156 Chapter 10
Thus, Euler ’ s concept of frequency, perhaps his most important contribution to the
theory of light, comes directly not just from the general analogy with sound but from the
specifically musical concept of a single pitch: as our perception of high or low pitches
depends on how many times a second our ear receives sound pulses, so do our eyes dis-
tinguish more or less frequent impacts by their color. When he goes on to address Newton ’ s
seminal finding that white light is a composite of many colors, Euler took his musical
analogy one step further by explicitly comparing a “ composite ray ” of light to a multinote
chord, under the implicit premise that the eye blends the “ notes ” of that chord into a single
perception of color (say, white), whereas the ear does not blend the chord tones but hears
them as separate, though perhaps related harmonically. Euler ’ s ingenious suggestion,
however, raises the unacknowledged question: do “ composite ” light rays themselves really
blend or do they keep separate their constituent chordal (and separately pure) “ notes? ”
Yet this bold application of music to harmony represents, for Euler, the power of the
sound – light analogy, when taken to its furthest extent.
In the remainder of his treatment of light, Euler continued to use the musical underpin-
nings of his theory to guide him, especially in difficult cases, such as the problem of the
colors of opaque bodies. Newton had somewhat tortuously argued that their colors tended
to come from iridescent layers, such as the colors in soap bubbles, a peacock ’ s tail, or thin
layers of air (as in Newton ’ s rings; see figure 8.4). Even the blue of the sky was, for
Newton, to be understood in terms of such seemingly evanescent phenomena.^10 Euler
emphasizes that Newton ’ s examples of iridescence have very different appearances and
colors when seen from different angles, whereas most opaque bodies do not exhibit any
such iridescence. Instead, Euler compares the vibrating particles of opaque bodies to a
number of taut strings, each one resonating only at its own particular frequency.^11 He
carries this comparison even further to hypothesize color overtones, analogous to those
produced by sounding bodies: “ Let us suppose that a ray representing a red color carries
f pulses to the eye in one second; and, just as in music sounds are held similar which have
vibrations, produced in the same length of time, that bear a double, quadruple, eight-fold
etc. ratio [to the main tone], so simple rays containing, in one second, 2 f , 4 f , 8 f etc. or ½ f ,
¼ f , ⅛ f etc. vibrations, will all be considered red. ”^12
Euler offers no proof of these assertions, other than the analogy with musical sounds,
which he seems to treat as established fact; one wonders whether he meant to go so far as
to include all the higher overtones of a given frequency, not just those corresponding to
octaves. Even more striking, he also includes under tones, those corresponding to fractional
underoctaves below the fundamental frequency f. This he could not have based simply on
acoustic theory; following Mersenne, only overtones had been experimentally recognized.
Because of this, we must conclude that Euler got his idea of undertones not from natural
philosophy as such but from music theory, particularly that of Rameau. Euler corresponded
with Rameau, who, though respectful, was critical of Euler ’ s degrees of agreeableness.
Euler was surely aware of Rameau ’ s concept of sous-entendre , which held that the hearer