Cubic lattice or A3 lattice ?

In a cubic lattice, the minimal vectors of the lattice (the neighbour of the origin) are the permutations of the vectors (1,0,0) and (-1,0,0). Those are the vertices of an octahedron. Taking the vertices of any triangular face and the origin (0,0,0), there is only two such tetrads useful in chord representation.

In a A3 lattice, the minimal vectors of the lattice are the permutations of the vector (-1,1,0,0). Those are the vertices of a cuboctahedron. Taking the vertices of any triangular faces and the origin (0,0,0,0), all these tetrads are truly useful in chord representation since that corresponds to a chord and its dual with their inversions.

I saw Gene mentioning An lattices recently but I'm not sure he has already defined it. So, I would say that An lattices are orthogonal complement of the vector (1,1,...,1) in Z¨^(n+1). Its points are all vectors (xi) in dimension n+1 where Sum xi = 0. It's a subspace of dimension n.

The An lattices are very useful not only to represent chords but also musical systems. In a step basis having the same dimension as the represented musical system, the minimal vectors are the unison vectors enclosing its fundamental domain. For instance, In the basis {16/15, 10/9, 9/8}, the minimal vectors of the A2 lattice are 81/80, 135/128, 25/24, 80/81, 128/135 and 24/25. The content of that hexagone (seen in the lattice of primal basis) is precisely the interval space generated by the tonic rotation in the Zarlino scale.

I mention yet I already resolved the following problem: How to determine easily (without computer) the polytope of minimum vectors in An lattices? As I already said, these polytopes, for A1 A2 A3 A4 ..., are segment, hexagone, cuboctahedron, prismatodecachoron, ...

Pierre

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> In a cubic lattice, the minimal vectors of the lattice (the neighbour of =
the origin) are the permutations of the vectors (1,0,0) and (-1,0,0).

What I meant by "cubic lattice" was Z^3, the 3D lattice of integers
[a,b,c]. In general, we have a lattice Z^n, and another lattice
Dn composed of integers [a1,a2,...,an] such that a1+a2+...+an is an even nu=
mber. D3 is isomorphic to A3, and is the face-centered cubic lattice. D3=A3 =
is the unique property of 3D lattices which makes this 7-limit thing work.

> I saw Gene mentioning An lattices recently but I'm not sure he has alread=
y defined it. So, I would say that An lattices are orthogonal complement of =
the vector (1,1,...,1) in Z¨^(n+1). Its points are all vectors (xi) in dimen=
sion n+1 where Sum xi = 0. It's a subspace of dimension n.

That's one way to define it.

> I mention yet I already resolved the following problem: How to determine =
easily (without computer) the polytope of minimum vectors in An lattices? As=
I already said, these polytopes, for A1 A2 A3 A4 ..., are segment, hexagone=
, cuboctahedron, prismatodecachoron, ...

Without a computer? The theta series might help.

Gene wrote:
What I meant by "cubic lattice" was Z^3, the 3D lattice of integers [a,b,c].
In general, we have a lattice Z^n, and another lattice Dn composed of integers
[a1,a2,...,an] such that a1+a2+...+an is an even number. D3 is isomorphic to A3,
and is the face-centered cubic lattice. D3=A3 is the unique property of 3D lattices
which makes this 7-limit thing work.
I was refering to octahedron used by Paul and Robert. About the particularity D3 = A3 and lattices
in general, I learned that past summer. I seeked confirmation for my sequence of polytopes. I found it
in a Sloane book about Packing Spheres and explored, after that, a bit of simple results got by Lie,
Cartan, Killing, Coxeter, Sloane, Conway, ... just to see the links with other math domains.

I had formulate my problem few years ago in terms of peripheal accordance graph of a non-degenerate
chordoid and produced the following graph corresponding to a prismatodecachoron:

I neglected to resolve that better (for the non-degenerate case is rejected by the fertility condition in
gammier systems) until I see that the non-degenerate case is more than the case of chords with relatively
prime elements. It's also the case of the S-matrix (giving the unison vectors from the set of steps) when
the gammier is simple, i.e. when dimension = amount of steps -- it's not the case, for instance, with the
Indian system having 4 steps in 5-limit (3D) or the ib1215 having 6 steps in 11-limit (5D).

Gene wrote:
Without a computer? The theta series might help.
I never looked at theta series. I imagine it would not be so useful for characterizing, in particular, the type of
cells bounded that kind of polytopes. I found they have in 2N and 2N+1 dimensions, N distinct type of cells.
Is it something appearing clearly in theta series?

For instance, the cuboctahedron of the minimal 4D vectors in A3 has 8 triangles and 6 squares, while the
cells of the primatodecachoron (yet the convex hull of vectors with norm = 2) in A4 are 20 tetrahedra and
30 triangular prisms.

I characterized the generalized cell of those polytopes as m-n-cells. An m-1-cell is simply a m-simplex while
a m-2-cell may be called a m-prism, i.e. a prism over a m-simplex. I don't know what are the name of the other
cells. Here is the correspondance:
1-1-cell = 1-simplex = point
2-1-cell = 2-simplex = segment
2-2-cell = 2-prism = square
3-1-cell = 3-simplex = triangle
3-2-cell = 3-prism = triangular prism
3-3-cell
4-1-cell = 4-simplex = tetrahedron
4-2-cell = 4-prism = tetrahedral prism
4-3-cell
4-4-cell
5-1-cell = 5-simplex = pentachoron
5-2-cell = 5-prism = pentachoral prism
5-3-cell
...
Is that conform to the standard English math rules?

