2 Introduction
As Horatio has ties to both Hamlet and Ophelia, music touches both natural philosophy
and mathematics. Accordingly, we will examine several critical shifts of understanding
in both these domains. Chapter 1 describes the most consequential move of all: ancient
Greek natural philosophy connected music with mathematics and astronomy within a
fourfold study, the quadrivium. This alliance involved both experiment and theory, and
hence positioned music at the frontier between the worlds of physical sensation and ideal
forms. During the subsequent fifteen centuries, music retained its central place among the
mathematical sciences, an essential component of what became “ liberal education, ” which
explicitly continued a program the Pythagoreans began, Plato systematized, and Boethius
transmitted to the West. These ancient developments are important not merely as historical
background, buried as hidden sediment under the surface of modernity, but as continually
active and reemergent forces that shaped and continue to shape “ modern ” science and
mathematics.
Chapter 2 considers the status of these forces as they seemed to a preeminent fourteenth-
century natural philosopher. Nicole Oresme used musical concepts as important elements
in his reexamination of the cosmos, its possible cycles, and their relation to arithmetic and
geometry. The case of Oresme shows how significant musical issues were well before the
advent of the “ new philosophy ” of the sixteenth century. Oresme ’ s contact with the “ new
music ” ( ars nova ) led him to refute the simplest versions of cosmic harmony and to
propose radical new alternatives that depend on the tension between arithmetic and geom-
etry. The underlying musical and cosmological considerations allow us to deduce his own
unstated conclusion in favor of geometry.
Disputes about the order of the planets had profound musical and cosmological implica-
tions, especially on the growing controversy over whether the seemingly immovable Earth
could be understood to move, as the Pythagoreans held. Though he had presented strong
arguments in favor of this view, Oresme himself finally accepted the geocentric account.
Chapter 3 investigates the role of music in this cosmological controversy as it came to a
head in the fifteenth century. The problem of a seemingly immovable yet moving center
such as the Earth parallels the musical problem of changing the usually fixed modal center
of a composition. Despite its proverbial impossibility, the theorist Heinrich Glarean drew
attention to just such a shift of mode in Josquin des Prez ’ s motet De profundis. If so, music
might show how the immovable could move, after all. Indeed, harmony became the defin-
ing issue on which Copernicus and those following him phrased their arguments about the
new cosmology. Musicians followed this controversy closely; in his 1588 polemic for a
revival of ancient musical practice, Vincenzo Galilei was among the first Italians to defend
the heliocentric view, many years before his celebrated son Galileo took up this cause,
again under the banner of harmony.
Several decades before Vincenzo Galilei ’ s surprising avowal, music influenced funda-
mental changes in the concept of number. Though irrational quantities had long been
excluded from arithmetic and harmonics, sixteenth-century musical theory and practice