

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
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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 12tone 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



KEYWORD

nonn,uned,obsc


AUTHOR



EXTENSIONS

This is very unclear. Which numbers refer to vertices of the pentagon and which are the vertices of the 7gon? 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



