Graph-theoretic properties of scales: Difference between revisions
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==Marvel11max7a== | ==Marvel11max7a== | ||
Like Gypsy, [[marvel11max7a|marvel11max7a]] is a [[Consonant_class_scale|consonant class scale]] for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}. | Like Gypsy, [[marvel11max7a|marvel11max7a]] is a [[Consonant_class_scale|consonant class scale]] for marvel, based on the fact that four 5/4s, two 6/5s and one 8/7 come to four times 225/224, and these are approximately the intervals in the second consonance class, in the order 5/4, 6/5, 5/4, 6/5, 8/7, 5/4, 5/4, leading to a scale which is the marvel tempering of 9/8, 5/4, 32/25, 3/2, 8/5, 15/8, 2. In 166et, this becomes 0, 28, 53, 60, 97, 113, 150, 166, with 11-limit consonance set {21, 23, 25, 28, 32, 37, 44, 48, 53, 58, 60, 69, 76, 81, 85, 90, 97, 106, 108, 113, 118, 122, 129, 134, 138, 141, 143, 145, 166}. | ||
The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6). | The first scale element, the approximate 9/8, is exceptional in having a consonant relation to all other notes. The other six notes make a symmetric graph in the form of an octahedron, and they, as well as the full scale graph, have the automorphism group of the octahedron, of order 48. The note pairs (0 6), (2 3) and (4 5) represent opposite verticies of the octahedron, not connected by a consonance, and give involutions in the automorphism group. The full automorphism group itself is generated by (2,3), (4,5), (2,4)(3,5) and (0,2)(3,6). | ||
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Elfjove7 has an edge and vertex connectivity of 3, a radius of 1 with a diameter of 2, and is of genus 1. | Elfjove7 has an edge and vertex connectivity of 3, a radius of 1 with a diameter of 2, and is of genus 1. | ||
[[File:elfjove7.png]] | |||
[http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/jupiter's%20rations.mp3 Jupiter's Rations] by Andrew Heathwaite | [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/jupiter's%20rations.mp3 Jupiter's Rations] by Andrew Heathwaite | ||