The seven-limit lattices

The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a lattice by putting a norm on the three dimensional real vector space we can regard them as living in. If we define the norm by

|| (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)

then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0), +-(0 1 0), +-(0 0 1), +-(1 -1 0), +-(1 0 -1) and +-(0 1 -1) at a distance of one from the unison, (0 0 0). These lie on the verticies of a cubeoctahedron, a semiregular solid. The lattice has two types of holes--the shallow holes, which are tetrahera and which correspond to the major and minor tetrads 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are octaheda and correspond to hexanies.

A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form x_i x_j, summed over all i <= j; moreover as an alternative approach we can use the Hahn norm in place of the Euclidean norm. In the two dimensional case of the 5-limit, this gives the plane lattice of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another family of lattices, called Dn, to which it also belongs as D3, theface-centered cubic lattice. If we take (b+c)^2+(a+c)^2+(a+b)^2 and expand it, we get 2 (a^2 + b^2 + c^2 + ab + ac + bc). If we therefore take our triples (a b c) and change basis by sending (1 0 0) to (0 1 1), (0 1 0) to (1 0 1), and (0 0 1) to (1 1 0), we have the lattice in terms of perpendicular coordinates, in which we may use ordinary Euclidean length. In this form, all distances are scaled up by a factor of sqrt(2), so that the 7-limit consonances become (+-1 +-1 0), (+-1 0 +-1), and (0 +-1 +-1), the verticies of a cuboctahedron in a more standard form. The lattice now may be described as triples of integers (a b c), such that a+b+c is an even number, and using the ordinary Euclidean norm of sqrt(a^2 + b^2 + c^2).

In this new coordinate system, the 4:5:6:7 tetrad consists of the notes (0 0 0), (1 0 0), (0 1 0), and (0 0 1); the centroid of this is (1/2 1/2 1/2); similarly the centroid of 1/4:1/5:1/6:1/7 is (-1/2 -1/2 -1/2). If we shift the origin to (1/2 1/2 1/2), major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This, once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form a lattice.

If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+c-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to a stellated hexany, or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.

If we look at twice the generators, namely [2 0 0], [0 2 0] and [0 0 2] we find they correspond to transposition up by 35/24 for [2 0 0], up 21/20 for [0 2 0], and up 15/14 for [0 0 2]. Temperaments where the generator can be taken as one of these three, such as miracle, are particularly easy to work with in terms of the lattice of chord relations because of this.

In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the An norm defined from x_j x_j is a norm defined by the inverse to the symmetric matrix of the quadratic form for the An norm, which normalizes to the square root of the quantity n times the sum of squares of x_i minus twice the product x_i x_j, for j > i. This defines the dual lattice An* to An. In the two dimensions of the 5-limit, A2 is isomorphic to A2* and the lattice of maps is a equilateral triangular ("hexagonal") lattice also. In the three dimensions of the 7-limit, we again have an exceptional situation, where A3* is isomorphic to the dual of D3, D3*. We have that the norm for A3* can be defined as the square root of (-x_1+x_2+x_3)^2 + (x_1-x_2+x_3)^2 +(x_1+x_2-x_3)^2, so if we change basis so that our basis maps are (-1 1 1), (1 -1 1) and (1 1 -1), then the norm becomes the usual Euclidean norm. If we take linear combinations with integer coefficents of these, we obtain all triples of integers which are either all even or all odd. The lattice with these points and the usual Euclidean norm is the body-centered cubic lattice.

It is easy to verify that the dot product of a triple of integers, either all even or all odd, times a triple of integers whose sum is even, is always even; and we get the precise relationship between mappings and note-classes by dividing by two, and taking the lattice of mappings to be triples of integers, plus triples of halves of odd integers. So for example the meantone mapping, (1 4 10), transforms to 1*(-1/2 1/2 1/2) + 4*(1/2 -1/2 1/2) + 10*(1/2 1/2 -1/2) = (13/2 7/2 -5/2), and the fifth class (1 0 0) to (0 1 1); taking the dot product of (13/2 7/2 -5/2) with (0 1 1) gives 1, as expected. However I think it is better to keep the coordinates as integers, and simply keep in mind that to get the mapping we now need to divide the dot product by two.

For any lattice, the isometries, or distance-preserving maps, which take the lattice to itself form a group, the group of affine automorphisms. It has a subgroup, called the automorphism group of the lattice, which consists of those affine automorphisms which fix the origin. In the case of D3, D3* and the cubic grid of tetrads, the automorphism group is the group of order 48 which consists of all permutations of the three coordinates and all changes of sign, and is called both the group of the cube and the group of the octahedron. It is easy to see that such a transformation takes triples with an even sum to triples with an even sum, and triples either even or odd to triples either even or odd. Hence it takes the cubic lattice of tetrads to itself, the face-centered cubic lattice of note-classes to itself, and the body-centered cubic lattice of mappings of note-classes to itself. The first two types of transformation includes the major/minor transformation, and can be regarded as a vast generalization of that. Robert Walker has a piece, Hexany Phrase, which takes a theme through all 48 resulting variations.

Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes interesting, since it sends one temperament to another while preserving 7-odd-limit (meaning, not including 9-odd-limit) harmony to itself. For example, the dominant seventh temperament, the {27/25, 28/25} temperament, and the {28/27, 35/32} temperaments can each be transformed to the others, as can septimal kleismic (the {49/48, 126/125} temperament) and the {225/224, 250/243} temperament, and hemifourths and the {49/48, 135/128} temperament. Temperaments with a period a fraction of an octave can also sometimes be transformed; for instance injera and the {50/49, 135/128} temperament.

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