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Norms and lattices | The cubic lattice of tetrads | Dual lattices | Transformations | Articles

Norms and lattices

Of the various norms which can be put on interval space which then make the monzos into a lattice, the most useful seem to be the L1 and L2 norms on the coordinates weighted by log2 of the primes. However, in the 5 and 7 limit cases, it is sometimes convenient, when emphasizing symmetry properties, to put a Euclidean norm on unweighted monzos, so that

|| |e2 e2 e5 e7> || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)

If T is the val T = <1 1 1 1| (note that this is not the JI point) then we may define a subspace of 7-limit interval space, symmetric interval class space, as the subspace of all vectors M in interval space such that <T|M> = 0, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in a two-dimensional subspace.

If |-x-y-z x y z> is any element of symmetric interval class space, then by definition || |-x-y-z x y z> || = sqrt(2) sqrt(x^2+y^2+z^2+xy+yz+zx) where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by |* x y z>.

The thirteen intervals of the 7-limit tonality diamond are represented by the unison |* 0 0 0> and twelve lattice points at a distance of one from the unison, given by ∓|* 1 0 0>, ∓|* 0 1 0>, ∓|* 0 0 1>, ∓|* 1 -1 0>, ∓|* 1 0 -1> and ∓|* 0 1 -1>. 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 octahedra and correspond to hexanies.

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, the face-centered cubic lattice.

The cubic lattice of tetrads

Suppose m0, m1, m2 and m3 are four monzos denoting four notes of a 7-limit tetrad, either otonal or utonal, with distinct pitch classes, so that all four chord elements are represented. The product of the four notes is represented by the monzo m = m0+m1+m2+m3, and the pitch class for m, which ignores the first coefficient m[1] defining octaves, is represented by the other three, m[2], m[3] and m[4]. This pitch class is uniquely associated to the tetrad, and can be used to name it. If the tetrad is otonal with the root being |* e3 e5 e7>, where the asterisk can be any integer value, then m = |* 4e3+1 4e5+1 4e7+1>. on the other hand, if it is utonal, with the fifth of the chord |* e3 e5 e7> then m = |* 4e3-1 4e5-1 4e7-1>. If we denote the 3-tuple [(m[3]+m[4]-2)/2 (m[2]+m[4]-2)/2 (m[2]+m[3]-2)/2] by [a b c], then if the tetrad is otonal, [a b c] = [e5+e7 e3+e7 e3+e5], whereas if it is utonal [a b c] = [e5+e7-1 e3+e7-1 e3+e5-1]. From this it follows that [a b c] is a triple of integers, and that a+b+c is even if the tetrad is otonal, and odd if the tetrad is utonal.

Hence every triple of integers refers uniquely to a 7-limit tetrad; this is the lattice of 7-limit tetrads. 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 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+b-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.

Dual lattices

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 coefficients 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 all even or all odd to triples either all even or all 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, dominant, the {36/35, 64/63} temperament, 5&10c, the {27/25, 28/25} temperament, and 5c&7d, the {28/27, 35/32} temperament, can each be transformed to the others, as can keemun (the {49/48, 126/125} temperament) with porky, the {225/224, 250/243} temperament, and godzilla, the {49/48, 81/80} temperament with superpelog the {49/48, 135/128} temperament. Temperaments with a period a fraction of an octave can also sometimes be transformed; for instance injera, the {50/49, 81/80} temperament and bipelog, the {50/49, 135/128} temperament.