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 A131979 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. 1
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS FORMULA 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}; MATHEMATICA 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] CROSSREFS Cf. A131213. Sequence in context: A201042 A142340 A185168 * A101496 A218490 A161696 Adjacent sequences:  A131976 A131977 A131978 * A131980 A131981 A131982 KEYWORD nonn,uned,obsc AUTHOR Roger L. Bagula, Oct 07 2007 EXTENSIONS 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 STATUS approved

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