Cube octahedron template
![cube octahedron template cube octahedron template](https://i.etsystatic.com/16285343/r/il/9660d0/1461973997/il_1588xN.1461973997_gqdn.jpg)
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. It has pentads, tetrads, and triads as well as hexanies and dekanies. In six dimensions the same construction gives the twenty-note eikosany, which is even richer in chords. 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. 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. If the 1/1 is 500 hertz, then 6/5 is 600 hertz, and so forth. The ratios' notation here shows the ratios of the frequencies of the notes. Since the harmonic construct as Erv called it as he did not consider it a scale and it does not have a 1/1 yet, any note chosen can be used to divide every note up to octave reduction. To make this into a conventional harmonic construct with 1/1 as the first note, all the notes are first reduced to the octave. 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. įor example, the face with vertices 3×5, 1×5, 5×7 is an otonal (major type) chord since it can be written as 5×(1, 3, 7), using low numbered harmonics. See also figure two of Kraig Grady's paper. In this 2D construction the interval relationships are the same.
![cube octahedron template cube octahedron template](https://img.wonderhowto.com/img/96/01/63456943797236/0/modular-origami-make-cube-octahedron-icosahedron-from-sonobe-units.w1456.jpg)
The hexany is the figure containing both the triangles shown as well as the connecting lines between them. Problems playing this file? See media help. Wilson also pointed out and explored the idea of melodic Hexanies. The points represent musical notes, and the three notes that make each of the triangular faces represent musical triads. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series. Each triad occurs just once with its inversion represented by the opposing 3 tones. It thus has eight just intonation triads where each triad has two notes in common with three of the other chords. The notes are arranged so that each point represents a pitch, each edge an interval and each face a triad. The hexany can be thought of as analogous to the octahedron. The possibility of an octave being a solution is not outside of Wilson's conception and is used in cases of placing larger combination product sets upon Generalized Keyboards. The notes are usually octave shifted to place them all within the same octave, which has no effect on interval relations and the consonance of the triads. For instance, the harmonic factors 1, 3, 5 and 7 are combined in pairs of 1*3, 1*5, 1*7, 3*5, 3*7, 5*7, resulting in 1, 3, 5, 7 Hexanies. It is constructed by taking any four factors and a set of two at a time, then multiplying them in pairs.
![cube octahedron template cube octahedron template](https://img.wonderhowto.com/img/79/97/63456968301334/0/modular-origami-make-cube-octahedron-icosahedron-from-sonobe-units.1280x600.jpg)
Simply, the hexany is the 2 out of 4 set. hexany is the name that Erv Wilson gave to the six notes in the 2-out-of-4 combination product set, abbreviated as 2*4 CPS. In this construction, the hexany is the third cross-section of the four-factor set and the first uncentered one. The numbers of vertices of his combination sets follow the numbers in Pascal's triangle. The Euler Fokker Genus fails to see 1 as a possible member of a set except as a starting point. While it is often and confusingly overlapped with the Euler–Fokker genus, the subsequent stellation of Wilson's combination product sets (CPS) are outside of that Genus. It achieves this by using consonant relations as opposed to the dissonance methods normally employed by atonality. It is referred to as an uncentered structure, meaning that it implies no tonic. In musical tuning systems, the hexany, invented by Erv Wilson, represents one of the simplest structures found in his combination product sets]]. Problems playing these files? See media help.