give new insights
into it, but I think this already defines the lattice geometrically
as a mathematical object for study.

Robert

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

> If one approaches it algebraicly I expect it would be quite complex...

Actually, I think that is the easiest approach. Simply define a quadratic form

Q(3^a 5^b 7^c) = a^2 + b^2 + c^2 + ab + ac + bc

and you have defined a Euclidean metric and hence the lattice. Nothing more needs to be done, and the generalization to higher dimensions is immediate.