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

The information related to $S$ , which is set forth by the quantification functions, is articulated in sheaves of function vector spaces $F$ on $GESPt (S)$ , whose sections $F(U )$ are linear combinations of determined systems of presence functions, content functions, or weight functions, respectively: Since gestalt neighborhoods stay constant for ’more or less small’ intervals, consider a sequence of neighborhood radii $e0 < e1,...< en$ for which $e0 > 0$ , and for all $ei > 0$ we consider also a quantification function $fei$ at $ei$ . We define a presheaf $F '$ through

F '(GESPt (S)) = R(fe0,fe1,...,fen)

where $R(fe0,fe1,...,fen)$ is the real vector space generated by the quantification function $fei$ evaluated at all $ei$ , and for any $' e > 0$ and gestalt $G$ in $P GES t (S)$

' ' P F (Ue'(G)) = F (GES t (S)| Ue'(G)).

The sheafification $F$ of $F '$ for presence, content, and weight, respectively, constitutes a system of local coordinate functions on motivic gestalt spaces.

4 The Flow Chart for the Calculations in the MeloTopRUBETTE

Our topological model of motivic analysis of music, introduced in Sections 2 and 3, is implemented in the MeloTopRUBETTE according to the flow chart shown in Figure 2. The ellipse of the flow chart is the link from the core program to the whole software RUBATO. It implies a change of programming language since Rubette’s core programs are normally written in java whereas the internal language of the RUBATO platform is the denotator language (Mazzola, 2002). Each plain box with plain lines represents some calculations from which meaningful objects for the model are created. These objects are sent back to RUBATO, and that is why the latter plain boxes are provided with an output denotator for visual and/or sonic representation on the RUBATO’s PrimaVista Browser (Mazzola, 2002, chapter 40). Ellipses are not present in the flow chart for a simpler vizualisation of the whole architecture. Plain boxes with dotted lines are mainly intermediate calculations for constructing the motivic topologies, and the plain ones without line are linear algebra operations. The lozenges are tests for stopping their respective loops. The white boxes are initialization of cycle variables.

There is a double input. On one side, input $(00)$ contains a score $S$ as a denoTeX or MIDI file. On the other side, input $(01)$ contains 20 input parameters: the score’s collection of motives parameters (e.g. minimal and maximal motif cardinality), the topological parameters (e.g. shape type and paradigmatic group), the similarity radii set parameters (in order to obtain relevant radii for evaluating the quantification functions), and the optional output parameters (e.g. with or without motivic evolution tree).