Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

We have the main

Theorem 4

C(G, F,w) = Z(G)(w(0)+ w(1),w(0)2 + w(1)2,...w(0)f + w(1)f)

and the

Corollary 1 The cardinality of the orbit space $2F/G$ is $Z(G)(2,...2).$

Generalizations of the main theorem by de Bruijn yield (for example) the orbit cardinalities of $k$ -element chords in $Z12$ under the groups $T12,TI12$ , and $-G-->L(Z12)$ :


$k =$	0	1	2	3	4	5	6	7	8	9	10	11	12

$T12$	1	1	6	19	43	66	80	66	43	19	6	1	1

$TI12$	1	1	6	12	29	38	50	38	29	12	6	1	1

$-G-->L(Z12)$	1	1	5	9	21	25	34	25	21	9	5	1	1

The same result also yields formulas for orbits of $(k,n)$ -series (Fripertinger, 1993b, Satz 2.2.5). We reproduce the particularly interesting list of orbits for $n = 12$ :


$k$	number of orbits of $(k,12)$ -series

2	6
3	30
4	275
5	2 000
6	14 060
7	83 280
8	416 880
9	1 663 680
10	4 993 440
11	9 980 160
12	(dodecaphonic series) 9 985 920

The huge number of isomorphism classes of local compositions of the most common type, and in spaces which are strong reductions of the >real parameter spaces< modulo octave or similar periodicities, preconizes the usage of statistical methods to control the variety of cases encountered in practical analyses. Even next generation computers cannot reach the calculation power to check all possible classes. We refer to specialized papers (Beran and Mazzola, 1999a,b, 2000) for this subject.

4.2 Standard Objects

General classification of local and global compositions has been attacked by the author since 1980 (Mazzola, 1981). First descriptions of algebraic schemes the rational points of which parametrize local and global compositions have been published