Combination Product Set

In music theory, the hexany is a six-note just intonation structure, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube. The notes are arranged so that every edge of the octahedron joins together notes that make a consonant dyad, and every face joins together the notes of a consonant triad.

This makes a "musical geometry" with the geometrical form of the octahedron. It has eight just intonation triads in a scale of only six notes, and each triad has two notes in common with three of the other chords, arranged in a musically symmetrical fashion due to the symmetry of the octahedron on which it is based. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series. The points represent musical notes and the three notes that make each of the triangular faces represent musical triads.

It's constructed by taking four musical intervals, one of which can optionally be the unison, and then combining them in pairs, in all possible ways. So for instance if you start with 1/1, 3/1, 5/1 and 7/1 then combine them in pairs you get 1*3, 1*5, 1*7, 3*5, 3*7, 5*7 and those are the notes of the 1, 3, 5, 7 hexany. The notes are often octave shifted to place them all within the same octave, which has no effect on interval relations and the consonance of the triads.

The 1, 3, 5, 7 hexany is found within any 3D cubic lattice of musical pitches, and so within the three factor Euler–Fokker genus based on a cube. If none of the intervals used to construct it is the unity then you need to go into four dimensions and the four factor Euler–Fokker genus based on a hypercube or tesseract. An example of this is the 3, 5, 7, 11 hexany. The result is still a three dimensional figure, the octahedron, with vertices 3*5, 3*7, 3*11, 5*7, 5*11, 7*11. However when you embed it in the four factor Euler Fokker genus and then represent this in 4D, the result is a 3D cross section of a hypercube. You can have 3D cross sections of a 4D shape much as you can obtain a triangle as a 2D cross section of a normal 3D cube.

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