Descartes’s Musical Apprenticeship 93
(which previous theorists treated first), Descartes shows that we feel number with the full
force of a martial drum.
Descartes observed that when a lute string is plucked, “ the force of its sound will set
in vibration all the strings which are higher by any type of fifth or major third, but nothing
will happen to those strings which are at the distance of a fourth or any other conso-
nance. ”^16 Here he made an important step toward the discovery of what later were called
overtones , showing the physical connection between the octave, fifth, and third, rather than
what the ancient theorists considered a purely numerical relation. Descartes correlated the
mathematical division of the octave with physical experience, giving the first example of
his new mathematical physics through the intermediacy of music.
Descartes continued to think about music long after his little Compendium , as shown
in his correspondence.^17 He often included music among other topics in physics and math-
ematics. Ten years later, in his first letter to Mersenne from Holland in September 1629,
Descartes begins by asking how one consonance can pass into another, such as “ might
offer all the diversity of music, ” then turns abruptly to a new “ part of mathematics I call
the science of miracles ” that could produce astonishing illusions.^18 His next letter describes
working through “ all the Meteores [celestial phenomena] ” such as parhelia (mock suns,
figure 6.3 ), pointing to what would become his work of that name, published in 1637 as
an appendix to his Discourse on Method , along with La Geometrie (on his algebraic
approach to geometry) and Dioptrique (optical phenomena and vision). Descartes tells
Mersenne that he is preparing “ a little treatise that will contain the explanation of the colors
of the rainbow, ” that archetypal miracle, but begs him “ not to speak of this to anyone in
the world, for I have decided to show this in public as a sample of my philosophy and to
hide myself behind the picture to hear what they will say of it. ”^19 To his friend, Descartes
discloses the masked persona he will don so that he can present his work, gauge its
response, yet remain safely hidden.
This letter goes on to discuss questions of rarefaction (which might be able to explain
the ancient concept of ether) and thence to a certain “ book of cameos [here meaning
monochrome painting on jewelry or enamel] and talismans, ” which he judges “ only con-
tains chimaeras, ” showing his disdain for the ordinary run of trompe-l ’ oeil or occult
“ wonders. ”^20 He then returns to the ways in which a unison could pass to a major versus
a minor third; Descartes shows a sensitive awareness of the relation of this particular issue
to the whole context of the composition in which it might occur, including the relative
motions of treble and bass. He seems familiar with contemporary musical practice, not
just theory in the abstract.
After this paragraph on music, Descartes turns to another question Mersenne posed:
How does a pendulum move in “ empty space, ” where the air gives no hindrance? Galileo ’ s
response was not published until 1632 and hence was unknown to these men. Descartes
makes a calculation dependent on the length of the string and arrives at fractions among