90 Chapter 6
According to Beeckman, Descartes told him, “ Really you are the only one who has
reawakened me from idleness. ”^4 Beeckman noted that Descartes “ says he had never found
someone, except for me, who is accustomed to study in the way I prefer and accurately
joins mathematics and physics. And for my part, I have never spoken with anyone apart
from him who studies in this way. ”^5 Aristotle had argued that mathematics applies to the
regular motions of the heavenly bodies, not to physis, the earthly realm of growth and
change, because mathematical concepts were inherently unchanging. Around 1600, Francis
Bacon coined the word “ mixed mathematics ” to describe mathematics applied to physical
problems; a similar term was used by Marin Mersenne, with whom Descartes also formed
an important relationship through discussion and correspondence.^6 Though Beeckman and
Descartes did not use the term “ mixed mathematics, ” they thought that understanding
nature required bringing mathematics together with natural philosophy in new ways, first
of all in the realm of music.
Beeckman was not really familiar with practical music, though interested in theoretical
questions, and he relied on Descartes for mathematical results beyond the elementary
level.^7 Descartes ’ s exposition in the Compendium is a fascinating mixture of old and new;
though Cohen has noted that it “ adheres so closely to the Renaissance style of music theo-
rizing, ” it also casts music in a new light by reconsidering its relation to mathematics and
to physical sound.^8
Descartes ’ s terse opening words signal a shift in thinking about music, “ whose object
is sound, ” as he puts it. As Suzannah Clark and Alexander Rehding observe, “ with only
slight exaggeration these four words sum up the impact of the scientific revolution on
music — the change from music as a divine force to music as a material phenomenon. ”^9
While agreeing with their general point, I think its causal force also works in reverse.
Because Descartes ’ s musical essay precedes the rest of his work, it is more coherent to
read it as a musical argument that contributed to the formation of the new natural
philosophy.
Descartes takes music as a perfect exemplar of his nascent project to “ accurately join
mathematics and physics, ” mediating between arithmetic and geometry and their physical
manifestations. As sound (rather than divine afflatus), music aims “ to please and to arouse
various emotions, ” which are empirical, sensual states, rather than idealized types. Their
intensity gives an observable magnitude of excitation, analyzable into “ differences of
duration or time, and its differences in tension from high to low. ” “ The quality of tone ”
of the sounding body he assigns to “ the domain of the physicists [ Physici ], ” one of the
first connections between physici and music. Until then, music ’ s place was next to astron-
omy, the changeless heavens, rather than physis , the sublunary realm of change.
Pleasure is essentially a geometrical magnitude, “ a proportional relation [ proportio ] of
some kind between the object and the sense itself. ” Within the limits set by our sensory
capacities, Descartes analyzes this pleasure-magnitude into sensations that do “ not fall on
the sense in too complicated or confused a fashion. ” Looking at an astrolabe ( figure 6.1 ),