Music and the Making of Modern Science

Euler: The Mathematics of Musical Sadness 139

But when we calculate the degrees associated with major versus minor triads, in general

and in the most common keys, the major triads are of lower degree (see box 9.1 ). If so,

we should always prefer music with major triads and avoid minor ones, which “ will be

almost painful. ”^18 Yet Euler knows that music contains many minor triads, and he points

out that “ the more easily we observe the order in a given thing, the simpler and more

perfect we consider it, and therefore we receive pleasure and delight from it. On the other

hand, if the order is discerned with difficulty and seems less simple and distinct, we per-

ceive something like sadness [ tristitia ]. In either case, as long as we sense order, the given

object pleases, and we conclude that the object has agreeableness. ”^19 Euler thus connects

the greater “ difficulty ” of minor intervals (and higher-degree “ dissonant ” intervals) with

their perceived affect of sadness. Hence, he explains the sadness of the minor mode (for

instance) as the direct correlate of its epistemological status: what is harder to know is felt

to be sad simply because we struggle to discern its order. In that sense, sadness seems to

be the felt effect of the pain we experience in the face of cognitive dissonance.

Euler interprets this sadness as part of the larger project of the pleasure conveyed by

music, which includes both happiness and sadness. He connects the mathematics of

sadness with the experience of drama: musical harmonies “ are like comedies and tragedies

all of which should be filled with agreeableness. The comedy should fill the spirit with

joy and the tragedy should convey sadness. Thus it is clear that something can please and

Figure 9.2

Euler ’ s table of the first ten degrees of agreeableness of musical intervals.