Unheard Harmonies 273
the color-theoretic episode in his work with Planck ’ s musical investigations of 1893 – 94,
though these occupied proportionately much less of Planck ’ s career than Schr ö dinger ’ s
work on color occupied him.
Schr ö dinger essentially carried forward the Young – Helmholtz three-color theory by
bringing to completion Helmholtz ’ s comparison of the manifolds of color and geometric
space. Helmholtz was moving toward determining the true geometry of color space,
which Schr ö dinger showed was a non-Euclidean manifold and even determined the exact
form of its distance function (its “ metric, ” in geometric terminology). The curved quality
of the lines of shortest length in this color space, implicit in Helmholtz ’ s work, came
forward explicitly in Schr ö dinger ’ s theory ( figure 18.1 ). Helmholtz and Riemann, in their
complementary ways, had gone from the complexities of color vision to a vision of a
dynamical space, which Riemann posited as curved. Their dialogue contrasted sound and
color perception to bring out the implicit geometry behind those modes of perception,
namely the manifolds involved in seeing and hearing. Riemann in particular envisaged
a new kind of geometric physics that would involve curved, multidimensional manifolds.
Einstein accomplished exactly that in his 1916 general theory of relativity, treating gravi-
tation and acceleration as manifestations of curved space-time. In 1920, directly influ-
enced by Einstein ’ s work, Schr ö dinger took the same curved manifolds Einstein had
drawn from Riemann and went back to the problem of color perception with which the
whole story had started. Schr ö dinger played in color theory the role Einstein played in
space-time geometry and gravitation: each applied Riemannian geometry to fulfill the
intuition of Helmholtz. Einstein showed that space-time, in the presence of matter, was
non-Euclidean; Schr ö dinger showed that human color perception also obeyed non-
Euclidean geometry.
In this d é nouement, it seems at first glance that these developments concerned only
visual perception, not sound. Yet despite Schr ö dinger ’ s disinterest, music in a particularly
Pythagorean vein shaped his greatest achievement, his wave equation. His work on color
theory prepared him to apply Riemannian, curved higher-dimensional manifolds to other
physical problems. And so when in 1926, just after his color theory work, he took up the
problem of the behavior of atoms and electrons, he was ready with the tools that had been
so successful with gravitation and now also with color perception.^6 What happened next
showed the enduring power of the musical shaping of physical law. Heisenberg ’ s matrix
mechanics had given numerically satisfactory predictions of atomic processes, but without
disclosing any intuitable — specifically, visualizable — mode of understanding these results.
Louis de Broglie had argued, by analogy, that, as light had particle as well as wave quali-
ties, so too should particles have analogous wave properties. In 1926, Schr ö dinger was
then a young theorist in Zurich; according to an eye-witness report by Felix Bloch, then
a graduate student, the senior professor Peter Debye said: “ Schr ö dinger, you are not
working right now on very important problems anyway. Why don ’ t you tell us some time
about that thesis of de Broglie, which seems to have attracted some attention. ”^7 At one of