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

this language was developed under constraints of object-oriented programming in a research grant (1992-1996) of the Swiss National Science Foundation, directed by the author, and targeted at the construction of a software platform for analysis and performance of musical scores. After the programming work, executed by the author’s assistant Oliver Zahorka and the author in C and Objective-C language Noll (2003) as well as Mazzola and Garbers (2001), the platform was named RUBATO $®$ ; several research reports (Mazzola and Zahorka, 1993-1995), papers (Mazzola and Zahorka, 1994), and free software units (Mazzola and Garbers, 2001) are available in this context. The Swiss RUBATO $®$ project was restartet in 1998 in a German research grant of the Vokswagen Foundation by a research group under direction of Thomas Noll (Noll, 2003) and is being ported to the Mac OS X platform on the basis of the Objective C language. Presently, the research group of the author at the Computer Science Department of the University of Zurich is also developing new software units of the RUBATO $®$ platform, based on Java/Java3D language. The latter implements the full functorial version of forms and denotators as well as the distributed components architecture (see Göller and Milmeister, 2004).

Independently of its different realizations on current software environments, the idea of such a platform is this: To implement a database management system (called PrediBase in the RUBATO $®$ terminology) which is based on the form and denotator data model, to implement a frame software (RUBATO $®$ ) which incorporates PrediBase, and to extend the frame software by dynamically loadable program units (such a unit is called $RUBETTE(R)$ in the RUBATO $®$ terminology) which implement different tasks for musical analysis, performance, composition, logical constructions, navigation, etc. So the entire platform is a universal, and indefinitely extendible tool for musicology in its valid form of an exact, operationalized science, where the experimental paradigm can be (and has been (Stange-Elbe, 2000; Fleischer Anja, 2000)) performed on an objective, distributed level of international collaboration. For basics on this concept, see Mazzola (2000).

3 Local and Global Compositions

Local and global compositions are a kind of musical varieties and describe the core theory of musical objects. Their cateogries share important properties which are also basic to the topos-theoretic evolution of the entire theory.

3.1 Categories of Local Compositions

Although the category of all denotators is defined (Mazzola, 2001b), we shall focus on the classically prominent subcategory of local compositions. These are the denotators $D : A~>F (x)$ the form $F$ of which is of power type. More precisely, we shall consider $A$ -addressed denotators with coordinates $x < @A × fun(S)$ , where form $S$ is called the ambient space⁷

⁷: If no confusion is likely, we identify $S$ with $fun(S)$ .

of $D$ . If there is a set $X < A@S$ such that $x = X@$ , the local composition is said to be objective, otherwise, we call it functorial.