Tuning
The four dimensions of the hypercube are usually tuned to distinct primes (sometimes to odd numbers) and a single step in each dimension corresponds to multiplying the frequency by that prime. The notes are then usually reduced to the octave (by repeated division by 2) using octave equivalence.
For example, for a 2 3 5 7 hexany, assign 2 3 5 7, to the four dimensions. Then to obtain the octahedron as a diagonal cross section of the hypercube, use all permutations of (1,1,0,0) as the coords. There for instance, (0,0,1,1) moves one step in the "5" dimension and one step in the "7" dimension and so would be tuned as 5×7.
So, to make the complete hexany, multiply the primes together in pairs to give six numbers: 2×3, 2×5, 2×7, 3×5, 3×7, and 5×7 (or 2×3×1×1, 2×1×5×1, 2×1×1×7, 1×3×5×1, 1×3×1×7 and 1×1×5×7). This shows the context in 4D.
In this picture of a hypercube, the six hexany vertices are shown in yellow, and four of these vertices are shown connected (in green). The other two vertices join to them to make the octahedron. It doesn't look like a perfect octahedron because we aren't used to interpreting 2D drawings of 4D pictures, but the "squashed" appearance is because it is rotated into the fourth dimension. All the quadrilaterals in this picture represent perfect squares, and you can see that all the sides of the octahedron are diagonals of perfect squares. This shows that its edges are all the same length (root two), which makes it a regular octahedron.
You can see the tetrahedral slices of the hypercube similarly - the red vertices can be joined together to make a regular tetrahedron, and the purple vertices likewise. So going from one of the blue points to the other you have 1 vertex, 4 for the red tetrahedron, 6 vertices for the yellow octahedron (hexany), 4 for the purple tetrahedron and 1 more vertex to make up the complete cube.
If one finds it a bit baffling that's to be expected — a few people, like Alicia Stott, have been able to think four-dimensionally but it is beyond most of us.
Then for example the face with vertices 3×5, 2×5, 5×7 is an otonal (major type) chord since it can be written as 5×(2, 3, 7), using low numbered harmonics. The 5×7, 3×7, 3×5 is a utonal (minor type) chord since it can be written as 3×5×7×(1/3, 1/5, 1/7), using low-numbered subharmonics.
Musical lattices are often constructed with the octave dimension omitted. Then the hexanies show up in the 3D lattices as octahedra between the alternating otonal and utonal tetrahedra (for tetrads). However the octave (2) dimension is shown in the diagram above to bring out its 4D context, and help make the connection with the Pascal's triangle construction via the hypercube.
To make this into a conventional scale with 1/1 as the first note, first reduce all the notes to the octave. Since the scale doesn't have a 1/1 yet, choose one of the notes, it doesn't matter which. Let's choose 5×7. Divide all the notes by 5×7 to get: 1/1 8/7 6/5 48/35 8/5 12/7 2/1 (up to octave reduction). The ratios notation here shows the ratio of the frequencies of the notes. So for instance if the 1/1 is 500 hertz, then 6/5 is 600 hertz, and so forth.
This figure shows the hexany in its more usual 3D representation:
Read more about this topic: Hexany