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

for which $G (- GESP t,n$ , form a topological base of $GESP t$ . With this topology (the structures leading to this topology be implicitly assumed), $P GES t$ is called a motivic gestalt space, for a score $S$ the relative space $P GES t (S)$ is called a motivic gestalt space on $S$ . We denote the $e$ -neighborhood of a gestalt $G$ in $P GES t (S)$ by

P SUe(G) := Ue(G) /~\ GES t (S),

and denote $mult(G)$ , called the multiplicity of $G$ , the number of motives in $M OT (S)$ in gestalt $G$ .

Theorem (Buteau, 1998). If the pseudo-metric $GdP t,n$ is a metric for all $n$ , then the canonical maps $g : M OT --> GESP t$ , and $g(S) : M OT (S) --> GESP (S) t$ are open continuous maps onto the quotients, and the topologies on $GESP t$ and $GESP (S) t$ are the quotient topologies.

Example. We exemplify the setup leading to a motivic topology. See Buteau and Mazzola (2000) for a more complete description. Consider the space ${O,P,D} R$ . We fix the parameters $O,P$ , and $D$ in a way which is standard in Mathematical Music Theory (Mazzola, 1990): For the pitch values, we select the usual gauge with $C4 = 0$ , and the chromatic pitch set being parameterized by the integers, i.e. $C#4 = Db4 = 1$ , $D4 = 2$ , etc. Duration values are taken by the prescription that $1$ in the $O$ -coordinate corresponds to the literal mathematical value of $4/4$ duration. The first tone of a score is given onset value 0. Consider Figure 1.

We suppose that our score $S$ contains only the eight notes from Bach’s Kunst der Fuge 8-tone Main Them as shown in Figure 1 top bars. First we have the set of the score’s notes:

1 1 1 1 3 1 { (0,2,2),(2,9,2),(1,5,2),(2,2,2),} S = 1 5 1 11 1 1 (2,1,2),(2,2,4),(4 ,4, 4),(3,5,2)

We select the collection $M OT (S)$ of motives for the score $S$ as containing all motives with cardinality between 2 and 4. Therefore, the collection $M OT (S)$ contains $(8)= 28 2$ motives of cardinality 2, $(8)= 56 3$ of cardinality 3, and $(8)= 70 4$ of cardinality 4, which makes a total of 154 motives.

Figure 1:

Examples of sets of notes which form a motif: $Motif1$ and $Motif2$ ; and which do not form a motif: $Set3$ .

We consider two motives $1 1 3 1 11 1 Motif1 = {(2,9, 2),(2,2,2),(4 ,4,4)}$ and $1 Motif2 = {(0,2,2),$ $1 3 1 (2,1, 2),(2,2,2)}$ from Figure 1. The abstract images of $Motif1$ and $Motif2$ are $Cont3(Motif1) = (-1,- 1,1)$ and $Cont3(Motif2) = (0,-1,- 1)$ . The gestalt for the counterpoint paradigmatic group $CP$ of $Motif1$ is the collection of all motives in $M OT3(S)$ such that their images through the mapping $Cont$ is one of the following four abstract motives: $(- 1,- 1,1)$ , $(1,1,- 1)$ , $(-1,1,1)$ , or $(1,-1,- 1)$ (corresponding respectively to the abstract motif $Cont3(Motif1)$ , its inversion, its retrograde, and its inversion composed with the retrograde). Using the MeloTopRUBETTE for identifying the motives together, we get the following number of gestalts: there are 2 gestalts of motif cardinality 2, 5 of cardinality 3, and 18 of cardinality 4.