So the counterpoint model not only exhibits a variety of fictitious counterpoint theories which could very well yield new, interesting counterpoint compositions. It also relates the existent counterpoint theory of the Fux dichotomy to its antipode, the major dichotomy , through the scales where the counterpoint has to be inserted, and thereby to a far-out music structure such as the melas from Indian raga tradition. It is not clear whether these intercultural relations can be made more realistic or whether they remain fictitious. Here, more research must be done. But it becomes evident that the extension of mathematical models could open not only new perspectives of historical developments, it could also unfold new perspectives of cultural specializations.

1.4 Performance

The author’s first steps in performance modeling were made 1989-1994 while programming the commercial musical composition software  for Atari computers (Mazzola, 1994). In ’s »AgoLogic« subroutine, a hierarchy of polygonal tempo curves can be defined and edited. The program uses the definition of musical tempo as a piecewise continuous map on the positive reals of symbolic time , measured in quarters and with values in the positive reals, measuring the tempo at symbolic (score) time in units of quarters per minute, , say. Mathematically, the tempo is the inverse derivative of the physical time as a function of symbolic time , as a function of symbolic time, i.e., . The program uses the calculation of physical time via the evident integration of . The hierarchical tempo structure implements the fact that musical tempo is not the same for all notes at a given score time. Rather is the tempo layered in a tree of successive refinements of local tempi. Typically, this looks like this: We are given a >mother tempo< curve , defined on the closed symbolic time interval . In a homophonic piano piece, this could be the global tempo which is played by the left hand. If the right hand should play a Chopin rubato during a subinterval of , then the tempo of the right hand will deviate from the mother tempo in this interval. However, at the start and end times , we ask the hands to coincide. So the daughter tempo of the right hand should have the same integral as the left hand with its mother tempo, i.e.,

By use of adaptation algorithms, the tempo hierarchy subroutine in the  software enables the graphically-interactive construction of such daughter curves, including an arbitrary number of sisters and of genealogical depth for daughters, granddaughters, great-granddaughters, etc. This means that interpretative time is encoded in a ramified tree of genealogical refinement of local tempi.

This first approach was successful on the time level. Therefore, the SNSF grant (1992-1996) for the RUBATO project (Mazzola and Zahorka, 1993-1995; Mazzola and Garbers, 2001) was designed to extend this approach to other parameters, such as pitch, duration, loudness, glissandi, and crescendi. But the  approach