Chords of Blackjack

The complete 7-limit o/u-tonalities, or tetrads, as well as the
complete 9-limit o/u-tonalities, form a rectangular lattice in a
natural way. We can reduce by a comma set in this lattice to minimize
the distance (Euclidean or what have you) of the chord from [0,0,0],
the major tonic. If we do this to the 16 tetrads of Blackjack, we find
we have two pairs of tetrads--{[1,1,-2], [-1,-2,1]} and {[-1,-1,2],
[1,2,-1]}--with identical Euclidean distances from [0,0,0], which
represent the same chord of Blackjack. Choosing +-[1,2,-1] as our
representative gives us this as the chords of Blackjack in reduced form:

{[1, 2, 0], [0, 0, 1], [-1, -1, 0], [1, 1, -1], [1, 2, 1], [0, 0, 0],
[0, 0, 2], [1, 2, -1], [1, 1, 0], [-1, -1, 1], [0, 1, -2], [-1, -2,
1], [0, 0, -1], [-1, -1, -1], [1, 1, 1], [0, 0, -2]}

Should anyone feel inspired to draw a 3D diagram of this, it would be
interesting to see what it looked like.