32 Chapter 2
contemporary Phillipe de Vitry.^21 A renowned scholar and friend of Petrarch, de Vitry had
written his Ars nova notandi in 1322 or 1323, setting forth the novel rhythmic procedures
of the “ new art. ” De Vitry ’ s own musical compositions are exemplars of this new idiom
( ♪ sound example 2.1). Oresme dedicated his mathematical work Algorismus proportio-
num to de Vitry, “ whom I would call Pythagoras if it were possible to believe in the return
of souls ... so that if it is agreeable to your Excellency you may correct that which I put
before you. For should it be approved by the authority of so great a man and corrected
after his examination [of it], everything that has been revised by your correction would be
an improvement. Then, if a disparager should open his mouth and set his teeth to rend
[my work] into pieces, he would not find [what he seeks]. ”^22 Oresme ’ s Algorismus pro-
portionum makes no overt reference to music; it treats the addition and subtraction of
“ rational ratios ” and “ irrational ratios, ” evidently as part of his larger project to understand
their relation. Oresme ’ s prologue considers de Vitry not only au courant with this advanced
mathematical investigation but capable of judging and correcting it, probably also of
approving and applauding it.
Thus, Oresme ’ s praise of the canticum novum accords with his dedication to the prince
of the new musical art. Given his acquaintance with de Vitry, Oresme surely knew his
treatise Ars nova notandi , which contains a considerable amount of mathematical detail
as part of its exposition of the new notational possibilities he exploits in his motets.^23 Both
men were part of larger currents of mathematical and musical speculation. At several
points, Oresme acknowledges the prior work of de Muris, de Vitry ’ s peer among the older
generation of music theorists. De Muris ’ s writings were also sources for the new musical
practice, especially his Ars novae musicae (1319), whose title also registers the sense of
musical innovation. As noted above, Oresme drew on de Muris ’ s work on commensurable
and incommensurable quantities.^24 These matters occupied Jewish as well as Christian
scholars. De Vitry asked Levi ben Gershon (Gersonides) to help him resolve a mathemati-
cal question bearing directly on music; in turn, Oresme used this musical result to make
an astronomical argument. This interchange illustrates the interweaving of mathematical,
musical, and astronomical issues in the works of these men.
According to Boethius, the basic musical ratios are superparticular , meaning that they
have the form n :( n + 1), such as 2:3 or 3:4. The more complex intervals derived from this
primal set involve only powers of 2 and 3. Besides these, de Vitry suspected that no other
compound superparticular intervals could exist. Having learned of Gersonides probably
from his Maaseh Hoshev ( Work of Calculation , 1341), a Euclidean compilation of results
in arithmetic, de Vitry asked Gersonides whether he could prove his conjecture, which he
did in his brief De numeris harmonices (1342).^25 De Vitry probably was interested in this
result less for its application to the ratios governing musical intervals, which were not
really under controversy at the time, than for its implications for rhythmic notation, the
subject of much controversy between the practitioners of ars antiqua and ars nova.^26 De
Muris had already set out the complicated rhythmic issue at stake; Gersonides ’ result