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

Note that, by the definition of $k SG(M )$ , each element of $M elk(M )$ not belonging to $k SG(M )$ forms an orbit of its own. Since this relation is determined by a group action, the $G$ -orbits from a partition of $k SG(M )$ . The corresponding set of equivalence classes will be denoted by $G\M elk(M )$ .

Definition 7 The orbit rate (OR) is defined as the ratio of the number of $G$ -orbits in $M el2(M )$ to the number of 3-element subsets of $M$ ,

#(G\(M--el2)(M-)) orG(M ) = l(M ) . 3

Again, the denominator serves as a normalization, this time considered to be the number of orbits for the action of the trivial group. As for the SR, there are of course higher analogues of the OR defined by

#(G\M el (M )) orkG(M ) =---(-----k)---. l(M ) k+ 1

The Proposition in Section 2 implies that the OR is generically equal to one.

Example 3 The following table contains the mean values of the OR for each of the three corpora considered and both paradigmatic groups.

Corpus	$T$	$CP$

Bach Chorale Lines	0.910	0.884
Baroni/Jacoboni Phrases	0.956	0.897
German Children Songs	0.963	0.864

These figures show that the chorale lines by Bach contain fewer orbits, consequently also longer ones, than the generated phrases. What needs to be explained by scrutinizing the orbit structures, is the remarkable change of order between the values passing from translations to the counterpoint group.

In general, a more detailed analysis will have to look more closely into the actual orbit structure of the pieces to reveal whether the statistical data cited above indeed hint at a compositional principle in action. Similar computations for $3 or$ and $4 or$ support the expectation that the orbit structure of $M el2(M )$ is actually much more varied than those of the $M elk(M )$ for larger $k$ . As of this writing, no general properties like estimates or asymptotic behavior similar to those for the SR are known in the case of the OR. In particular, combing the study of the OR with that of the SR wil lead to a clearer picture of the amount as well as the nature of symmetry within a given melody or a sample under consideration. (See also the remark at the end of Section 5.)