Euler: The Mathematics of Musical Sadness 137
Euler illustrates his reasoning with a diagram ( figure 9.1 ) showing “ the pulses in the
air as dots placed in a straight line. The distances between the dots correspond to the
intervals of the pulses. ” He sees this diagram as visualizing their degree of understand-
ability and hence agreeableness. At the same time, though, this diagram represents the
coincidences between the “ pulses ” and hence represents geometrically the interrelation
between sound waves. Implicitly, Euler ’ s two different meanings converge: agreeableness
correlates with the alignment of the two wave-forms, which Hermann von Helmholtz made
explicit in his physical theory of consonance over a century later (with due acknowledg-
ment to Euler).^15 In his Tentamen , Euler restricted himself mainly to the traditional just
intonation using simple whole-number ratios, not the newer temperaments intended to
allow free modulation between all keys (see box 4.2).
Within these limitations, Euler ’ s quest for a precise degree of agreeableness informs his
mathematical rankings. In light of this, he chooses the degree always to be integral, never
fractional, “ since in this case the ratio would be irrational and impossible to recognize, ”
implying an underlying rationality to the felt quality of agreeableness. He sets out the
result in a table that goes far beyond the traditional set of musical ratios ( figure 9.2 ). Euler ’ s
mathematical schema leads him to include ratios that have no precedent in traditional
music theory; Zarlino, for instance, argued that only numbers up to six (the senario , as he
called them) are allowable in musical ratios, but Euler makes a case for going past six. In
so doing, Euler makes consonance and dissonance really a matter of degree, as opposed
to the traditional tendency to distinguish sharply between them. He is led to this notably
innovative step by his mathematics, which phrases both in the same general language of
ratios, as well as by his awareness of the expressive power of dissonance.
Euler thus found a new numerical index that, to some extent, correlates with traditional
(and aural) judgments of relative consonance but is far more precise: the lower the degree,
the more agreeable the sound. Yet in his system an interval between two notes can have
the same degree as a triad , which has a more fundamental status in harmony. Worse,
Euler ’ s scheme assigned the same degree to the most familiar triadic harmony (like
C – E – G) as to the dissonant major seventh chord (C – E – G – B) ( ♪ sound example 9.1).^16
Still, Euler ’ s numerical rankings illuminate a long-standing theoretical problem: the status
of the minor mode. After its discovery by Mersenne and Descartes, music theorists realized
that the overtone series provided a natural justification for the major mode because the
first six overtones sound a major triad. Yet the minor mode had no such acoustical justi-
fication. Further, why does the major mode sound “ happy, ” the minor mode “ sad ”? From
where, exactly, does the minor mode derive its origin and its emotive power?
Euler argues that “ everything pleases us in which we perceive perfection to exist, and
so we are pleased more when we observe more perfection. On the other hand, we are
displeased by those things in which we perceive a lack of perfection or much imperfec-
tion. ”^17 Hence, we should absolutely prefer music that keeps to the lowest degrees of
agreeability (in his scale) and be displeased by any deviation toward the higher degrees.