Hearing the Irrational 57
Figure 4.1
Jacques Lef è vre d ’ É taples ’ s diagram from Elementa musicalia (1496), demonstrating that the interval Ab : bc can
be divided geometrically exactly in half ( bg ). If the two collinear segments Ab and bc are in the ratio Ab : bc ::
8:9, then (Euclid, Elements 6:3) erecting the perpendicular bisector bg on Ac gives the mean proportional Ab : bg ::
bg : bc. Thus, a string of length bg would sound an exact semitone higher than string Ab ; because 8: bg :: bg :9,
in modern notation, bg = 72 , hence the “ ratio ” of a semitone is 8: 72.division, despite its mathematical irrationality, in all senses. To be sure, Euclid ’ s Elements
used “ irrational proportion ” to denote “ incommensurable quantities. ”^10 Though Oresme
referred to rational and irrational proportions, he never connected number with irrationality
because “ no irrational ratio is found in numbers. ”^11 Yet neither Euclid nor Oresme ever
overstepped the boundary between rational and irrational, as Stifel does. For instance,
Stifel divides the tone into equal semitones following a Euclidean construction ( figure
4.1 ).^12 Likewise, Stifel applies various arithmetic operations to musical proportions, noting
that “ in these ways irrational proportions of irrational terms can be computed by rational
numbers through this beautiful reckoning, ” including his explicit halving of the tone ( figure
4.2 ).^13 To my knowledge, this is the earliest printed statement that combines a rational
(arithmetic) proportion with its irrational (geometric) mean. Acknowledging the contro-
versy about them, Stifel still asserts that “ these halvings are so certain that no one can
deny them. ... A tone can be divided by an uncertain number and that is constituted by
no assembly of units, that is, by an irrational number. ”^14 Thus, in this musical context,
Stifel treats these irrational “ halved ratios ” as though they are as valid as rational
proportions.
Yet when Stifel returns to the larger question of “ the essence of irrational numbers, ” his
attitude shifts: