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

In other words, the inequality $d(A)(B) < e$ means that the set $A$ lies within the $e$ -neighborhood of the set $B$ , i.e., the union of all $e$ -balls around points of $B$ . Note that this also holds true, if all the points of $A$ belong to the $e$ -ball of a single point in $B$ .

Corollary 1 For every melody $M$ and any paradigmatic group $G$ , the Hausdorff distance $d H$ defines a metric on the family of all non-empty subsets of $M$ , $P*(M )$ . Consequently, the sets $S (M ) G$ (and $^S (M ) G$ ) of (maximal) symmetry patches are metric subspaces of the space $(P*(M ),d ) H$ .

To obtain an overall view of these metric spaces might be worth trying. In particular, comparing $e$ -balls around points with $e$ running through the set of ordered distances between points should lead to insights into the actual situation of modification, enlargement, or reduction in a melody. In general, the huge number of subsets will make it difficult to arrive at definite conclusions this way. (For a method to deal with this problem, the study of motivic evolution trees, see Buteau (2001).)

The Hausdorff distance will be used here mainly to take into account the relative position of the subsets in a melody. Another natural field of application for this measure of proximity is melodic similarity. This appears to be an interesting topic in its own right with considerable relevance (see Selfridge-Fields, 1998).

Example 4 As an illustration, we mention the results of the computation of mutual Hausdorff distances between the phrases generated by Baroni and Jacoboni on the one hand, and between the Bach chorale lines, on the other. On the whole, the former stay “closer together” than the latter. With the normalization mentioned in Section 2, the mutual Hausdorff distances of the Bach chorale lines extend up to 123.6 with 135 out of the 1,830 exceeding 77.5, whereas all of the Hausdorff distances of the generated phrases are smaller than 77.5. Hence, the melodies from the Bach sample do not only have “more” (in the sense of larger SRs) and “higher” (in the sense of smaller ORs) symmetry, but are also more “diverse” (in the sense of larger mutual distances). Further distinguishing features will arise in connection with the combinatorial characteristics to be introduced in the next section.

Now we turn to the definition of a slight variation of the melodic weight first introduced in Mazzola and Zahorka (1996).

Definition 9 Let $M$ be a melody and $e > 0$ a real number. For each non-empty subset $A < M$ the Hausdorff $e$ -presence and the Hausdorff $e$ -content of $A$ are defined by

Hpres (A) = {B (- P*(M )|d(A)(B) < e} e

and

* Hconte(A) = {B (- P (M )|d(B)(A) < e}.

Thus the Hausdorff $e$ -presence of $A < M$ comprises all subsets of the melody $M$ in whose Hausdorff $e$ -neighborhood $A$ is contained, whereas the Hausdorff $e$ -content of $A$ consists of all subsets of $M$ which are contained in the Hausdorff $e$ -neighborhood