Structure metric: Difference between revisions
Wikispaces>genewardsmith **Imported revision 567146411 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 567251741 - 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>2015-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-20 16:12:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>567251741</tt>.<br> | ||
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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> | ||
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An interesting example of this is given by the [[https://en.wikipedia.org/wiki/Hexany|hexany]], 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]]. If we set f(1) = 1, f(15/14) = 9/8, f(5/4) = 6/5, f(10/7) = 5/4, f(3/2) = 9/5 and f(12/7) = 15/8, then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-2 are the same as for the hexany; this scale, the [[hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space. | An interesting example of this is given by the [[https://en.wikipedia.org/wiki/Hexany|hexany]], 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]]. If we set f(1) = 1, f(15/14) = 9/8, f(5/4) = 6/5, f(10/7) = 5/4, f(3/2) = 9/5 and f(12/7) = 15/8, then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-2 are the same as for the hexany; this scale, the [[hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space. | ||
Even though the [[Graph-theoretic properties of scales|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. | Even though the [[Graph-theoretic properties of scales|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[Nova]], which is isometric with star and has an isomorphic graph, is similar. | ||
=Invariants= | =Invariants= | ||
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An interesting example of this is given by the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Hexany" rel="nofollow">hexany</a>, 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]]. If we set f(1) = 1, f(15/14) = 9/8, f(5/4) = 6/5, f(10/7) = 5/4, f(3/2) = 9/5 and f(12/7) = 15/8, then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-2 are the same as for the hexany; this scale, the <a class="wiki_link" href="/hexagon">hexagon</a>, is isometric to the hexany. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both <a class="wiki_link" href="/27edo">27edo</a> and <a class="wiki_link" href="/31edo">31edo</a>, for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.<br /> | An interesting example of this is given by the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Hexany" rel="nofollow">hexany</a>, 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]]. If we set f(1) = 1, f(15/14) = 9/8, f(5/4) = 6/5, f(10/7) = 5/4, f(3/2) = 9/5 and f(12/7) = 15/8, then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-2 are the same as for the hexany; this scale, the <a class="wiki_link" href="/hexagon">hexagon</a>, is isometric to the hexany. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both <a class="wiki_link" href="/27edo">27edo</a> and <a class="wiki_link" href="/31edo">31edo</a>, for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.<br /> | ||
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Even though the <a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales">group of the graph</a> is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13.<br /> | Even though the <a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales">group of the graph</a> is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, <a class="wiki_link" href="/star">star</a> has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. <a class="wiki_link" href="/Nova">Nova</a>, which is isometric with star and has an isomorphic graph, is similar.<br /> | ||
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