Descartes’s Musical Apprenticeship 95
a plucked string, which he clearly considers a direct continuation of the problem of the
pendulum swinging in a vacuum. What follows gives a window into his emergent thought
process, especially its twists and turns in the face of difficulty and paradox. Descartes
treats the string as making “ turns and returns ” away from its equilibrium position in a
vacuum, each point along the string behaving essentially like the pendulum he had just
considered. He concludes that the string ’ s vibrations would damp down geometrically: if
its first vibration had amplitude 4, the next would have amplitude 2; if it began with mag-
nitude 9, then would follow 6, 4, ... — again those musical numbers, here illustrating a
quintessentially musical phenomenon. Then he hesitates: “ I said in vacuo , but in air I
believe that [the successive vibrations] will be a little slower at the end than at the begin-
ning because, the movement having less force, it will not overcome the resistance of the
air so easily. ”^23 Air is evidently more resistive than empty space, but suddenly he seems
to recall the Aristotelian paradoxes of motion in a void, as he continues: “ However, I am
not sure of this and perhaps also the air, on the contrary, may help [the string] at the end
because the movement is circular. ”
This moment of confusion, especially in this proudly lucid mind, gives invaluable evi-
dence of his struggle to make physical and mathematical sense of a phenomenon that, so
far, had purely been treated arithmetically by music theory. In the face of this hornet ’ s nest
of problems, Descartes ’ s response is telling: “ But you can experience [ experimenter ] it
with the ear by examining whether the sound of a string thus plucked is higher or lower
at the end than at the beginning, for if it is lower, that means that the air slowed it; if it is
higher, then the air made it move faster. ” His purely mathematical arguments rely on a test
by experiment, and a musical one at that. In the course of this letter, the problem of the
plucked lute string that had begun as a purely musical phenomenon passes through a
middle stage of mathematization (the analogy with a chain of tiny linked pendula), and
finally returns to the realm of musical experience: the ear can test the exact influence of
the resistive air.
Descartes ’ s letter continued to discuss the vibrating string, but the remainder of his text
has not survived. His next letter (December 18, 1629) returns to these matters in even
greater detail. Their interchange about optical phenomena now includes Descartes ’ s doubts
about Mersenne ’ s claim to have seen a colored “ crown ” around a candle flame, as if it
were a miniature mock sun. Descartes also asks Mersenne, as a cleric, about the possible
danger in speculating about natural philosophy in directions contrary to Aristotle, “ for
it is almost impossible to express another philosophy without it immediately seeming
against the faith. ”^24 He is worried about whether there might be anything “ determined
by religion regarding the extension of created things, namely whether they are finite
or rather infinite, and whether in all those lands one calls imaginary spaces there might
be true and created bodies, for as yet I have not wanted to touch this question. ”^25
This hypothetical question gives a preview of Descartes ’ s nascent project, to propose
a new (and distinctly post-Aristotelian) natural philosophy disguised as the description