was, of course, a central tenet of Lutheran theology in Bach's time and remained a guiding force in German musical thought and philosophy through the end of the last century. As in the animation of Herr Gott, nun schleuss den Himmel auf, its attainment was hypothetically possible but never achieved.

The possibility of perfection inhered in the static rationalization of the musical work, not in the temporal experiences of composition, performance, or listening. The dynamic element of time,

to which the expressive aspect of music seems to be so inextricably linked, is better accommodated by models of process and procedure than by those of a complete, fixed product. It proves, on a closer historical examination than can be given here, that the mathematical paradigm has usually been heavily allied with abstract notions of musical material rather than with KlangArt in the direct sense.

Virtual music belongs to a separate realm from that of abstract rationalizations of musical form

and the like. At best, paradigms based on abstract rationalizations incorporate what is hypothetically dynamic into what is actually static. The ancient idea of 'sciences' of motion still holds something of value, for it concentrates on the ever-changing constellations instead of the snapshot view of the heavens. Yet our world and our extra-musical assumptions differ in profound and fundament always from those of antiquity.

One irony of our times is that having gained the power systematically to examine the higher

mathematical phenomena long presumed to exist in music (especially the music of Bach), we choose to direct out energies elsewhere, not because mathematics is out of fashion but because the idea of perfection can only be realized in fixed states and finished products, while the existing tools of technology are better suited to revealing what is endlessly mutable.

References

Alexander Russell Brinkman, Johann Sebastian Bach's Orgelbüchlein: A Computer-Assisted Study of the Melodic Influence of the Cantus Firmus on the Contrapuntal Voices, Ph. Dissertation, Eastman School of Music 1978

Alexander Russel Brinkman, Johann Sebastian Bach's Orgelbüchlein, in: Music Theory Spectrum 2, 1980, p. 46-73

Nicholas Cook, Structure and Performance Timing in Bach's C major Prelude (WTCI): An Empirical Study, in: Music Analysis 6/3,1987, p. 257-272

David Cope, Experiments in Musical Intelligence (EMI): Non-Linear Linguistic-based Composition, in: Interface 18,1989, p. 117-139

David Cope, Computers and Musical Style, Madison and Oxford 1991

David Cope, Recombinant Music: Using the Computer to Explore Musical Style, in: IEEE Computer, 24/7, 1991, p. 22-8