Rank-3 temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 141050747 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 141051507 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 06: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 06:44:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>141051507</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></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> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]] | ||
triples of integers (a b c). We can make this into a [[ | by putting a [[http://en.wikipedia.org/wiki/Normed_vector_space|norm]] on the three dimensional real we can regard them as living in. If we define | ||
by putting a | |||
the norm by | the norm by | ||
|| (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc) | || (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc) | ||
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temperament.<P ALIGN="CENTER"><P ALIGN="CENTER"><A HREF="home.htm">home</A></BODY></HTML></pre></div> | temperament.<P ALIGN="CENTER"><P ALIGN="CENTER"><A HREF="home.htm">home</A></BODY></HTML></pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Planar Temperament</title></head><body>The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Planar Temperament</title></head><body>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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a> <br /> | ||
triples of integers (a b c). We can make this into a | by putting a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">norm</a> on the three dimensional real we can regard them as living in. If we define <br /> | ||
by putting a | |||
the norm by<br /> | the norm by<br /> | ||
|| (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)<br /> | || (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)<br /> | ||
then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0), <br /> | then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0), <br /> | ||
+-(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 <br /> | +-(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 <br /> | ||
lie on the verticies of a &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | lie on the verticies of a &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:77:http://en.wikipedia.org/wiki/Cuboctahedron --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow">http://en.wikipedia.org/wiki/Cuboctahedron</a><!-- ws:end:WikiTextUrlRule:77 -->&quot;&gt;cubeoctahedron&lt;/A&gt;, a semiregular <br /> | ||
solid. The lattice has two types of holes--the shallow holes, which are &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | solid. The lattice has two types of holes--the shallow holes, which are &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:78:http://en.wikipedia.org/wiki/Tetrahedron --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow">http://en.wikipedia.org/wiki/Tetrahedron</a><!-- ws:end:WikiTextUrlRule:78 -->&quot;&gt;tetrahera&lt;/A&gt; <br /> | ||
and which correspond to the major and minor &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | and which correspond to the major and minor &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:79:http://tonalsoft.com/enc/tetrad.htm --><a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow">http://tonalsoft.com/enc/tetrad.htm</a><!-- ws:end:WikiTextUrlRule:79 -->&quot;&gt;tetrads&lt;/A&gt; 4:5:6:7 and <br /> | ||
1/4:1/5:1/6:1/7, and the deep holes which are &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | 1/4:1/5:1/6:1/7, and the deep holes which are &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:80:http://en.wikipedia.org/wiki/Octahedron --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow">http://en.wikipedia.org/wiki/Octahedron</a><!-- ws:end:WikiTextUrlRule:80 -->&quot;&gt;octaheda&lt;/A&gt; and <br /> | ||
correspond to &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | correspond to &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:81:http://tonalsoft.com/enc/hexany.htm --><a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">http://tonalsoft.com/enc/hexany.htm</a><!-- ws:end:WikiTextUrlRule:81 -->&quot;&gt;hexanies&lt;/A&gt;.<br /> | ||
A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form <br /> | A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form <br /> | ||
x_i x_j, summed over all i &lt;= j; moreover as an alternative approach we can use the &lt;A HREF=&quot;hahn.htm&quot;&gt;Hahn <br /> | x_i x_j, summed over all i &lt;= j; moreover as an alternative approach we can use the &lt;A HREF=&quot;hahn.htm&quot;&gt;Hahn <br /> | ||
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to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, <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 /> | 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&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | family of lattices, called Dn, to which it also belongs as D3, the&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:82:http://en.wikipedia.org/wiki/Crystal_structure --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">http://en.wikipedia.org/wiki/Crystal_structure</a><!-- ws:end:WikiTextUrlRule:82 -->&quot;&gt;face-centered <br /> | ||
cubic lattice&lt;/A&gt;. 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). <br /> | cubic lattice&lt;/A&gt;. 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). <br /> | ||
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 <br /> | 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 <br /> | ||
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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 /> | 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+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) <br /> | 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) <br /> | ||
if a+b+c is odd. Each unit cube corresponds to a &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | if a+b+c is odd. Each unit cube corresponds to a &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:83:http://tonalsoft.com/enc/stellat.htm --><a class="wiki_link_ext" href="http://tonalsoft.com/enc/stellat.htm" rel="nofollow">http://tonalsoft.com/enc/stellat.htm</a><!-- ws:end:WikiTextUrlRule:83 -->&quot;&gt;stellated hexany&lt;/A&gt;, <br /> | ||
or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.<br /> | or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.<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 /> | 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 /> | ||
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relations because of this.<br /> | relations because of this.<br /> | ||
In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the <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 &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | An norm defined from x_j x_j is a norm defined by the inverse to the symmetric matrix of the &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:84:http://en.wikipedia.org/wiki/Quadratic_form --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quadratic_form" rel="nofollow">http://en.wikipedia.org/wiki/Quadratic_form</a><!-- ws:end:WikiTextUrlRule:84 -->&quot;&gt;quadratic <br /> | ||
form&lt;/A&gt; for the An norm, which normalizes to the square root of the quantity n times the sum of squares of x_i <br /> | form&lt;/A&gt; for the An norm, which normalizes to the square root of the quantity n times the sum of squares of x_i <br /> | ||
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 /> | 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 /> | ||
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becomes the usual Euclidean norm. If we take linear combinations with integer coefficents of these, we obtain all <br /> | becomes the usual Euclidean norm. If we take linear combinations with integer coefficents of these, we obtain all <br /> | ||
triples of integers which are either all even or all odd. The lattice with these points and the usual Euclidean <br /> | triples of integers which are either all even or all odd. The lattice with these points and the usual Euclidean <br /> | ||
norm is the &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | norm is the &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:85:http://en.wikipedia.org/wiki/Crystal_structure --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">http://en.wikipedia.org/wiki/Crystal_structure</a><!-- ws:end:WikiTextUrlRule:85 -->&quot;&gt;body-centered cubic lattice&lt;/A&gt;.<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 /> | 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 /> | of integers whose sum is even, is always even; and we get the precise relationship between mappings and note-classes <br /> | ||
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to itself, and the body-centered cubic lattice of mappings of note-classes to itself. The first two types of transformation <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 /> | includes the major/minor transformation, and can be regarded as a vast generalization of that. Robert Walker has <br /> | ||
a piece, &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | a piece, &lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:86:http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase --><a class="wiki_link_ext" href="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase" rel="nofollow">http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase</a><!-- ws:end:WikiTextUrlRule:86 -->&quot;&gt;Hexany Phrase&lt;/A&gt;, which takes <br /> | ||
a theme through all 48 resulting variations.<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 /> | Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes <br /> | ||