login
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
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.
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 * A335894 A101496 A218490
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