Some numerical results
In order to apply this Theorem one has to know the cycle indices of
G and Sk. The formulae for the
cycle indices of Cn, Dn and
Sk are well known. (See [2][10].) The cycle index of Aff
(1,Zn) is computed in [17]. All these cycle index methods are
implemented in SYMMETRICA [16], a computer algebra system
devoted to combinatorics and representation theory of the symmetric
group and of related groups. Using this program system for twelve
tone music the numbers of C12,
D12 and Aff (1,Z12)-mosaics of
size k were computed. (See table 1.)
Number of mosaics in twelve tone music.
| G\k |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
| C12 |
1 |
179 |
7254 |
51075 |
115100 |
110462 |
52376 |
13299 |
1873 |
147 |
6 |
1 |
| D12 |
1 |
121 |
3838 |
26148 |
58400 |
56079 |
26696 |
6907 |
1014 |
96 |
6 |
1 |
| Aff (1,Z12) |
1 |
87 |
2155 |
13730 |
30121 |
28867 |
13835 |
3667 |
571 |
63 |
5 |
1 |
In conclusion there are 351773 C12-mosaics,
179307 D12-mosaics and 93103 Aff
(1,Z12)-mosaics in twelve tone music. A SYMMETRICA
program for computing these numbers can be written in the following
way:
main()
{
OP a,b,c,d,e,f,g;
INT i;
anfang();
a=callocobject(); b=callocobject(); c=callocobject();
d=callocobject(); e=callocobject(); f=callocobject();
g=callocobject();
scan(INTEGER,a);
zykelind_Cn(a,b);
/*
zykelind_Dn(a,b);
zykelind_aff1Zn(a,b);
*/
m_i_i(0L,f);
for (i=1L;i<=S_I_I(a);++i)
{
m_i_i(i,c);
zykelind_Sn(c,d);
debruijn_all_functions(b,d,e);
sub(e,f,g);
printf("Number of %d mosaics in an %d-scale: ",i,S_I_I(a));
println(g);
copy(e,f);
}
printf("Number of mosaics in an %d-scale: ",S_I_I(a));
println(f);
freeall(a); freeall(b); freeall(c); freeall(d);
freeall(e); freeall(f); freeall(g);
ende();
}
harald.fripertinger "at" uni-graz.at, May 10,
2016
