In advance of the International Computer Music Conference, NYC 2010, here is "Harmonic Hypercube [7-9-11-13]"

"a systematic articulation of the 4-dimensional harmonic hypercube [7-9-11-13] applied to pitch with complementary inversion. 1/1=293664764 Hz. @ 600 bpm = 51.2 seconds."

[An odd companion piece to the work Harmonic Hypercube [6-8-9-12] - as heard at Vox Novus' 60x60, presented as part of the Vermillion Mix and Canadian Mix 2010].

Given the harmonic matrix is a 2-dimensional plane H[n,n], then transposing every element within the matrix by every element of the same matrix results in the harmonic hypercube H[n,n]*H[n,n]=H^2. H^2 is a 4-dimensional object. Obviously, it has no real world physical representation. In the above video representation, the changing floor of the harmonic matrix represents the transposition levels of the hypercube, in this case, 7,9,11,13.

Harmonic cubes are somewhat simpler but arbitrary, as the harmonic matrix is multiplied by only a single row from a harmonic matrix [as opposed to the entire matrix itself].

For example taking the basic major-minor matrix [4,5,6]/[4,5,6], gives

4/4 5/4 6/4

4/5 5/5 6/5

4/6 5/6 6/6

If we transpose these 9 elements by the 9 elements, a hypercube of 81 elements is the result.

To form a harmonic cube, we choose only a single row, for example [4/4 5/4 6/4]:

4/4 5/4 6/4 20/16 25/16 30/16 24/16 30/16 36/16

4/5 5/5 6/5 20/20 25/20 30/20 24/20 30/20 36/20

4/6 5/6 6/6 20/24 25/24 30/24 24/24 30/24 36/24

These ratios can be reduced, but in general, I remain opposed to simplifying the already overly simple.