melody, rhythm, large musical forms (in particular the classical sonata theory), including their classification, and the paradigmatic semiotics of musical structures as described by Ruwet (Ruwet, 1972) and Nattiez (Nattiez, 1975).

This approach included satisfactory theorems which model modulation, counterpoint, and string quartet theory in coincidence with the classical knowledge and tradition, which yield classification for some interesting global structures (Mazzola, 1990a, 1991), and which have been operationalized in music composition software (Mazzola, 1994) and corresponding CDs (Jan, 1991; Beran and Mazzola, 1992; Beran, 2000; Mazzola, 1990b). It was however incomplete and too narrow in its concept framework for many musical problems. Here are some critical points:

• The Yoneda point of view was not properly developed. This defect became virulent after Noll’s reconstruction of Riemann harmony (Noll, 1995).
• The development of the music platform RUBATO for analysis and performance (Mazzola and Zahorka, 1994; Mazzola and Garbers, 2001) enforced a critical review of music data models for universal purposes from score representation to performance (Mazzola and Zahorka, 1993-1995) and the definition of an extended concept framework the elements of which were described in Mazzola and Zahorka (1993-1995); Zahorka (1995) and implemented in RUBATO’s PrediBase DBMS.
• The complexity of musical performance asked for concepts and methods from differential geometry, such as vector fields and their integration (ordinary differential equations and associated numerical Runge-Kutta-Fehlberg methods), Lie derivatives, and method of characteristics in partial differential equations.
• The differenciation between mathematical fiction and musical facticity had to be explicated and led to the concept of textual and paratextual predicates (Mazzola, 1998, 2000). At this point, logical and geometric perspectives were forced to unite. This approach is centered around topos-theoretic construction of musical predicates by means of logical and geometric operations and also targets at the design of universal composition tools. Presently, several research groups (e.g. TU Berlin, IRCAM Paris, U Osnabrück, UNAM Mexico, ETH Zürich, U Zürich) are collaborating in the theoretical and software design of these extensions.

So this second status report will be centered around the most important improvements and extensions of the theory since the early nineties. Again, this report is paralleled by an upcomping book The Topos of Music (Mazzola, 2001b) with extensive discussions of the old and new topics. A history of mathematical music theory has however not been written, and this report is just a flash on the ongoing process.

The report first deals with mathematical models in music, discussing the methodological background, then illustrating it by three classical models: modulation, counterpoint, and performance. The second section introduces the concept framework of forms and denotators, including their operationalization on the RUBATO