music, but the astronomical association lingered in the Baroque age. Great value was now attached to the expressive nature of music, and it was the duty of the composer to ignite in mortal man a direct perception of these heavenly hosts and thereby enable the hearer to receive their spiritual messages.

In the Enlightenment, the dynamic element that was captured in the metaphor of the spheres was somewhat diminished by more mundane, practical conceptions of music that have come down to our own time as proofs of the 'mathematical' nature of music. In 1714 Bach's contemporary Leibniz wrote that music is a kind of counting performed by the mind without knowing that it is counting. Writing in the late nineteenth century, Debussy maintained that music is the arithmetic of sounds, as optics is the geometry of light. As recently as 1974 Gustav Holst (best known for his orchestral suite called The Planets) said: When I am composing, I feel just like a mathematician.

The idea has persisted in the twentieth century that the music of J. S. Bach represents a paramount expression of mathematical principles. This notion appears to be bound up with a logical approach that largely excludes both temporal information and aural experience. It is therefore wholly different from the ancient analogy of astronomical motion.

At a general level, many of Bach's collections of instrumental works systematically explore a set of relationships. These relationships are sometimes based on principles of sound (e.g., the exploration of bodies of diverse timbral composition in the Brandenburg Concertos), sometimes on principles of compositional technique (e.g., the simple schemes of imitation surveyed in the two-part inventions, complex aspects of contrapunctal technique in the Art of Fugue, melodic elaboration in the Goldberg Variations), and sometimes of both simultaneously (e.g., the possibilities offered by the control of temperament in conjunction with improvisatory and contrapuntal techniques in the Well-Tempered Clavier). To the extent that Bach's works focus on compositional technique, they are indestructible when translated to an electronic context. To the extent that they focus on particular sonorities, the electronic means to do them justice do not fully exist at this time (I refer here to deficiencies in simulating the sound of stringed instruments and obsolete winds).

One might indeed argue that a systematic approach to sound is not the equivalent of a mathematical approach to composition. Yet to a generation of computer scientists and artificial intelligence researchers, the music of Bach has offered a special challenge to achieve a new level of understanding of the intricacy of design. Douglas Hofstadter's virtuoso study Gödel, Escher, Bach draws many parallels between the processes by which physical, temporal, and visual space may be represented. Where others find clarity, Hofstadter finds a significant degree of ambiguity, but this ambiguity results from processes that many would describe as mathematical. In current computer-based research projects concerned with the music of Bach and other common-practice composers it appears that as computer technologies evolve and applications multiply, so also paradigms for musical understanding evolve and multiply. Apart from essays such as Hofstadter's, mathematical paradigms are curiously few.