 Hexany

In music theory, the hexany is a sixnote just intonation scale, 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 it is based on.
The Hexany is the invention of Erv Wilson^{[1]} and represents one of the simplest structures found in his Combination Product Sets, which are obtained as successive cross sections of an ndimensional cube. The numbers of vertices follow the numbers in Pascal's triangle. The hexany is the third cross section of the fourdimensional cube. "Hexany is the name that Erv Wilson gave to the six notes in the 2outof4 combination product set (abbreviated as 2\4 CPOS)."^{[2]}
Contents
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 the vertices are shown connected (in green). The other two vertices join to them to make the octahedron. All the sides of the octahedron are diagonals of squares, so are the same length (root two), so it is a regular octahedron — the "squashed" appearance is because it is rotated into the fourth dimension.
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 fourdimensionally 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 lownumbered 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:
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 crosssections of a fivedimensional cube give two versions of the dekany – a tennote scale rich in tetrads, triads and dyads, which also contains many hexanies.
In six dimensions the same construction gives the twentynote eikosany which is even richer in chords. It has pentads, tetrads, and triads as well as hexanies and dekanies.
In the case of the threedimensional cube, it is usual to consider the entire cube as a single eightnote scale, the octany – the crosssections then are 1, 3 (triad), 3 (another triad), 1, taken along any of the four main diagonals of the cube.
Coordinates for the Pascal's triangle of combination product sets
First row (square):
00
10 01
11Second row (cube or octony):
000
100 010 001 triad (triangle)
110 101 011 triad (triangle)
111Third row (hypercube)
0000
1000 0100 0010 0001 tetrad (tetrahedron or 3simplex)
1100 1010 1001 0110 0101 0011 hexany (octahedron)
1110 1101 1011 0111 tetrad
1111The octahedron there is the edge dual of the tetrahedron, or rectified tetrahedron
Fourth row (5dimensional cube)
00000
10000 01000 00100 00010 00001 pentad (4simplex or pentachoron  four dimensional tetrahedron)
11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 2)5 dekany (10 vertices, rectified 4simplex)
00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 3)5 dekany (10 vertices)
01111 10111 11011 11101 11110 pentad
11111The rectified 4simplex for the dekany is also known as the dispentachoron
Fifth row (6dimensional cube
000000
100000 010000 001000 000100 000010 000001 hexad (5simplex or hexateron  five dimensional tetrahedron)
110000 101000 100100 100010 100001 011000 010100 010010 010001 001100 001010 001001 000110 000101 000011 2)6 pentadekany (15 vertices, rectified 5simplex)
111000 110100 110010 110001 101100 101010 101001 100110 100101 100011 011100 011010 011001 010110 010101 010011 001110 001101 001011 000111 eikosany (20 vertices birectified 5simplex)
001111 010111 011011 011101 011110 100111 101011 101101 101110 110011 110101 110110 111001 111010 111100 4)6 pentadekany (15 vertices)
011111 101111 110111 111011 111101 111110 hexad
111111It is easy to see that the geometric figure for the dekany is the edge dual of the 4simplex and the one for the pentadekany is the edge dual of the 5simplex.
To see this, in the figure of the octahedron in the hypercube, scale the entire figure by 1/2 about the origin (blue vertex). The octahedron vertices will move to the midpoints of the original tetrahedron edges (joining the red vertices in the figure).
So  similarly the dekany vertices when scaled by 1/2 move to the midpoints of the 4simplex edges, and the pentadekany vertices move to the midpoints of the 5simplex edges, and so on in all the higher dimensions.
The eikosany vertices when scaled by 1/3 move to the centres of the 2D faces of the 5simplex. To see that, note that in a 3D cube, 111 when scaled by 1/3 moves to the midpoint of 100 010 001 (each edge vector subtends the same distance along the long diagonal of the cube). So 11100 moves to the centre of the equilateral triangle with coords 10000 01000 00100 and similarly for all the other eikosany vertices.
So  the geometric figure for the eikosany is the 2D face dual of the 5simplex or birectified 5simplex. Similarly for the 3)7, 3)8 etc figures in all higher dimensions.