The number of vertices of an m-n-cells is mn and the cells of the minimal vectors polytope, at the center of the
lattice A_(N-1) using vectors of dimension N, are those having m + n = N. For instance, cuboctahedron has
3-1-cells (triangles) and 2-2-cells (square), while prismatodecachoron has 4-1-cells (tetrahedron) and 3-2-cells
(triangular prisms). As dimension grows, other kind of cells appears. In dimension 8, there are 4 kind of m-n-cells,
the 7-1, 6-2, 5-3 and 4-4 one.

It could be continued in tuning-math.

Pierre

Pierre wrote: ( ... It's me. It seems I'm reduced to discuss with me )
For instance, the cuboctahedron of the minimal 4D vectors in A3 has 8 triangles and 6 squares, while the
cells of the primatodecachoron (yet the convex hull of vectors with norm = 2) in A4 are 20 tetrahedra and
30 triangular prisms.
Yupps! It's naturally wrong.These are 10 tetrahedra and 20 triangular prisms. So, I will go more systematically.

--------------------------------------------------------------------------------

Let denote
E(k+1) the Euclidean space of dimension k+1 with standard basis,

Ak the lattice of vectors (xi) whose elements xi are integers with sum zero.
What is the amount of each type of cells at the boundary of the chordo-polytope of Ak, which is the
convex hull of its minimal (norm = 2) vectors?

I don't demonstrate now. I give only results. The cells are the m-n-cells where m + n = k + 1. The amount of
m-n-cells is C(n, m+n) if m = n and the double if not. I recall C(a,b) is easily computed with the Pascal triangle:
1
1 2 1
1 3 3 1
1 4 6 4 1
...
The first cases are:
Segment with 2 cells in 2D
1-1-cells amount = C(1,2) = 2 points
Hexagone withf 6 cells in 3D
2-1-cells amount = 2 C(1,3) = 6 segments
Cuboctahedron with 14 cells in 4D
3-1-cells amount = 2 C(1.4) = 8 triangles
2-2-cells amount = C(2,4) = 6 squares
Prismatodecachoron with 30 cells in 5D
4-1-cells amount = 2 C(1,5) = 10 tetrahedra
3-2-cells amount = 2 C(2,5) = 20 triangular prisms
62 cells in 6D
5-1-cells amount = 2 C(1,6) = 12 pentachora
4-2-cells amount = 2 C(2,6) = 30 tetrahedral prisms
3-3-cells amount = C(3,6) = 20
126 cells in 7D
6-1-cells amount = 2 C(1,7) = 14
5-2-cells amount = 2 C(2,7) = 42
4-3-cells amount = C(3,7) = 70
254 cells in 8D
7-1-cells amount = 2 C(1,8) = 16
6-2-cells amount = 2 C(2,8) = 56
5-3-cells amount = C(3,8) = 112
4-4-cells amount = C(4,8) = 70
...
In dimension N, the total amount of cells = 2^N - 2. So
2, 6, 14, 30, 62, 126, 254, ...

--------------------------------------------------------------------------------

Hoping someone could appreciate, I recall that in a basis of steps (so linearly independent)
-- for instance, 16/15, 10/9, 9/8 -- the An lattice is precisely the kernel of unison vectors giving
the periodicity blocks -- here A2 generated by 81/80 and 25/24 -- and the standard metric
is a similitude of the appropriate one I talked about on tuning-math, fitting the bilinear form on
the nearest unison vectors -- here F(x,y) = 3x^2 + 3xy + 13y^2 = 7^2 at the nearest unison
vectors.

It's only one of numerous deep aspects concerning the music at paradigmatic viewpoint.

Inversely, the sequence of polytopes I named chordo-polytopes, which are pure mathematical
objects derived from the An lattices, are simply the sequence of chordoids generated by the
standard basis (as chords) of the Euclidean spaces. So, the chordoid theory I developed few
years ago, specifically for the musical needs, appears now a much more fundamental object.

Pierre

I lost all my graphics and templates when the power supply of my computer exploded few months ago.
I used an image of my website to show clearly the cuboctahedron generated by the 7th chord. The
background represents the Blues gammier with its four characteristic seventh chords. I added the edges
of the cuboctahedron. As minimal vectors, at the center of the A3 lattice in the basis = 7th chord, all the
vertices might be on the sphere of radius sqrt(2).

Pierre

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Pierre wrote: ( ... It's me. It seems I'm reduced to discuss with
me )

maybe this would be better discussed on tuning-math. however, you may
want to search this list for "dispentachoron" . . .

Paul wrote:
maybe this would be better discussed on tuning-math. however, you may
want to search this list for "dispentachoron" . . .
I don't have any objection to pursue the matter on tuning-math if someone want to answer there.

The prismatodecachoron I talked about was
(o)----o-----o----(o)
[Small] prismatodecachoron [5]
Alternative names:
Runcinated 5-cell (Norman W. Johnson)
Runcinated pentachoron
Runcinated pentatope
Runcinated [four-dimensional] simplex
Spid (Jonathan Bowers: for small prismatodecachoron)

Symmetry group: [[3,3,3]], the extended pentachoric group, of order 240

Schläfli symbol: t0,3{3,3,3}

Elements:
Cells: 10 tetrahedra, 20 triangular prisms
Faces: 40 triangles (all joining tetrahedra to triangular prisms), 30 squares (all joining triangular prisms to triangular prisms)
Edges: 60
Vertices: 20
See http://members.aol.com/Polycell/section1.html

Since naming is harder when dimension grows, I suggest chordo-pentatope, chordo-hexatope, chordo-heptatope, ...

Pierre