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

3 Presence, Content, and Weight Functions

Recalling from Buteau (1998) motivic gestalt spaces are of type $T0$ ⁶

⁶: Recall that a topological space $X$ is a $T0$ -space iff for each pair $x,y$ of distinct points in $X$ , there exists a neighborhood of one point to which the other odes not belong; $X$ is a $T1$ -space if for each point $x (- X$ then for all other points $y (- X$ there exists a neighborhood of $x$ not containing $y$ ; and $X$ is a $T 2$ -space if for all pairs of disjoint points $x$ and $y$ there exist disjoint neighborhoods of $x$ and $y$ . We say (Buteau, 1998) that gestalt motivic spaces are ’almost $T1$ ’ in the sense that for each gestalt $G (- GESPt$ and for all gestalts $H$ such that $H /] G$ : there exists a neighborhood of $G$ not containing $H$ .

, ‘almost’ of type $T1$ , and, if $M OT (S)$ contains motives of different cardinalities, not at all of type $T2$ (Hausdorff), which excludes any intuitive representation of the topological structure. Therefore, in order to provide us with a more geometric picture of the motivic spaces and motivic gestalt spaces on a score $S$ , we introduce real-valued functions ( $presence,content$ , and $weight$ ) which account for the topological relations of these spaces. Observe that we have to take into account the intrinsic neighborhood asymmetry between gestalts. For more details about quantitative functions, refer to Buteau (2001).

The ’presence’ of a gestalt is the magnitude of its neighborhood and its ’content’ is the frequency of its appearance in other motives’ neighborhoods: First consider two gestalts $G$ and $H$ in $P GES t (S)$ with $P G (- GES t,n$ , and a neighborhood radius $e > 0$ . If $H (- SUe(G)$ , then one measures the presence of gestalt $G$ in gestalt $H$ (or, inversed roles: $H$ being contained in $G$ ) by the intensity integer

' ' P Inte(H|G) = card{H [ H |H (- GES t,n /~\ SUe(G)}.mult(H)

Since the higher the cardinality difference between $G$ and $H$ is, the higher is the probability that $Inte(H |G) /= 0$ , we weight the intensity by the factor $1/2(card(H)-card(G))$ . The presence and the content of $G$ at radius $e > 0$ in the score are defined⁷

⁷: There are more parameters in the general definition (Buteau, 2001) of these two functions.

sum (card(H)-card(G)) Presencee(G) := 1/2 .Inte(H |G) H( - GESPt (S)

sum (card(G)-card(H)) Contente(G) := P 1/2 .Inte(G |H) H (- GES t (S)

and the weight of gestalt $G$ at radius $e$ is

W eighte(G) := Presencee(G) .Contente(G)

Note that these (Mazzola) quantification functions⁸

⁸: These functions were introduced by Guerino Mazzola.

, presence, content, and weight, at radius $e > 0$ and defined on gestalts can be extended to motives, such as $M W eighte : M OT(S) --> R,$ and from which we then define the weight of a note $n$ of the score $S$ at radius $e > 0$ :

sum N W eighte(n) := M W eighte(M ). n (- M( - MOT (S)