140 Chapter 9
evoke joy, and something else can please and bring sadness. ”^20 His paradoxical context
points to Aristotle ’ s discussion of the “ joy of tragedy, ” the tragic pleasure that purifies the
soul through pity and fear.^21 In Euler ’ s account, sadness and joy both have a precise numeri-
cal degree: mathematics is capable of rendering these seemingly incommensurable mental
states literally commensurable by calculating their common measure. The sadness we
experience hearing minor harmonies is not simply dismay at “ contemplation of the imper-
fect, ” as we would feel hearing a blatantly wrong note. Instead, we locate the minor har-
monies (and dissonances in general) as parts of the larger perfection of the whole musical
edifice. “ For music, since it tries to please, neither intends nor is capable of much sadness.
Thus sadness simply involves more difficult perception of perfection or order and differs
from joy only in degree. ”^22 By providing commensurate degrees for both joy and sadness,
Euler shows the basis on which the mind can integrate and reconcile them in the overall
pleasure conveyed by the entire musical work. In so doing, he also illuminates the nature
of tristitia by revealing it as the sensation of the mathematical mind laboring to understand
difficult ratios.
Even as it struggles, the mind experiences the concomitant pleasure of connecting its
complex labors with the relative resolution felt in simpler states, which are then perceived
as joy. This does not mean that we simply suffer through the sad parts in order to enjoy
the relief of their ending, as if they were a kind of toothache whose passing gives us the
relative pleasure of anguish ended. Though he does not spell it out, Euler ’ s argument
clearly implies that a mind capable of contemplating complex things has a more
intense response to the work of music than would be felt by a less percipient — and less
intelligent — hearer.
This analysis of mental mathematical activity also informs Euler ’ s parallel inquiry why
“ barbarians get little or no enjoyment from our music, ” whether that pleasure comes “ of
familiarity alone ” or because “ there is far more order and agreeableness in our music, of
which only the least part is perceived by the barbarians. ”^23 Though he acknowledges the
power of familiarity, ultimately his argument puts much more weight on the trained mind ’ s
ability to discern order. More complex ratios may indeed weary and sadden us, but they
are the indispensable grounds for our experience of joy, which we know through our
mathematical , even calculational, faculties. Euler thus identifies mathematical awareness
as the core of our ability to experience joy and sadness, whose inherent nature in fact
requires the connected experience of both. The young mathematician here seems to antici-
pate and to welcome his coming lifetime of struggle and triumph, with all the sadness and
joy his endeavors will entail. He implicitly places these under the aegis of music by treat-
ing “ music as a part of mathematics, ” connected just as are joy and sadness.
In the remainder of his Tentamen , Euler gives evidence of such mutual interactions
between music and mathematics. At many points, he goes into considerable musical
detail.^24 Not content only to schematize degrees of agreeableness, he gives detailed theo-
retical examples of increasingly complicated harmonic structures to illustrate his funda-