158 Chapter 10
succeeding thirty years. For instance, he changed his mind several times about whether
red or violet had the higher frequency; his voltes faces caused some amusement among
savants. But his main initiatives turned from theory to practice; he demonstrated the pos-
sibility of a lens that could be free of chromatic aberration, which would be of great
importance for all optical instruments, such as eyeglasses, telescopes, and microscopes.
This aspect of his work culminated in his magisterial General Theory of Dioptrics (1765).
His work on acoustics itself continued with studies of the propagation of sound in the
atmosphere (1759). During those decades, Euler also returned a number of times to musical
questions, demonstrating that his interest was not merely a youthful fancy but a continuing
preoccupation alongside his other work in mathematics and natural philosophy.
As with his later work on optics, Euler ’ s interests in music became more practical,
devoted more to issues closer to the composition and performance of music than to its
theoretical foundations. His 1764 paper “ On the True Character of Modern Music ” took
up a theme already broached in his 1730 Tentamen , the possibility of using intervals having
the number 7 in their ratios.^15 Euler sets forth a contrast between “ ancient ” and “ modern ”
music, which he evidently assumed would be clear to his readers. Though he does not
specify the exact chronology or stylistic periods, his distinction seems very close to that
of Vincenzo Galilei, contrasting the serene polyphonic practice of composers like Pal-
estrina to the expressive, monophonic art of the early operas, reviving the fabled powers
of ancient Greek music. Among composers or theorists, Euler refers only to Rameau, a
preeminent “ modern. ” Euler ’ s own preferences emerge in his characterization of modern
music as “ sublime, because its character consists in a higher degree of harmony, ” compared
to ancient music as “ common [ commune ], ” in the sense of adhering to common harmonic
practice. Yet he never cites a single musical example that would give specific insight into
his compositional tastes. Disconcertingly, his sole musical example is a formulaic cadence
that violates elementary rules of voice-leading by allowing parallel octaves ( figure 10.3a ;
♪ sound example 10.1). Were these solecisms just typos, or did the great mathematician
finally have a tin ear?^16 Or was he quoting crude hymnody he remembered from the Cal-
vinist services of his childhood?
Euler lived in an age of vivid musical controversy, between French and Italian styles,
between partisans of ancient and modern practices. A full study of the issues involved
would call for another book; in our present context, Euler is offering an implicit defense
of the modern practice of Rameau and others by explaining its modernity as a new
freedom in the use of dissonance. Euler perceptively locates this modern dissonance in
what he himself and his contemporaries were beginning to call the chord of the dominant
seventh ( figure 10.3b ; ♪ sound example 10.2). Euler argued that the aural pleasure of the
progressions enabled by this chord justifies its dissonance. Quoting Rameau, Euler notes
that this dissonance serves to alert the hearers to the key they are in; in figure 10.3a , the
tritone B – F in the penultimate dominant seventh chord tells us that we are in C major