The Dream of Oresme 33
confirmed the superiority of the ars nova notation over its older rival. Thus, musical ques-
tions had led to a question in arithmetic , whose result then bolstered one side in the
antecedent musical controversy. Oresme used the implications of this result to argue for a
change in the fundamental concept of the harmony of the spheres, both musically and
astronomically.
In his Tractatus de commensurabilitate , before the concluding dialogue between Arith-
metic and Geometry, Oresme investigates the precise relation between the relative ratios
of revolution of heavenly bodies and their conjunctions, the occasions at which they would
occupy the same apparent position in the sky. His Proposition 11 demonstrated that the
number of such conjunctions in any one revolution is given by the difference between the
two terms of the ratio between the velocities of the two bodies, here assumed to be rational.
He then notices the deep astronomical and musical problems this result implies:If the ratio of velocities of any two celestial mobiles were in any of the principal harmonic ratios in
music, namely the diapason [octave, 1:2], diapente [fifth, 2:3], diatesseron [fourth, 3:4], and tone
[8:9], which make a concord or harmony, the mobiles will never conjunct except in one place only,
since the least numbers of such a ratio differ only by a unit. As an example, if the mean motion of
Mars were exactly twice the speed of the sun ’ s mean motion, there would never be a middling
conjunction of these two bodies for [they would conjunct] in only one place, or point. ... Since no
configuration consisting of two motions is found to occur in only one point of the sky, [it follows]
as a consequence that no two celestial motions have velocities related in a principal harmonic ratio.
Therefore, if celestial bodies in motion produce a harmony, it is not necessary [to assume] that such
a harmony arises from the velocities of their motions, but perhaps it stems from some other source
for other reasons, as will be seen later.^27Here Oresme relies on a long-known finding of observational astronomy that the conjunc-
tions of the planets occur at different points in the sky, not just one. Where the theorem
of Gersonides proved that the “ principal ” harmonic ratios were all superparticular, Oresme ’ s
corollary now showed that any putative celestial music based on those intervals was
incompatible with the visible evidence of astronomy , based on the common assumption
that the “ harmonies ” related celestial velocities. This argument showed that the musical
program of harmonizing the cosmos mathematically contradicted its own starting point,
as it had been commonly understood. Oresme then generalized these results to several
moving heavenly bodies or to a single body moving in several ways at once.
This, then, is the framing question that surrounds the debate between Arithmetic and
Geometry: what is to become of the music of the spheres in light of these results? On the
face of it, they seem to contradict the common assumption of rational (commensurable)
relations between planetary orbits. As the patroness of this assumption, Arithmetic protests
that “ a position contrary to ours would destroy the beauty of the universe and detract from
the goodness of the gods. ”^28 Though she also mentions Oresme ’ s result that “ if the motions
are incommensurable, it is impossible that they all return [to the same place] on their
circles, ” she seems to ignore Oresme ’ s corollary to Proposition 11, which effectively