Euler: The Mathematics of Musical Sadness 135
pleasing. In the whole discussion, I have necessarily had a metaphysical basis, wherein
the cause is contained why a piece of music can give one pleasure and the basis for it
is to be located, and why a thing to us pleasing is to another displeasing. ”^8 By Euler ’ s
time, music was well on its way to its present status as a fine, rather than liberal, art,
grouped with painting and architecture rather than with mathematics. Not satisfied with
the classical accounts, Euler wanted to find new principles connecting mathematics
with music and pleasure.
Euler begins his Tentamen by reviewing his earlier work on the physical basis of sound,
understood as “ the perception of successive pulses which occur in the air particles situated
around the ear. ”^9 He reviews the mathematics of strings, vibrating bodies in general, and
his own “ entirely new theory of sounds provided by wind instruments. ”^10 Though he takes
note of the Pythagorean teachings about musical ratios, he seeks to put them on a new
mathematical basis. As he noted in a 1752 letter to the great composer and theorist Jean-
Philipe Rameau, “ the Pythagoreans were early misled in their numbers and treated them
capriciously, as when they maintained that only superparticular ratios furnished conso-
nances, a principle devoid of all foundation, and in this regard the Aristoxenians were right
to mock their false theory. ”^11 Where Boethius had assumed that simple ratios like 1:2
(octave) were more perfect than complex ones like 243:256 (semitone), Euler wished to
demonstrate that they were more pleasurable and to calculate the exact degrees of pleasure
involved.
The difference in these fundamental categories reveals the profound shift toward an
aesthetics premised on sentiment and pleasure, rather than pure order and its concomi -
tant goal of moral perfection. In his reply to Euler, Daniel Bernoulli expressed some
puzzlement:I cannot readily divine wherein that principle should exist, however metaphysical, whereby the
reason could be given why one could take pleasure in a piece of music, and why a thing pleasant
to us, may for another be unpleasant. One has indeed a general idea of harmony that it is charming
if it is well arranged and the consonances are well managed, but, as it is well known, dissonances
in music also have their use since by means of them the charm of the immediately following con-
sonances is brought out the better, according to the common saying opposita juxta se posita magis
elucescunt [opposites placed together shine brighter]; also in the art of painting, shadows must be
relieved by light.^12Bernoulli shares the common presupposition that pleasure is fundamental, but he does not
see how it could be mathematically or metaphysically grounded beyond itself. As if
expressing the widely shared admiration of sentiment, Bernoulli relies on purely intuitive
notions of pleasure through contrast, which his example from painting underlines; if so,
the fine arts all share this fundamentally nonmathematical reliance on contrast. His invoca-
tion of charm sustains this aesthetics of ineffable sentiment.
Euler seeks to combine what his friend considered two rather antithetical approaches.
To find a mathematics of sentiment, Euler had to make a new construct, for the traditional