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

on the space $ROP...$ of notes as the group of affine transformations generated by all the translations $e(o,p)$ of onset $o$ and pitch $p$ , the horizontal reflexion $U$ (the inversion), and the vertical reflexion $K$ (the retrograde), i.e., more precisely $CP = <e(o,p),U,K >o,p (- R$ where if $(o',p',...) (- ROP...$ , then $e(o,p) .(o',p',...) = (o'+ o,p'+ p,...)$ , $U .(o',p',...) = (o',-p',...)$ , and $K .(o',p',...) = (-o',p',...)$ ; and for all $g (- CP$ and for all $M = {m1,...,mn}( - M OTn$ : $g.M := {g.mi |mi (- M }$ ;
and on $Zn(n-1)/2$ as the canonically induced action, i.e., for all abstract motives $b = (b1,...bn(n-1)/2) (- Zn(n-1)/2$ : $e(o,p) .b = b$ , $U .b = - b$ , and $K .b = (- bn,...,-b1,-bn+(n-1),...,-bn+1,...,- bn(n-1)/2)$ .

One also supposes that $P$ acts as a group of isometries⁵

⁵: Note that this hypothesis is natural: the distance between two motives should not change when applying an element of $P$ to both of them.

(preserving metrical distances) on each $Gt,n$ , which clearly induces an action by isometries on $M OTn$ . We call the inverse image $t- 1(P .tn(M ))$ of a $P$ -orbit of a motive’s abstract motif $tn(M )$ its gestalt:

GesP (M ) = t-1(P .tn(M )), M (- M OTn. t

The gestalts define a partition of the total space $M OT$ , the set of gestalts is denoted by $P GES t$ . It is evidently the disjoint union of the classes $P GES t,n$ in $M OTn$ . If $S$ is a score, then its trace in $M OT(S)$ is denoted by $GESPt (S)$ , and the projections are denoted by $g : M OT --> GESPt$ , and $g(S) : M OT (S) --> GESPt (S)$ . We have a (family of) pseudo-metric(s) $GdPt = (GdPt,n)n (- N$ with $GdPt,n$ on $GESPt,n$ , which is defined by

GdPt,n(GesPt (M1),GesPt (M2)) = infp (- P dt,n(p.M1,M2).

Now we are ready to set forth the topological framework. For a given data set as described above, suppose that $e > 0$ is a real number, and that $M (- M OTn$ . Then we set

Ue(M ) = {N | E N' (- M OTn, N '< N : infp (- P dt,n(p.M, N ') < e},

and call it the $e$ -neighborhood of $M$ . Observe that at this point we link motives of different(!) cardinalities. If our setup fulfills the inheritance property (Buteau, 1998, 2001), the system of $e$ -neighborhoods defines a base for a topology on $M OT$ . For example, the contour type together with the Euclidean metric fulfills the inheritance property. The space $M OT$ with this topology is called the motivic space, its relativization to $M OT (S)$ is called the motivic space on $S$ .

We now introduce a topology on $GESPt$ . To this end, we need a relation on gestalts corresponding to the submotif relation on motives: If $G$ and $G'$ are two gestalts, we say that $G'$ is a small gestalt of $G$ , in signs: $G'[ G$ , iff there are motives $M (- G$ , $M ' (- G'$ such that $M '< M$ . Then the sets

' P ' P ' Ue(G) = {H | E H (- GES t,n, H [ H : Gd t,n(G,H ) < e},