Relationship To Pascal's Triangle
The complete row of Pascal's triangle for the hypercube in this construction runs 1 (single vertex), 4 (tetrahedron tetrad), 6 (hexany), 4 (another tetrad), 1. The idea generalises to other numbers of dimensions – for instance, the cross-sections of a five-dimensional cube give two versions of the dekany – a ten-note scale rich in tetrads, triads and dyads, which also contains many hexanies.
In six dimensions the same construction gives the twenty-note eikosany which is even richer in chords. It has pentads, tetrads, and triads as well as hexanies and dekanies.
In the case of the three-dimensional cube, it is usual to consider the entire cube as a single eight-note scale, the octany – the cross-sections then are 1, 3 (triad), 3 (another triad), 1, taken along any of the four main diagonals of the cube.
Read more about this topic: Hexany
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