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

in addition, weight the individual contribution by a factor inversely proportional to the square of the one-sided distance. Through this modification, the sets containing extra tones lying further away contribute less than those with additional tones close to the given set.

Definition 11 Let $M$ be a melody, $e > 0$ any real number, and $G$ a paradigmatic group. For each non-empty subset $A < M$ denote the $G$ -orbit by $G(A)$ and the isotropy group for the action of $G$ on $M elk(M )$ , $k = l(A)$ , by $GA$ . Then we define the numbers $G Hp e(A)$ and $G Hce(A)$ by

G sum sum 1 Hp e(A) = d2(B)(A), A' (- G(A) B( - Hprese(A') G sum sum ---1---- Hc e(A) = ' ' d2(A)(B). A (- G(A) B( - Hconte(A )

The Hausdorff melodic weight of the subset $A$ is then defined as

G G G Hw e(A) = Hp e(A) .Hce (A) .#(GA).

5 Combinatorial Characteristics

There is a single elementary observation lying at the basis of this section.

Lemma 2 Let $M$ be a melody and $G$ a paradigmatic group. Then each subset of a symmetry patch (with at least two elements) is itself a symmetry patch.

The restriction concerning the number of elements is only due to the definition of a symmetry patch in which, for natural reasons, we required $k > 0$ . The property stated in the lemma for the set $SG(M )$ of all symmetry patches of a melody with respect to some paradigmatic group is the decisive requirement defining a simplicial complex. This simple fact opens the topic of symmetry in melodies to the methods of combinatorial topology. Much of what has been said so far could be interpreted from this point of view. Instead, this section will deal with one of the central structures constructed for a simplicial complex: its homology groups. Since the definition and properties will not be needed here, we refer the reader to Spanier (1966), Chapters 3 and 4, for a concise introduction into the field as well as all relevant facts and methods. For the purposes of this note it suffices to state that the corresponding algebraic procedure associates with each finite simplicial complex $K$ finitely many abelian groups $Hk(K)$ , $k = 0,1,...,n$ , which are finitely generated. In particular, the ranks of these groups, rk $(Hk)$ , will be of interest in the sequel.

Definition 12 A simplicial complex $K$ consists of a set $V (K)$ of vertices and a set $S(K)$ of finite non-empty subsets of $V(K)$ called simplices such that

Each set consisting of exactly one vertex is a simplex.