Planar lattice of 7-limit major/minor pairs

Here's something which strikes me as very interesting and potentially
useful. I've talked a lot about the 3D cubic lattice of 7-limit
tetrads, which has the property that adjacent lattice points represent
tetrads with an interval in common, and all such common interval
pairings are represented by adjacent tetrads.

The major/minor pairings work as follows: for a lattice point [a,b,c],
if a+b+c is even, so that the tetrad is otonal, the minor version of
the chord is [a-1,b,c]. If a+b+c is odd, so that the tetrad is utonal,
the major version of the chord is [a+1,b,c]. Hence, the first coordinate
is the one which relates major and minor.

Suppose we take the sublattice generated by the other two coordinates,
[0,b,c] for b, c integers. This represents an infinitesimal proportion
of the total number of tetrads; however, if we breed temper, that is,
temper out 2401/2400, it now represents a subgroup of index two. That
is, half of all tetrads are of this form. If [0,a,b] represents a major
tetrad (a+b is even) then [-1,a,b] represents the corresponding minor
tetrad. If it represents a minor (utonal) tetrad. then [1,a,b] is the
corresponding major tetrad. If we put the major and minor tetrads
together, we can define a lattice whose points represent major/minor
pairs of tetrads, breed tempered. The remarkable thing now is that
*every* tetrad pair is represented by a point on this lattice. A tetrad
can be defined by a triple [major,a,b] or [minor,a,b], where a and b
are integers. Adjacent lattice points associate pairs where the major
tetrad of one lattice point shares an interval with the minor tetrad of
the other, and vice-versa. Of course, the major/minor pairs themselves
share the interval 1-3/2.

>Here's something which strikes me as very interesting and potentially
>useful. I've talked a lot about the 3D cubic lattice of 7-limit
>tetrads, which has the property that adjacent lattice points represent
>tetrads with an interval in common, and all such common interval
>pairings are represented by adjacent tetrads.
>
>The major/minor pairings work as follows: for a lattice point [a,b,c],
>if a+b+c is even, so that the tetrad is otonal, the minor version of
>the chord is [a-1,b,c]. If a+b+c is odd, so that the tetrad is utonal,
>the major version of the chord is [a+1,b,c]. Hence, the first coordinate
>is the one which relates major and minor.
>
>Suppose we take the sublattice generated by the other two coordinates,
>[0,b,c] for b, c integers. This represents an infinitesimal proportion
>of the total number of tetrads; however, if we breed temper, that is,
>temper out 2401/2400, it now represents a subgroup of index two. That
>is, half of all tetrads are of this form. If [0,a,b] represents a major
>tetrad (a+b is even) then [-1,a,b] represents the corresponding minor
>tetrad. If it represents a minor (utonal) tetrad. then [1,a,b] is the
>corresponding major tetrad. If we put the major and minor tetrads
>together, we can define a lattice whose points represent major/minor
>pairs of tetrads, breed tempered. The remarkable thing now is that
>*every* tetrad pair is represented by a point on this lattice. A tetrad
>can be defined by a triple [major,a,b] or [minor,a,b], where a and b
>are integers. Adjacent lattice points associate pairs where the major
>tetrad of one lattice point shares an interval with the minor tetrad of
>the other, and vice-versa. Of course, the major/minor pairs themselves
>share the interval 1-3/2.

This is cool.

-Carl