Hexany

Regular Octahedron.

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.

The Hexany is the invention of Erv Wilson[1] and represents one of the simplest structures found in his Combination Product Sets. The numbers of vertices of his combination sets when set out as subdivisions of a Euler-Fokker genus follow the numbers in Pascal's triangle. IN this construction, the hexany is the third cross section of the four factor genus. "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)."[2]

Tuning

This shows the three dimensional version of the hexany.

Orthographic projection.

Here is another diagram showing how the hexany can be found in the three factor Euler Fokker genus. The hexany is the figure containing both the triangles shown as well as the connecting lines between them.

Note - in this 3D construction it is not visually a perfect octahedron - it is somewhat "squashed". But that is an inessential difference as the interval relationships are the same. See also figure 2 of Kraig Grady's paper.[3]

In its most general form the hexany is embedded in a four factor Euler–Fokker genus, or geometrically, a hypercube, also called a tesseract. The four dimensions of the hypercube are often tuned to distinct primes (sometimes more generally to odd numbers) to achieve a hexany with maximally consonant triads. A single step in each dimension corresponds to multiplying the frequency by that prime (or odd number). The notes are then usually reduced to the octave (by repeated division by 2) using octave equivalence. However hexanies will occur naturally in any tuning that has four generators.

The 1, 3, 5, 7 hexany can also be shown as embedded in a four factor Eular Fokker genus, by using a 2 instead of a 1 as one of the factors. This makes no difference when it is reduced to the octave. So 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 diagram the central hexany has its vertices in yellow and they are connected as shown as a square with the two remaining vertices joined to them to make a regular octahedron. It looks distorted for much the same reason that the squares of a cube look distorted in 2D projection.

One of the tetrads is shown with red vertices and one is shown with purple vertices. Each red vertex is joined to all the other red vertices to make a regular tetrahedron. Similarly each blue vertex is joined to all the other blue ones to make another regular tetrahedron. Those lines are not shown as they would make the diagram confusing to read.

The hexany looks distorted because of issues with drawing a 4D shape in perspective in 3D but is in fact a perfect octahedron and the tetrads are perfect tetrahedra.[4]

In this picture of a hypercube, the six hexany vertices are shown in yellow, and four of these vertices are shown connected (in green). The other two vertices join to them to make the octahedron. It doesn't look like a perfect octahedron because we aren't used to interpreting 2D drawings of 4D pictures, but the "squashed" appearance is because it is rotated into the fourth dimension. All the quadrilaterals in this picture represent perfect squares, and you can see that all the sides of the octahedron are diagonals of perfect squares. This shows that its edges are all the same length (root two). The points are also all in the same 3D plane as they are all at a distance of root two from the origin. So it can't be folded into the fourth dimension in any way, so it is a regular octahedron, and a 3D figure just tilted into the fourth dimension (just as the 3D cube above shows two triangles tilted into the third 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.

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 low-numbered 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.

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.[5] 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.[5]

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.

Coordinates for the Pascal's triangle of combination product sets

First row (square):
00
10 01
11

Second row (cube or octony):
000
100 010 001 triad (triangle)
110 101 011 triad (triangle)
111

Third row (hypercube)
0000
1000 0100 0010 0001 tetrad (tetrahedron or 3-simplex)
1100 1010 1001 0110 0101 0011 hexany (octahedron)
1110 1101 1011 0111 tetrad
1111

The octahedron there is the edge dual of the tetrahedron, or rectified tetrahedron

Fourth row (5-dimensional cube)
00000
10000 01000 00100 00010 00001 pentad (4-simplex or pentachoron - four-dimensional tetrahedron)
11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 2)5 dekany (10 vertices, rectified 4-simplex)
00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 3)5 dekany (10 vertices)
01111 10111 11011 11101 11110 pentad
11111

The rectified 4-simplex which is the mathematical name for the geometrical shape of the dekany is also known as the dispentachoron

Fifth row (6-dimensional cube
000000
100000 010000 001000 000100 000010 000001 hexad (5-simplex 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 5-simplex)
111000 110100 110010 110001 101100 101010 101001 100110 100101 100011 011100 011010 011001 010110 010101 010011 001110 001101 001011 000111 eikosany (20 vertices birectified 5-simplex)
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
111111

There's an interesting connection with the 4-simplex. This is the four dimensional generalization of the triangle and the tetrahedron, and is got by adding a fourth point equidistant from all the vertices of a tetrahedron in the fourth dimension and the result is the 4-simplex, and if you add a fifth point similarly in the fifth dimension and you get a 5-simplex and so on.

The connection is that the geometric figure for the dekany is the edge dual of the 4-simplex. Similarly the geometrical figure for the pentadekany is the edge dual of the 5-simplex. I.e. you can make a dekany by joining together the midpoints of the edges of the 4-simplex, and similarly for the pentadekany and the 5-simplex.

Scale the yellow vertices of the hexany by 1/2 relative to the blue vertex. They will move to the mid points between the red dots of the tetrad. This shows that the octahedron is the "edge dual" of the tetrahedron. By a similar construction it's easy to see that the dekany is the edge dual of a 4-simplex and the pentadekany is the edge dual of a 5-simplex. By rather similar reasoning it's also possible to show that the eikosany is the face dual of the 5-simplex

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 4-simplex edges, and the pentadekany vertices move to the midpoints of the 5-simplex 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 5-simplex. 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 face dual of the 5-simplex or birectified 5-simplex. That's the dual of its 2D faces as of course it also has 3D and 4D facets.

It is a similar picture 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 7-simplex, or 3-rectified 7-simplex (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

  1. ↑ Chalmers, John H. (1993). Divisions of the Tetrachord: a Prolegomenon to the Construction of Musical Scales, p.116. Frog Peak Music. ISBN 978-0-945996-04-0.
  2. ↑ (1993). Musicworks, Issues 55-60, p.43. Music Gallery.
  3. ↑ Grady, Kraig (1991). "Ervin Wilson's Hexany" (PDF). Just Intonation. 7 (1): 8–11.
  4. ↑ Wilson, Erv. "D'Alessandro, Like a Hurricane, Figures 6b and 6c" (PDF). Xenharmonikon. 12: 10.
  5. 1 2 Wilson, Erv. "D'Alessandro, Like a Hurricane, Figures 6b, 6c, 19, 20 and 20b" (PDF). Xenharmonikon. 12: 10,21.

Further reading

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