Eleanor Selfridge-Field

The Music of J. S. Bach:

Paradigms for the Age of Technology

Abstract

Facile equations between music and mathematics have been drawn for more than two millennia. It would seem that the tools of the past 20 years would be especially adept at providing specific examples of this assumed correspondence. When we look, however, at the paradigms currently being implemented in computer applications to studies of historical repertories, mathematical models are seldom found. This is especially surprising for the music of J. S. Bach, which is often taken as the supremely 'mathematical' repertory of the European past. Current paradigms are examined with reference to the particular aspects of musical experience to which they seem best to pertain.

Some Paradigms of the Past

Like all analogies, the exact correspondences become faint under scrutiny. 'Mathematics' has been a general rubric under which one may find discussions of logic, systematic procedure, numerical operations, geometrical relationships, and, in the musical analogy, the simple process of counting. 'Music' has had as its main manifestations the logical work conceived as an orderly whole and studied from the written score, the sounding work perceived through the act of performance, and the expressive result as received by the listener. There are myriad couplings of these phenomena, and some of the equations are obviously more apt than others.

The ancient Greeks took great care in differentiating these many phenomena. Harmony was seen to be related to arithmetic by Aristotle and modes were seen to influence behavior by Plato. Much of the 'mathematics' discussed in ancient writings is, however, related to physics, that is to tuning systems. In Roman times, music was likened to astronomy. The science of moving bodies and the science of moving tones were seen as natural allies in the quadrivium. It is important to remember that in this culture the composer and the performer were normally one. The joint experience of composition and performance was founded on a commanding knowledge of musical scales and thus grounded in the physical world.

The 'Music of the Spheres' metaphor that dominated the Renaissance thinking retained but reworked the astronomical association. Often articulated by priests, this metaphor acquired a Christian overlay, for the angels who dominated the spheres made their own heavenly music. The great sixteenth-century theorist Gioseffo Zarlino was extraordinarily facile in drawing parallels between such things as the ordering of the modes and the ordering of the planets. The Copernican revolution changed the underlying context of many of the constructs of Zarlino and other philosophers of