In
musical tuning systems, the hexany, invented by
Erv Wilson
Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexican/ American (dual citizen) music theorist.
Early life
Ervin Wilson was born iColonia Pacheco a small village in the remote mountains of northwest Chihuahua, Mexico, where he lived u ...
, represents one of the simplest structures found in his combination product
sets.
It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this by using consonant relations as opposed to the dissonance methods normally employed by atonality. 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. The Euler Fokker Genus fails to see 1 as a possible member of a set except as a starting point. The numbers of
vertices of his combination sets follow the numbers in
Pascal's triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
. In this construction, the hexany is the third cross-section of the four-factor set and the first uncentered one. 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.
Simply, the hexany is the 2 out of 4 set. It is constructed by taking any four factors and a set of two at a time, then multiplying them in pairs. 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. 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. 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 hexany can be thought of as analogous to the
octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
. The notes are arranged so that each point represents a pitch, each
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
an interval and each face a triad. It thus has eight just intonation triads where each triad has two notes in common with three of the other chords. Each triad occurs just once with its inversion represented by the opposing 3 tones. 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. Wilson also pointed out and explored the idea of melodic Hexanies.
Tuning
This shows the three dimensional version of the hexany.

The hexany is the figure containing both the triangles shown as well as the connecting lines between them.
In this 2D construction the interval relationships are the same. See also figure two of Kraig Grady's paper.
For 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
harmonic
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
s. 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
subharmonic
In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones mus ...
s.
To make this into a conventional harmonic construct with 1/1 as the first note, all the notes are first reduced to the octave. 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. The ratios' notation here shows the ratios of the frequencies of the notes. If the 1/1 is 500 hertz, then 6/5 is 600 hertz, and so forth.
In music
Composers including
Kraig Grady
Kraig Grady (born 1952) is a US-Australian composer/ sound artist. 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 D ...
,
Daniel James Wolf
Daniel James Wolf (born September 13, 1961 in Upland, California) is an American composer.
Studies
Wolf studied composition with Gordon Mumma, Alvin Lucier, and La Monte Young, as well as musical tunings with Erv Wilson and Douglas Leedy an ...
, and
Joseph Pehrson have used pitch structures based on hexanies.
See also
*
Euler–Fokker genus
References
Further reading
*
*
* (see the Background section)
*
External links
"Some hexany and hexany Diamond Lattices (and Blanks)" '. Original hexany papers showing different facets and configurations, not assembled by Erv Wilson (1967 on)
''Anaphoria.com''
''RobertInventor.com''. With a hexany you can turn around and click on any of its vertices, edges, or faces to hear the chords.
"Combination-Product Set Patterns" ''Xenharmonikon IX'' (1986) by Kraig Grady.
''Anaphoria.com''.
''Music and Virtual Flowers''. Intro. to musical geometry.
"Unusual musical scales", ''Dave Keenan's Home Page''. Dave Keenan's Dekany tumbling in 4 dimensions — as a musical Excel spreadsheet
{{Musical tuning
Multi-dimensional geometry
Just tuning and intervals
Hexatonic scales
Hexachords