Euler: From Sound to Light 157
would “ sup-pose ” or “ hear below ” the lowest notated pitch the fundamental bass needed
to make sense of an inverted chord, in which that fundamental note may be buried in the
middle. In Rameau ’ s own theoretical writings, it is not clear whether he took this sous-
entendre to be essentially an action of judgment, though perhaps almost unconscious,
whereby the hearer gravitates toward the fundamental bass, “ the musician ’ s invisible guide,
which has always directed him in all his musical works without his having yet noticed
it. ”^13 In this reading, sous-entendre is an act of implication , of discerning a note that is not
physically there but whose presence is implied by the other sounding pitches. Alternatively,
the sous-entendre could be interpreted as responding to a subtle physical phenomenon of
undertones, vibrations below the fundamental pitch of a string precisely on the analogy
of overtones.
The notion of undertones went on to a long history of its own; as late as 1875, the
musicologist Hugo Riemann made delicate nighttime experiments trying to hear the under-
tones of a piano.^14 Though this extreme view of audible undertones eventually fell into
disrepute, its significance here is independent of experimental judgment. Euler was moved
to assert color overtones and undertones simply on the authority of Rameau ’ s harmonic
theory and its agreement with Euler ’ s sense of mathematical symmetry. The only compa-
rable example of a natural philosopher being so swept up by the prior force of musical
theorizing may have been Newton himself trying to impose the scale on the spectrum.
Though Euler proposed his color undertones already in his 1744 summary announce-
ment, he reiterated and expanded them in his 1746 “ New Theory. ” Just as a musical octave
includes many microtonal pitches, some of which have not been named, so too the color
spectrum contains many unnamed colors, along with those whose names reflect their
“ musical ” interrelation. This may have been Euler ’ s version of Newton ’ s musical spec-
trum; Euler seems to imply, as Newton had, that the well-known colors correspond to the
principal notes of the musical scale. In his 1744 “ Thoughts, ” Euler treated the higher
“ octaves ” of color as appearing brighter and more vivid than the lower. By his 1747 “ New
Theory, ” Euler is aware of Newton ’ s mistake: the extreme visible frequencies in sunlight
differ less than by a factor of two, the octave factor Newton had implicitly assumed in his
color scale. Thus, Euler recognizes that the naked eye cannot discern an “ over-red ” that
would be an “ octave ” 2 f above the frequency of ordinary red, f.
Even though the naked eye could not discern these “ derivative colors ” that represent
octaves above or below those we see, Euler considered that such undercolors or overcolors
might have some experimental reality, perhaps observable through the phenomena of reso-
nance or sympathetic vibration: a body resonating to red might also perforce resonate to
over-red, which shares the same submultiple frequencies as red. Essentially, Euler ’ s
musical theorizing drew him to propose the physical existence of what we now call infrared
and ultraviolet light, demonstrated experimentally around only 1800, as we shall see.
Euler ’ s “ New Theory ” became the most influential work on the theory of light for the
rest of the century. He himself returned to various optical topics many times over the