7-limit symmetrical lattices: Difference between revisions
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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. | 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.</pre></div> | 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. | ||
=Articles= | |||
* [[http://tonalsoft.com/monzo/lattices/lattices.htm|Harmonic Lattice Diagrams]] by Joseph L. Monzo [[http://www.webcitation.org/5xeHLDlo9|Permalink]] | |||
* [[http://www.huygens-fokker.org/docs/fokkerpb.html|Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes]] by Adriaan Fokker [[http://www.webcitation.org/5xeGyOWPA|Permalink]] | |||
* [[http://x31eq.com/lattice.htm#7limit|Octave Equivalent Music Lattices]] by Graham Breed [[http://www.webcitation.org/5xeJ48wUh|Permalink]]</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>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 /> | <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>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 /> | ||
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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, <a class="wiki_link_ext" href="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase" rel="nofollow">Hexany Phrase</a>, which takes a theme through all 48 resulting variations.<br /> | 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, <a class="wiki_link_ext" href="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase" rel="nofollow">Hexany Phrase</a>, which takes a theme through all 48 resulting variations.<br /> | ||
<br /> | <br /> | ||
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.</body></html></pre></div> | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Articles"></a><!-- ws:end:WikiTextHeadingRule:0 -->Articles</h1> | |||
<ul><li><a class="wiki_link_ext" href="http://tonalsoft.com/monzo/lattices/lattices.htm" rel="nofollow">Harmonic Lattice Diagrams</a> by Joseph L. Monzo <a class="wiki_link_ext" href="http://www.webcitation.org/5xeHLDlo9" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerpb.html" rel="nofollow">Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes</a> by Adriaan Fokker <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGyOWPA" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://x31eq.com/lattice.htm#7limit" rel="nofollow">Octave Equivalent Music Lattices</a> by Graham Breed <a class="wiki_link_ext" href="http://www.webcitation.org/5xeJ48wUh" rel="nofollow">Permalink</a></li></ul></body></html></pre></div> | |||