Unheard Harmonies 279
the themata that Gerald Holton identified as the enduring, recurrent motifs of science, the
theme of harmony may be the most pervasive and perhaps the deepest.^15 We have seen
this harmony go from the almost audible music of the spheres, to the soaring (but inau-
dible) polyphony Kepler found in the planetary songs, to the imaginary (yet visible) strains
of Planck ’ s cosmic harmonium. In between, Chladni, Wheatstone, Faraday, and Helmholtz
sought harmony in the gritty actualities of human hearing; we are only at the beginning
of exploring how complex data (such as from stock markets, galaxy surveys, climatic
studies, and seismometers) might be grasped through hearing them, via “ sonification ”
rather than visualization.^16 Beyond overtly musical meanings, “ harmony ” became a touch-
stone for scientists, a way of stating their deepest, largest goals, their shared (or contested)
senses of the explanatory order they sought. Still, the elusive sense of world-harmony has
become a rather ghostly apparition, now so dispersed throughout the mathematical struc-
ture as no longer to be recognizable as music. In that sense, the Pythagorean dream suc-
ceeded to such an extent that it subsumed and even evaporated its own musical content.
If, as John Keats wrote, “ heard melodies are sweet, but those unheard / are sweeter, ” can
unhearable harmonies be the sweetest of all? 17 Yet what is left of the archaic quest for
world-harmony if its connection with music becomes purely metaphorical?
For example, consider how twentieth-century string theory began with a mathematical
relation between physical processes, the Veneziano amplitude (1968), which unified
several important theoretical features of the behavior of high-energy particles. Many
physicists were struck with the beauty of that expression, whose few symbols related
several different kinds of physical processes and satisfied a surprising number of theoreti-
cal desiderata. Even more amazing, this amplitude was a function Euler had devised long
before in an utterly different context.^18 It seemed miraculous that this relic from “ classical ”
mathematical physics could be the key to understanding high-energy behavior in the
extreme quantum realm, two and a half centuries later. A number of theorists extended the
Veneziano amplitude into the dual resonance model, hoping that it would develop into a
theory of the strongly interacting particles. Then in 1969 – 70, Yoichiro Nambu, Holger
Nielsen, and Leonard Susskind independently suggested that the dual resonance model
could be understood as describing the behavior of relativistic strings. Each found his way
to this idea through mathematical analogies.
For his part, Nambu reexpressed the Euler beta function in an alternative mathematical
form and then realized that its spectrum of possible values “ immediately suggested a one-
dimensional harmonic oscillator system, like an oscillating string of some length moving
in four dimensions. ” He began hypothetically and analogically, using such phrases as
“ suggested, ” “ could be labeled, ” and “ could be interpreted. ” Gradually, as the analogy
cohered more compellingly, he shifted to a more direct mode of expression: the number
of states “ reminds ” him of the Hagedorn model of hadrons, in which the rising mass
spectrum of these strongly interacting particles recalls the rising spectrum of a vibrating
string. Where Hagedorn thought of the interacting hadrons as “ fireballs ” having a common