Geometry of 9-limit tetradic harmony

A 7-limit octave equivalence class is defined by the odd number 3^a 5^b
7^c, and so can be represented by the vector [a b c]. Consider the major
tetrad, 1-5/4-3/2-7/4, represented by
[0 0 0]-[0 1 0]-[1 0 0]-[0 0 1]. Taking the sum of this (which
corresponds to taking the product 3*5*7=105) gives us [1 1 1], which we
can use to represent the tetrad. The inverse
chord, 1-8/7-4/3-8/5 ~ 1-1/3-1/5-1/7, is a minor tetrad, represented by
[-1 -1 -1]. Multiplying
all four notes of a tetrad by any 7-limit interval will change the sum by
a power of four of that interval, or in terms of our vector notation,
will change it so as to leave it unchanged modulo four. Hence a major
tetrad is represented by a vector of three odd numbers, each of which is
of the form 4n+1, or 1 mod 4; and a minor tetrad by four odd numbers,
each of the form 4n-1, or -1 mod 4.

We've used up only a small proportion of the lattice of integral
3-vectors, so there is plenty of room to represent more types of tetrads.
The subminor tetrad, 1-7/6-3/2-5/3 adds up to
[-1 1 1], and so we can represent subminor tetrads by vectors equal
modulo 4 to [-1 1 1], or in other words, of the form [4n-1 4n+1 4n+1].
Inverting the subminor tetrad gives a supermajor tetrad 1-6/5-4/3-12/7
which adds up to [1 -1 -1], and we can represent all supermajor tetrads
by vectors of this form; for intense 1-9/7-3/2-9/5 is [5 -1 -1].

We still have plenty of space left, and can fill a small further amount
with the two so-called "asses" given by 1-3-5-5/3 and 1-3-7-7/3, which
are represented by [0 2 0] and [0 0 2] respectively. I'd be interested to
hear what else people think should be represented.

Let us call a tetrad interval adjacent if it shares an interval, or in
other words at least two notes, with another tetrad, and note adjacent if
it shares a single note. For major and minor tetrads, the difference
vectors defining interval adjacency are:

[2, 2, -2] [2, -2, 2] [-2, 2, -2] [-2, -2, 2] [2, -2, -2] [-2, 2,2]

[2, 0, 0] [-2, 0, 0] [0, -2, 2] [0, 2, -2] [4, -2, -2] [-4, 2, 2]

where the first row sends major to minor, and vice versa, and the second
sends to subminor/supermajor. If the tetrad is major, then
[2 0 0] attaches to the unique subminor tetrad with *three* notes in
common (for intense, 1-5/4-3/2-7/4 to 9/8-5/4-3/2-7/4), while if the
tetrad is minor, [-2 0 0] attaches to the unique supermajor tetrad with
three notes in common; for 1-6/5-3/2-12/7, represented by [3 -1 -1], this
would be 1-6/5-4/3-12/7 represented by [1 -1 -1].

For subminor/supermajor tetrads, the interval adjacency vectors are:

[2, 2, -2] [2, -2, 2] [-2, 2, -2] [-2, -2, 2] [6, -2, -2] [-6, 2, 2]

[2, 0, 0] [-2, 0, 0] [0, -2, 2], [0, 2, -2] [4, -2, -2] [-4, 2, 2]

By symmetry the second line, moving from sub/super to major/minor, is the
same for both, but the subminor/supermajor line involves more movement
along the chain of fifths.

For note adjacency, all four types of triads have different sets, though
with an approximate 2/3 overlap and 20 connections in common. These are,
in Maple readable form:

# note adjacency of major tetrad

majn := {[6, -2, -2], [6, 0, -4], [8, -2, -2], [4, 0, 0], [4, 2, -2], [6,
-4,
0], [2, 0, -4], [2, 4, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4,
0],
[2, -4, 4], [0, -2, -2], [0, 0, -4], [0, 0, 4], [0, 4, -4], [0, 4, 0],
[-2,
6, -2], [0, -4, 0], [0, -4, 4], [-2, -2, 6], [-2, 0, 4], [-2, 4, 0], [-4,
0
, 4], [-4, 2, -2], [-4, 4, 0], [-4, 6, -2], [-2, -2, -2], [-6, 4, 0],
[-4, -
2, 2], [-4, -2, 6], [-4, 0, 0], [-6, 0, 0], [-6, 0, 4]}:

# note adjacency of minor tetrad

minn := {[-6, 2, 2], [-8, 2, 2], [6, 0, -4], [4, 0, 0], [4, 2, -2], [6,
-4, 0]
, [2, 0, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4, 0], [0, 0, -4],
[0
, 0, 4], [0, 4, -4], [0, 4, 0], [0, -4, 0], [0, -4, 4], [-2, 0, 4], [-2,
4,
0], [-4, 0, 4], [-4, 2, -2], [-4, 4, 0], [-6, 4, 0], [-4, -2, 2], [-4, 0,
0]
, [-6, 0, 4], [6, 0, 0], [4, 2, -6], [2, 2, -6], [2, 2, 2], [4, -6, 2],
[2,
-6, 2], [0, 2, 2], [-2, 4, -4], [-2, -4, 4]}:

# note adjacency of subminor tetrad

subn := {[6, 0, -4], [8, -2, -2], [4, 0, 0], [4, 2, -2], [6, -4, 0], [2,
0, -4
], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4, 0], [0, -2, -2], [0, 4,
-4],
[0, -4, 4], [-2, 0, 4], [-2, 4, 0], [-4, 0, 4], [-4, 2, -2], [-4, 4, 0],
[-4
, 6, -2], [-6, 4, 0], [-4, -2, 2], [-4, -2, 6], [-4, 0, 0], [-6, 0, 4],
[6,
0, 0], [-2, 4, -4], [-2, -4, 4], [8, -4, 0], [8, 0, -4], [10, -2, -2],
[2, -
2, -2], [-8, 0, 4], [-8, 4, 0], [-6, -2, 6], [-6, 6, -2]}:

# note adjacency of supermajor tetrad

supn := {[-10, 2, 2], [-8, 2, 2], [6, 0, -4], [4, 0, 0], [4, 2, -2], [6,
-4, 0
], [2, 0, -4], [2, 4, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4,
0], [
2, -4, 4], [0, 4, -4], [0, -4, 4], [-2, 0, 4], [-2, 4, 0], [-4, 0, 4],
[-4,
2, -2], [-4, 4, 0], [-6, 4, 0], [-4, -2, 2], [-4, 0, 0], [-6, 0, 0], [-6,
0
, 4], [6, 2, -6], [6, -6, 2], [-2, 2, 2], [4, 2, -6], [4, -6, 2], [0, 2,
2]
, [8, -4, 0], [8, 0, -4], [-8, 0, 4], [-8, 4, 0]}:

It would be interesting to see a three-dimensional diagram, with four
colors for the four different types of tetrads, and three colors for
lines connecting 3, 2, and 1 note adjacencies, of this geometry. I might
also point out that a harmony vector plus a 4-et interval uniquely
determines a 7-limit JI interval, but this is a large topic.