7-limit symmetrical lattices: Difference between revisions
Wikispaces>genewardsmith **Imported revision 230633574 - Original comment: ** |
Wikispaces>clumma **Imported revision 245784087 - Original comment: spelling and linebreaks** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:clumma|clumma]] and made on <tt>2011-08-13 15:17:39 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>245784087</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>spelling and linebreaks</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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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>. | 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|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 [[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| | The thirteen intervals of the 7-limit [[Tonality Diamond|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 [[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|octahedra]] and correspond to [[http://tonalsoft.com/enc/hexany.htm|hexanies]]. | ||
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]]. | 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]]. | ||
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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. | 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 | 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 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 [[http://en.wikipedia.org/wiki/Crystal_structure|body-centered cubic lattice]]. | ||
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. | 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. | ||
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If |-x-y-z x y z&gt; is any element of symmetric interval class space, then by definition || |-x-y-z x y z&gt; || = 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&gt;.<br /> | If |-x-y-z x y z&gt; is any element of symmetric interval class space, then by definition || |-x-y-z x y z&gt; || = 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&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 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"> | 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 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">octahedra</a> and correspond to <a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">hexanies</a>.<br /> | ||
correspond to <a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">hexanies</a>.<br /> | |||
<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 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>.<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 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>.<br /> | ||
| Line 58: | Line 55: | ||
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.<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 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.<br /> | ||
<br /> | <br /> | ||
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 <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 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 | 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 <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 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 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">body-centered cubic lattice</a>.<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 /> | <br /> | ||
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.<br /> | 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.<br /> | ||