Music and the Making of Modern Science

Planck’s Cosmic Harmonium 267

sounding (by thermal excitation) and resounding (by sympathetic vibration) a whole spec-

trum of pitches. By preferring the Helmholtzian and musical term “ resonator ” over the

more neutral “ oscillator, ” Planck indicates the acoustic origins of his thought experiment:

he is doing for electromagnetic vibrations conceptually what Helmholtz did for sound

experimentally.

The climax of this extended comparison between sound and electromagnetic vibrations

connects Planck ’ s central innovation and his involvement in tuning controversies. Unlike

the purely continuous possibility of reeds or tubes resonating at any arbitrary length,

Planck tunes his resonators in an analogous way to the Eitz harmonium he had spent so

long studying. That is, at a crucial point in his calculations he allows the resonators

to have only discrete possible energies, just as the harmonium allows only discrete mul-

tiples of the fundamental unit of pitch. For the Eitz harmonium, this unit was 1041 of an

octave; for ordinary “ tempered tuning, ” it was the equal-tempered semitone, 121 of an

octave. In contrast, the “ natural tuning ” had no obvious “ fundamental unit, ” for it envis-

aged different-sized versions of semitones (major or minor, for instance) and was inher-

ently an unequal temperament. Planck ’ s 1893 argument had led him to accept the

“ unnatural ” equal-tempered scale, even though its number of divisions depended on con-

vention. In his work on black-body radiation, using his resonator, Planck essentially argued

for the “ equal temperament ” of electromagnetic radiation, that is, the equally spaced

quanta of his expression E = h ν , where E is their energy, ν their frequency, and h the

constant that sets their spacing or “ temperament. ”

AB

Figure 17.5

(a) Heinrich Hertz ’ s illustration of his oscillator, which produces a rapidly oscillating electric spark across the

gap near B ( ♪ sound example 17.6). (b) Harmonium reeds, as illustrated by Helmholtz, which have an essentially

identical structure to the vibrators Hertz used to interrupt his electric spark.