Graph-theoretic properties of scales: Difference between revisions
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==Archchro== | ==Archchro== | ||
[[Archchro]] is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of [[Archytas' Chromatic]]. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}. | [[Archchro]] is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of [[Archytas' Chromatic]]. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}. | ||
Archchro has four maximal cliques, the pentads 02346, 02356, 12346, and 12356. Its graph has edge, vertex and algebraic connectivity all 5, a radius of 1 and a diameter of 2, and is of genus 1. | |||
=Eight note scales= | =Eight note scales= | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Archchro"></a><!-- ws:end:WikiTextHeadingRule:20 -->Archchro</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Seven note scales-Archchro"></a><!-- ws:end:WikiTextHeadingRule:20 -->Archchro</h2> | ||
<a class="wiki_link" href="/Archchro">Archchro</a> is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of <a class="wiki_link" href="/Archytas%27%20Chromatic">Archytas' Chromatic</a>. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}.<br /> | <a class="wiki_link" href="/Archchro">Archchro</a> is the tempering in hemif, a 2.3.7.11.13 subgroup scale, of <a class="wiki_link" href="/Archytas%27%20Chromatic">Archytas' Chromatic</a>. In the 58et tuning, it has steps of 0, 3, 10, 24, 34, 37, 44, 58, with a consonance set {6, 7, 10, 11, 13, 14, 17, 20, 21, 24, 27, 31, 34, 37, 38, 41, 44, 45, 47, 48, 51, 52}. Archchro is intransitive, with one orbit consisting of notes 2, 3, and 6, and the other of notes 0, 1, 4 and 5. The three note orbit is permuted by the group of the triangle, which is to say, the symmetric group S3, and the four note orbit by the group of the square. Together these give an automorphism group of order 6 X 8 = 48. The eight permutations fixing 2, 3, and 7 are {0123456, 0123546, 1023456, 1023546, 4523016, 4523106, 5423016, 5423106}, and the six fixing 0, 1, 4 and 5 are {0123456, 0126453, 0132456, 0136452, 0162453, 0163452}.<br /> | ||
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Archchro has four maximal cliques, the pentads 02346, 02356, 12346, and 12356. Its graph has edge, vertex and algebraic connectivity all 5, a radius of 1 and a diameter of 2, and is of genus 1.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Eight note scales"></a><!-- ws:end:WikiTextHeadingRule:22 -->Eight note scales</h1> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Eight note scales"></a><!-- ws:end:WikiTextHeadingRule:22 -->Eight note scales</h1> | ||