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

of $A$ . Choosing now the tolerance $e$ appropriately and comparing the cardinalities of the presence as well as the content for all the subsets of a melody already expresses an important hierarchical relation among them. Taking, e.g., $e = V~ 2$ allows for a difference of at most one time and one pitch unit between the tones of two subsets to be counted in this respect.

Given, in additon, a paradigmatic group $G$ , presence and content of some $A < M$ should also take into account the corresponding sets at each point in the $G$ -orbit of $A$ . This leads to the following notions.

Definition 10 Let $M$ be a melody, $e > 0$ any real number and $G$ a paradigmatic group. For each non-empty subset $A < M$ we denote the $G$ -orbit through $A$ in $M elk(M )$ , $k = l(A)$ , by $G(A)$ . Then the Hausdorff $e$ -presence with respect to $G$ and the Hausdorff $e$ -content with respect to $G$ of $A$ are defined by

G * ' ' Hprese(A) = {B (- P (M )|d(A )(B) < efor someA (- G(A)}

and

HcontGe(A) = {B (- P*(M )|d(B)(A') < efor someA'( - G(A)}.

It has to be noted that the original definition of Mazzola and Zahorka (see Mazzola and Zahorka, 1996) based presence and content on the so-called Gestalt (pseudo-)distance, i.e. the orbit distance for the chosen paradigmatic group. The distance of one subset of the melody to another is taken to be the infimum of the distances to all transformed copies of the second in the plane, not just those contained in the melody.

Before going on, three important points have to be mentioned. First, there are “too many” subsets contributing to presence and content. On the one hand, every superset of a set in the presence also belongs to the presence. On the other hand, along with any set the content contains each of its subsets . Hence, both of these sets are highly redundant. To limit both, the amount of information and of computation, presence and content should be reduced. One way to proceed could consist in keeping only the minimal (with respect to inclusion) sets in the presence and the maximal in the content to form what could be called the upper and lower frontier of the given set. A simpler possibility reduces presence and content by just restricting the subsets to be included to those not having more elements than a prescribed upper bound or not having less than a prescribed lower bound, respectively (see Mazzola and Zahorka, 1996; Buteau, 2001; Buteau and Mazzola, 2000). Plausible candidates for these bounds are, e.g., $#(A) + 1$ or $#(A) + 2$ for the upper one and $#(A) - 1$ or $#(A) - 2$ for the lower one. In the latter case, even 2 or 3 could be worth considering. Again it has to be stressed, that all this will have to be decided by actual computations for musical examples which have not been carried out, yet.

The second point to be recalled here is the phenomenon of non-trivial orbits containing only a single subset. In order to include this kind of contribution into the definition of the weight, we will introduce the cardinalty of the isotropy group as a factor.

Finally, we want to alter the weight definition of Mazzola and Zahorka by not only counting sets in the presence and content for a given subset of the melody but,