|
present operationalization of our motivic theory implements modular algorithms, making possible to modify the model during its empirical testing process. Let us briefly recall our mathematical model on motivic analysis of music. In Mazzola (1990), a mathematical theory of musical structures was developed which is based on local charts and global discrete varieties of tones in specific parameter spaces. In this approach, a motif is a local chart of a specific type. A motivic interpretation of a musical composition is a global variety, i.e. an interpretation of the score by a selected covering system of motives. Topological spaces built on motivic interpretations are called motivic spaces. They are defined from orbits of motives under actions of transformation groups, from metrical similarity between motives with same cardinality, and from submotif relations. It is important to observe that this topological structure is defined on the set of all motives (with different cardinalities) of a score. The motivic space of a score is however non-intuitive (not Haussdorff, only of type The problem of geometrization of motivic spaces deals with the main problem of motive theory: the exhibition of germinal motives. It has been further investigated and led to the concept of a motivic evolution tree (MET), the graphical representation of an overall spectrum of a score’s motivic structure (motivic space) (Buteau and Mazzola, 2000). The first trace of a sheaf-theoretic perspective related to the MET is given in Buteau (2001). In fact, the sheaves give rise to coordinate functions (the global sections) which yield embeddings of motivic topologies in real vector spaces. First empirical investigations (Buteau and Mazzola, 2000; Buteau, 2001) on small compositional units, such as the main theme of Bach’s ”Kunst der Fuge”, demonstrated the viability of our approach. The present MeloTopRUBETTE opens the path to deeper and more meaningful empirical investigations. A major improvement of the MeloTopRUBETTE against the MeloRUBETTE is the interactive control of the ongoing computational process making possible to extend and to improve the model ”on the flight”. We also mention that the contour similarity ”theory” of the American Set Theory is a special case of our implementation, in which their contour similarity concepts are extended to a 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. See section 5 for a more detailed comparison between these two RUBETTEs. 2 Motivic SpacesWe shortly recall the construction of a motivic space of a score which is here simplified, and for which details can be read in Buteau (1998), Buteau (2001), Mazzola (2002). The formal definition of a motivic space presupposes a motif concept. We want to restrict our attention to a minimal parameter setup in order to make the essential |
). For this reason, a more geometric perspective has been realized in the original MeloRUBETTE by giving each motive, and thereby each tone of the score, a weight that corresponds to its topological ”presence” and ”content”, see chapter 22.9 in