7-limit symmetrical lattices

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Of the various [[http://mathworld.wolfram.com/VectorNorm.html|norms]] which can be put on [[Monzos and Interval Space|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 //unwieghted// monzos, so that

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

If T is the [[Vals and Tuning Space|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 s two-dimensional subspace.

If |-x-y-z z 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|tonality diamond]] are represented by the unison |* 0 0 0> and twelve 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 [[http://en.wikipedia.org/wiki/Cuboctahedron|cubeoctahedron]], a semiregular solid. The lattice has two types of holes--the shallow holes, which are [[http://en.wikipedia.org/wiki/Tetrahedron|tetrahera]] and which correspond to the major and minor [[http://tonalsoft.com/enc/tetrad.htm|tetrads]] 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are [[http://en.wikipedia.org/wiki/Octahedron|octaheda]] and 
correspond to [[http://tonalsoft.com/enc/hexany.htm|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 [[http://en.wikipedia.org/wiki/Crystal_structure|face-centered cubic lattice]]. 

The 4:5:6:7 major 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+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 [[http://tonalsoft.com/enc/stellat.htm|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 [[http://en.wikipedia.org/wiki/Quadratic_form|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 [[http://en.wikipedia.org/wiki/Crystal_structure|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, [[http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase|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.

Original HTML content:

<html><head><title>The Seven Limit Symmetrical Lattices</title></head><body>Of the various <a class="wiki_link_ext" href="http://mathworld.wolfram.com/VectorNorm.html" rel="nofollow">norms</a> which can be put on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> 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 <em>unwieghted</em> monzos, so that<br />
<br />
|| |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)<br />
<br />
If T is the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> T = &lt;1 1 1 1| (note that this is <em>not</em> the JI point) then we may define a subspace of 7-limit interval space, <strong>symmetric interval class space</strong>, as the subspace of all vectors M in interval space such that &lt;T|M&gt; = 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 s two-dimensional subspace.<br />
<br />
If |-x-y-z z y z&gt; is any element of symmetric interval class space, then by definition<br />
<br />
|| |-x-y-z x y z&gt; || = sqrt(2) sqrt(x^2)y^2+z^2+xy+yz+zx)<br />
<br />
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&gt;.<br />
<br />
<br />
The thirteen intervals of the 7-limit <a class="wiki_link" href="/Tonality%20Diamond">tonality diamond</a> are represented by the unison |* 0 0 0&gt; and twelve twelve lattice points at a distance of one from the unison, given by +-|* 1 0 0&gt;, +-|* 0 1 0&gt;, +-|* 0 0 1&gt;, +-|* 1 -1 0&gt;, +-|* 1 0 -1&gt; and +-|* 0 1 -1&gt;. These lie on the verticies of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow">cubeoctahedron</a>, a semiregular solid. The lattice has two types of holes--the shallow holes, which are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow">tetrahera</a> and which correspond to the major and minor <a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow">tetrads</a> 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow">octaheda</a> and <br />
correspond to <a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">hexanies</a>.<br />
<br />
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 <br />
to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, <br />
in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another <br />
family of lattices, called Dn, to which it also belongs as D3, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>. <br />
<br />
The 4:5:6:7 major tetrad consists of the notes |* 0 0 0), |* 1 0 0&gt;, |* 0 1 0&gt;, and |* 0 0 1&gt;; the centroid of this is |* 1/2 1/2 1/2&gt;; similarly the centroid of 1/4:1/5:1/6:1/7 is |* -1/2 -1/2 -1/2&gt;. If we shift the origin to |* 1/2 1/2 1/2&gt;, major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 <br />
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 <br />
of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This, <br />
once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form <br />
a lattice.<br />
<br />
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) <br />
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) <br />
if a+b+c is odd. Each unit cube corresponds to a <a class="wiki_link_ext" href="http://tonalsoft.com/enc/stellat.htm" rel="nofollow">stellated hexany</a>, <br />
or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.<br />
<br />
If we look at twice the generators, namely [2 0 0], [0 2 0] and [0 0 2] we find they correspond to transposition <br />
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 <br />
taken as one of these three, such as miracle, are particularly easy to work with in terms of the lattice of chord <br />
relations because of this.<br />
<br />
In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the <br />
An norm defined from x_j x_j is a norm defined by the inverse to the symmetric matrix of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quadratic_form" rel="nofollow">quadratic form</a> 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 &gt; i. This defines the dual lattice An* to An. In the two dimensions of <br />
the 5-limit, A2 is isomorphic to A2* and the lattice of maps is a equilateral triangular (&quot;hexagonal&quot;) 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 <br />
norm is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">body-centered cubic lattice</a>.<br />
<br />
It is easy to verify that the dot product of a triple of integers, either all even or all odd, times a triple <br />
of integers whose sum is even, is always even; and we get the precise relationship between mappings and note-classes <br />
by dividing by two, and taking the lattice of mappings to be triples of integers, plus triples of halves of odd <br />
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 <br />
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) <br />
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.<br />
<br />
For any lattice, the isometries, or distance-preserving maps, which take the lattice to itself form a group, <br />
the group of affine automorphisms. It has a subgroup, called the automorphism group of the lattice, which consists <br />
of those affine automorphisms which fix the origin. In the case of D3, D3* and the cubic grid of tetrads, the automorphism <br />
group is the group of order 48 which consists of all permutations of the three coordinates and all changes of sign, <br />
and is called both the group of the cube and the group of the octahedron. It is easy to see that such a transformation <br />
takes triples with an even sum to triples with an even sum, and triples either even or odd to triples either even <br />
or odd. Hence it takes the cubic lattice of tetrads to itself, the face-centered cubic lattice of note-classes <br />
to itself, and the body-centered cubic lattice of mappings of note-classes to itself. The first two types of transformation <br />
includes the major/minor transformation, and can be regarded as a vast generalization of that. Robert Walker has <br />
a piece, <a class="wiki_link_ext" href="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase" rel="nofollow">Hexany Phrase</a>, which takes <br />
a theme through all 48 resulting variations.<br />
Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes <br />
interesting, since it sends one temperament to another while preserving 7-odd-limit (meaning, not including 9-odd-limit) <br />
harmony to itself. For example, the dominant seventh temperament, the {27/25, 28/25} temperament, and the {28/27, <br />
35/32} temperaments can each be transformed to the others, as can septimal kleismic (the {49/48, 126/125} temperament) <br />
and the {225/224, 250/243} temperament, and hemifourths and the {49/48, 135/128} temperament. Temperaments with <br />
a period a fraction of an octave can also sometimes be transformed; for instance injera and the {50/49, 135/128} <br />
temperament.</body></html>