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%I #17 Mar 30 2014 13:50:01
%S 1,3,5,7,8,2,3,5,7,9,2,4,5,7,10,2,4,6,7,12,1,8,9,12,1,2,4,6,2,3,5,7,9,
%T 2,4,5,7,10,2,4,6,7,12,3,8,9,12,1,2,4,6,1,3,4,6,2,4,5,7,10,2,4,6,7,12,
%U 5,9,10,11,1,2,4,6,1,3,4,6,1,3,5,6,11,2,4,6,7,12,7,10,11,12,1,3,5,7,8,1,8,9
%N A graph substitution group based on a heptagon (Church Music 7 tones; white piano keys) and a pentagon (flats and sharps: black piano keys): the polygons are tied together with 5 connections beside chord connections 12 X 12 matrix substitution with polynomial: 8 + 4 x - 4 x^2 + 9 x^3 - 141 x^4 + 196 x^5 + 35 x^6 - 259x^7 + 265 x^8 - 156 x^9 + 58 x^10 - 12 x^11 + x^12.
%C As D_6 is two like interconnected hexagons in an hexagonal prism, this figure is an unexpected asymmetry break to that: {6,6}->{7,5}. This sequence has the virtue of tying music theory to both graph theory and a spatial model in group theory. The sequence gives a type of mathematical "model" for 12-tone music theory. It is interesting to note that: binomial(12,8)=495 and dimension of E_8*E_8=496.
%F 12 Substitutions of the form: 1->{1, 3, 5, 7, 8}; 2->{1, 2, 4, 6}; 3->{2, 3, 5, 7, 9}; 4->{1, 3, 4, 6}; 5->{2, 4, 5, 7, 10}; 6->{1, 3, 5, 6, 11}; 7->{2,4, 6, 7, 12}; 8->{1, 8, 9, 12}; 9->{3, 8, 9, 12}; 10->{5,9, 10, 11}; 11->{6, 10, 11, 12}; 12->{7, 10, 11, 12};
%t Clear[s] s[1] = {1, 3, 5, 7, 8}; s[2] = {1, 2, 4, 6}; s[3] = {2, 3, 5, 7, 9}; s[4] = {1, 3, 4, 6}; s[5] = {2, 4, 5, 7, 10}; s[6] = {1, 3, 5, 6, 11}; s[7] = {2, 4, 6, 7,12}; s[8] = {1, 8, 9, 12}; s[9] = {3, 8, 9, 12}; s[10] = {5, 9, 10, 11}; s[11] = {6, 10, 11, 12}; s[12] = {7, 10, 11, 12}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]; aa = p[4]
%Y Cf. A131213.
%K nonn,uned,obsc
%O 1,2
%A _Roger L. Bagula_, Oct 07 2007
%E This is very unclear. Which numbers refer to vertices of the pentagon and which are the vertices of the 7-gon? Once this is straightened out, the entry needs to be edited in the same way that I edited A131213. - _N. J. A. Sloane_, Jan 25 2012
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