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

facts clear, and we refer the reader to the end of this section for a simple example leading to a motivic topology. First consider the space ${O,P,D,L,G,C} ~ 6 R = R$ of tone parameterization for which the parameters are respectively onset, pitch, duration, loudness, glissando, and crescendo, and consider also the canonical projection

{O,P,D,L,G,C} {O} pO : R --> R

on the axis of onset events. Denote $ROP... (_ R{O,P,D,L,G,C}$ the space of notes parameterized by at least onset and pitch parameters. A motif $M = {m1, ...,mn}$ is a non-empty finite subset in $ROP...$ such that the canonical induced projection $M --> pO(M )$ is a bijection.

A motif is therefore a finite set of notes $mi = (oi,pi,...)$ such that only one note is heard at a given onset. A submotif $M '$ of a motif $M$ is a motif such that $M '< M$ . The set of all motives is denoted by $M OT$ , which is the disjoint union of subsets $M OTn$ of all motives $M$ of cardinality $card(M ) = n$ . If we are given a score $S$ , a collection of motives with all notes living in $S$ is denoted by $M OT (S)$ , which is the disjoint union of the subsets

M OTn(S) = M OT (S) /~\ M OTn.

Motives are always mapped to abstractions, for example for contour information. This means that we have a family $t = (tn)$ of maps

tn : M OTn --> Gt,n

into mutually disjoint²

²: In the general theory, disjointness is not mandatory, however.

sets $Gt,n$ of abstract motives of abstract cardinality $n$ . The family $t$ is called the shape type, whereas the elements of $Gt,n$ are called abstract motives of abstract cardinality $n$ . A typical map is the contour type $t = Cont$ (which corresponds in the American Set Theory to the COM matrix). We have

GCont,n = Zn(n-1)/2,

and if $M = {m1,m2,...,mn}( - M OTn$ with notes $mi = (oi,pi,...)$ for which onsets $o1 < ...< on$ , we set $Contn(M ) = (D12,D13,...D1n,D23,...,D(n- 1)n)$ the vector with $Dij = 1$ if the pitch difference $(pj - pi)$ of notes $mj$ and $mi$ in $M$ is positive, $0$ if the difference is null, and $- 1$ if it is negative; see Buteau (2001) for further examples. On each space $Gt,n$ of abstract motives of abstract cardinality $n$ , we suppose we have a pseudo-metric³

³: Recall that a pseudo-metric for a set $X$ is defined as a metric $d$ for $X$ but for which the axiom $A x,y (- X : d(x,y)= 0==> x= y$ does not have to be fulfilled.

$d (x ,x ) n 1 2$ , for example the Euclidean metric on $G Cont,n$ . Call the family $d = (d ) n n (- N$ a pseudo-metric on the shape type $t$ . This induces a pseudo-metric (family) $dt = (dt,n)n (- N$ , which on each $M OTn$ is defined by $dt,n(M1, M2) = dn(t(M1),t(M2))$ .

Finally, we suppose that there is a pair of $tn$ -equivariant group actions $P × M OTn --> M OTn$ , $P × Gt,n --> Gt,n$ for each $n$ . Following Nattiez and Ruwet (Nattiez, 1975), the group $P$ ⁴

⁴: We take the liberty to use the same notation as for the note parameter ’Pitch’ since no confusion is likely.

is called the paradigmatic group. The typical example is the pointwise action of the affine counterpoint group $CP$ , which acts