More on chord geometry

I posted a while back on the geometric relations between 9-limit tetrads,
using a system of coordinates defined by the vector sum of the exponents,
or in other words, the product of the notes of the chord in
exponent-vector form. This form of the coordinates allows one to find the
vector form easily from the JI form, is generalizable to any p-limit, and
works in the 5-limit also. However, there is a coordinate transformation
which is special to the 7-limit which is more or less essential to
understanding the how the 7-limit chord geometry works.

The coordinates I've used may be related to the exponent coordinates of
the notes by the observation that it is simply the centroid of the notes,
times a scalar product of four. Since the notes themselves have a
symmetric Euclidean metric given by the quadratic form Q(3^a 5^b 7^c) =
a^2+b^2+c^2+ab+ac+bc, the chords inherit this metric. If we look at the
interval adjacency vectors for major/minor, minor/major connections,
namely [-2,2,2],[2,-2,2],[2,2,-2],[2,-2,-2],[-2,2,-2],[-2,-2,2] we find
that these are either of opposite sign or are orthogonal. We may
therefore select three of these to give us an orthogonal basis; if we
also shift the coordinate center to the major tetrad [1,5/4,3/2,7/4] we
may define a coordinate transformation which takes [a,b,c] to [(b+c-2)/4,
(a+c-2)/4, (a+b-2)/4], the inverse of which takes [a,b,c] to
[-2a+2b+2c+1, 2a-2b+2c+1, 2a+2b-2c+1]. This now makes the major and minor
tetrads represented as the points of a cubic lattice, a nice feature
which is unique to the 7-limit (the 5-limit gives a hexagonal tiling, but
not a lattice; the lattice appears because of a special property of the
7-limit note lattice, which belongs to two different classes of lattices
at once.)

The pumps are particularly easy to find and understand in this coordinate
system; for example
[3 4 1][3 3 1][3 2 1][2 2 1][2 1 1][1 1 1][0 1 1][0 0 0] is a 1029/1024
pump I gave previously; in this form it is easy to see how we can obtain
other such pumps.

We again can represent subminor and supermajor tetrads, which transform
to non-lattice points. In particular,
1-7/6-3/2-5/3 is represented by [0 -1/2 -1/2] and 1-9/7-3/2-9/5 by [-1
1/2 1/2]

We can also represent complete 9-limit harmonies, basing ourselves on the
major quintad
1-9/8-5/4-3/2-7/4 and its minor quintad inverse; the same coordinates
serve for these. Formerly, given a 4-et value (that is, the value h4(q)
for h4 = [4,6,9,11], which reduces mod 4 to [0 2 1 3], a complete set of
representatives mod 4) and a chord, we could reconstruct the note. Now we
may similarly use a 5-et value, based on h5 = [5,8,12,14], and because
h5(1)-h5(9/8)-h5(5/4)-h5(3/2)-h5(7/4) is 0-1-2-3-4, a complete set of mod
5 residues, we can again recover the note from a chord and h5 value.
(Similar comments apply to triadic harmony using h3 and 13-limit harmony
using h7.)
In terms of these chords, there are 12 interval adjacencies from major to
minor, and 12 from minor to major.

The symmetrical 7-limit note lattice has "deep holes" which are
geometrically octahedra, and musically hexanies. The dual cubic lattice
of 7-limit tetrads just discussed has eight-tetrad cubes as holes; if we
look at the corresponding notes we have a 14-note stellated octahedron.
Fans of superparticular ratios may be interested to hear that considered
as a scale, this has all of its steps superparticular rations, though of
highly variable size:

1-21/20-15/14-35/32-9/8-5/4-21/16-35/24-3/2-49/32-25/16-105/64-7/4-15/8