Introduction 3
called for irrational and rational quantities on an equal footing. Chapter 4 discusses three
central figures in this story: the German mathematician Michael Stifel, who was the first
person to use the phrase “ irrational numbers ” in the course of his exposition of music, but
who then hesitated to grant those numbers full reality; Girolamo Cardano, celebrated
physician-polymath, who gave such quantities even greater significance in his musical
writings; and Nicola Vincentino, a composer obsessed with reviving ancient Greek quarter
tones who found himself in need of what he called “ irrational proportions ” to define these
unfamiliar intervals. Each of these three men was involved with practical music to a degree
correlated with their respective reliance on irrational numbers.
Johannes Kepler, more than anyone, incorporated music into the foundations of his
innovative astronomy. Chapter 5 relates his interest in musical practice to his novel
approach to its theory, which moved him to reject algebraic results that contradicted
musical experience. Kepler ’ s search for cosmic polyphony points to Orlando di Lasso ’ s
In me transierunt as a moving expression of the “ song of the Earth, ” down to the melodic
spelling of the Earth ’ s song. Kepler presents both cosmos and music as essentially alive
and erotically active, based on his sexual understanding of numbers. The pervasive dis-
sonance of the cosmic harmonies reflects the throes of war and eros. Like Oresme, Kepler
realized the essential incompleteness of the cosmic music, which may never reach a final
cadence, a universal concord on which the world-music could fittingly end. Kepler treats
this as an indication of divine infinitude, inscribed in the finite cosmos.
Ren é Descartes began his career writing about music, which affected his innovative
natural philosophy throughout its development. Chapter 6 reads his correspondence with
Marin Mersenne as tracing the interaction between musical, mathematical, and philosophi-
cal themes. Musical observations led to Descartes ’ s initial observations of the overtones
of vibrating strings, which in turn led to wider considerations of mechanics, motion in a
vacuum, and eventually to his continuum theory of the universe. His theories emerged in
constant dialogue with musical issues and problems.
Music was central to the natural philosophy of Mersenne. Chapter 7 begins with his
musical arguments for heliocentrism, against the hermetist Robert Fludd. Mersenne used
musical devices to make pioneering measurements of the frequency of vibrating strings
and of the speed of sound. Mersenne ’ s work on overtones both profited from and struggled
with his musical preconceptions, as did his attempt to incorporate atomism into his account
of vibrating bodies.
Though Isaac Newton walked out of the only opera he ever attended, music had a sig-
nificant place in his work.^4 His early notebooks show his close study of music theory and
his attention to matters of “ elegancy ” in practice. A decade later, he applied the musical
scale to define the colors in the spectrum. Though he had initially assumed that color
spanned a perfect octave from red to violet, chapter 8 discusses Newton ’ s subsequent
realization that it actually spanned a smaller interval. Yet he did not realize that the depar-
ture from an octave implied strong evidence for the wave nature of light.