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

is the dimension of the output vectors and the entries of $M$ are either 0 or 1. The usage of $M$ will become clear in the following type-related specifications of Fold.

Figure 5:

The folder matrix on the PrimaVista Browser’s 3D input interface the so called Di.

8.4 Simple Denotator Folding

If $F : Id.Simple(V)$ is a simple form with module $V$ , the denotators $D : 0@F (D1,D2, ...,Dm)$ are essentially elements of $V$ . In this case, the module-theoretic framework of our implementation guarantees a real number $Fi = Fold(Di)$ , and the linear ordering among the $Di$ , as it is supposed to be given on the module $V$ , is preserved by the very construction of $Fold$ on $V$ .

8.5 Limit

Since in the topoi of presheaves, limits are given as subsets of cartesian products, we may restrict this case to a cartesian product form $F = V × V × ...× V 1 2 k$ , see Mazzola (2002a) for the topos-theoretical background of detonators. In this case, each denotator $D i$ is a $k$ -tuple $D = (D ,D ,...,D ) i i,1 i,2 i,k$ of denotators in the respective factor forms. For each index $j = 1,2,...,k$ , we denote by $Dj = {D ,D ,...,D } 1,j 2,j m,j$ the projection of $D$ onto its $j$ th component in the form space $V j$ . By recursion, we may fold each $Dj$ according to a default $1× m j$ folder matrix in $V j$ . This yields a sequence of real numbers $F old(Dj) = (Dj1,Dj2, ...,Djm)$ which also preserves the respective linear ordering. Now, for each row $r = 1,2,...,n$ of the folder matrix $M$ , the $1$ entries define a subsequence of vectors of folded denotators $(F old(Dt(r,1)),...,Fold(Dt(r,dr)))$ , i.e., $m$ vectors in $d r$ -space. According to section