On the algorithm level, we introduced a significative more efficient calculation step: Calculations extend to classes (gestalts) of motives, thereby reducing considerably the amount of calculations. The reduced calculations are distance values for each pair of gestalts with same cardinality, as well as the evaluation of each quantification function (on the set of all gestalts instead of all motives). Because of the gained efficiency we could implement the evaluation of quantification functions at a collection of neighborhood radii compared to the evaluation at only one radius for MeloRUBETTE. As a consequence we can also calculate dimensions of quantification function spaces which is of course not possible with the MeloRUBETTE.

We also generalized the MeloRUBETTE by offering 9 shape types compared to 3 in the MeloRUBETTE. In particular, the ”diastematic index” shape type from the MeloRUBETTE has been refined (as the contour shape type) to yield a motivic space, a topological structure which is impossible to define for the ”diastematic index” shape type. Moreover, with this rich variety of shape types and the easy access to define new shape types, the contour similarity ”theory” of the American Set Theory is a special case of our implementation, since their contour similarity concepts are, through our model, extended (see Buteau and Mazzola, 2000) to a (motivic) topology on the space of all motives of a score, i.e. a structure in which a similarity concept between motives of different cardinalities is introduced.

There are also more possibilities of paradigmatic groups, as well as of distances. Moreover the quantification functions, which have been extended to gestalts, are implemented as their generalization. A special case is the Mazzola quantification functions as defined in section 3 and as implemented in the MeloRUBETTE.

Finally, the most significant improvement of the MeloTopRUBETTE is the enrichment of the output: In the MeloRUBETTE, the unique output consisted of weights on notes of the analyzed score, whereas in the MeloTopRUBETTE, we propose weights on notes, weights on motives and on gestalts, the evaluated qualitative function necessary for the motivic evolution tree, the dimensions of real vector spaces of the presence, content, or weight functions, dimensions of respective stalks of related function sheaves, and, a very useful feature, a motif and gestalt information interactive window which outlines e.g. the motif’s notes in the score, its weights at all radii, its gestalt (class of motives), its image in the motivic evolution tree, etc.

6 Acknowledgments

This research is supported by SNSF grant 2100-065265.01/1. My thanks go to Guerino Mazzola for his continuous support and encouragement.

References