Similarly in eight dimensions, the figure you get using all permutations of 4 out of 8 is the 3D face dual of the 7simplex, or 3rectified 7simplex (since 1111 scaled by 1/4 moves to the centre of the 3D regular tetrahedron face 1000 0100 0010 0001), and so on.
Composers
Composers including Kraig Grady, Daniel James Wolf, and Joseph Pehrson have used pitch structures based on hexanies.
See also
References
 ^ Chalmers, John H. (1993). Divisions of the Tetrachord: a Prolegomenon to the Construction of Musical Scales, p.116. Frog Peak Music. ISBN 9780945996040.
 ^ ^{[author missing]} (1993). Musicworks, Issues 5560, p.43. Music Gallery.
Further reading
 Grady, Kraig (1991), "Ervin Wilson's Hexany", Just Intonation 7 (1): 8–11
 Schiemer, Greg, "Tempered Dekanies: Chorus effect using microtonal intervals based on just intonation" (^{[dead link]}), Proceedings of the 7th International Conference on Music Perception and Cognition, Sydney, 2002: 300–302, http://www.uow.info/crearts/staff/schiemer/ICMPC7.pdf (see the Background section)
External links
 "Some Hexany and Hexany Diamond Lattices (and Blanks)", The Wilson Archives. Original Hexany papers showing different facets and configurations, not assembled by Erv Wilson (1967 on)
 "The Wilson Archives", Anaphoria.com
 "Hexany", RobertInventor.com. With a hexany you can turn around and click on any of its vertices, edges, or faces to hear the chords.
 "CombinationProduct Set Patterns", Xenharmonikon IX (1986) by Kraig Grady.
 "Eikosany Papers", Anaphoria.com.
 "Musical Geometry", Music and Virtual Flowers. Intro. to musical geometry.
 "Alicia Boole Stott", The MacTutor History of Mathematics archive. Biography of Alicia Stott
 "The Tumbling Dekany", "Unusual musical scales", Dave Keenan's Home Page. Dave Keenan's Dekany tumbling in 4 dimensions — as a musical Excel spreadsheet
Musical tunings Measurement Just intonation Euler–Fokker genus · Harmonic Scale · Harry Partch's 43tone scale · Hexany · Pythagorean tuning · Scale of harmonicsTemperaments IrregularTraditional nonWestern Chinese musicology · Dastgah · Maqam (Arabian maqam · Makam · Mugham · Muqam) · Octoechos · Pelog · Raga (Carnatic rāga) · Slendro · TetrachordNonoctave Categories: Multidimensional geometry
 Just tunings
 Musical scales
Wikimedia Foundation. 2010.
Look at other dictionaries:
Musical tuning — Open string redirects here. For the concept in string theory, see String (physics). Two differently tuned thirds: Just major third … Wikipedia
Kraig Grady — is a US Australian composer. He has composed and performed with an ensemble of microtonal instruments of his own design and also worked as a shadow puppeteer, tuning theorist, filmmaker, world music radio DJ and concert promoter. His works… … Wikipedia
Joseph Pehrson — (b. Detroit, 1950) is an American composer pianist who has written works for a wide variety of media including orchestra and chamber works. They have been performed at numerous venues including Merkin Hall, Weill Recital Hall, Symphony Space in… … Wikipedia
Euler–Fokker genus — Rectangular parallelepiped. In music theory and tuning, an Euler–Fokker genus (plural: genera), named after Leonhard Euler and Adriaan Fokker,[1] is a musical scale in just intonation whose pitches can be expressed as … Wikipedia
Just intonation — In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic… … Wikipedia
Octahedron — For the album by The Mars Volta, see Octahedron (album). Regular Octahedron (Click here for rotating model) Type Platonic solid Elements F = 8, E = 12 V = 6 (χ = 2) Faces by sides … Wikipedia
Musical scale — For psychoacoustic scale, see bark scale and mel scale. Pattern of whole and half steps in the Ionian mode or major scale on C … Wikipedia
Meantone temperament — is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal … Wikipedia
Well temperament — (also circular or circulating temperament) is a type of tempered tuning described in 20th century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach s famous composition, The Well Tempered … Wikipedia
Whole tone scale — Qualities Number of pitch classes 6 Maximal evenness Degenerate well formed collection In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole … Wikipedia