268 Chapter 17
To be sure, this analogy is not made explicit in any of Planck ’ s own writings and must
be read judiciously: the equally spaced energy quanta need to be distinguished from
equally spaced semitones in frequency. But the implicit analogy is strong and, in retro-
spect, less surprising, given Planck ’ s intensive work on the Eitz harmonium and on the
issue of natural versus tempered tuning in the year just preceding his return to the ther-
modynamics of electromagnetic radiation. Comparing these two consecutive projects,
Planck ’ s black-body “ harmonium ” is set up so that, in terms of a “ tuning pitch ” ν , every
“ note ” on that instrument sounds integer multiples of E = h ν. In that sense, the analogy
is quite exact between harmonium and black body: parallel to Eitz ’ s 104 keys per octave,
Planck ’ s constant h sets the minimum spacing between adjacent “ notes ” on the atomic
harmonium.
Planck also kept before him the question of the arbitrariness of the “ standard pitch ” for
his electromagnetic “ harmonium. ” In 1899, he realized that his assumption of the “ tuning
constant ” h led, along with the Newtonian gravitational constant G and the speed of light
c , to a fundamental and universal set of units that “ necessarily retain their meaning for all
times and for all civilizations, even extraterrestrial and non-human ones, and therefore
[should] be designated as ‘ natural units. ’ ”^32 The exalted universality of this realization
moved him at the time to tell his young son Erwin that he had made a great discovery,
comparable to those made by Newton or Copernicus.^33 By this, Planck seemed to mean
the disclosure of these “ natural, ” universal units, more than his success in fitting the
experimental data for black-body radiation per se. From the point of view of the extended
analogy with a cosmic harmonium, he had discovered the natural wavelength of that instru-
ment, the “ Planck length ” he calculated to be 4.13 × 10 – 35 m. Corresponding to this uni-
versal length, Planck also calculated a universal time unit, 5.391 × 10 – 44 sec. Though he
did not mention it, its inverse should then be a universal “ Planck frequency, ” 1.855 × 10 43
Hz. If we return to Young ’ s attempt to state the “ pitch ” of a light vibration in musical
terms, Planck ’ s cosmic harmonium is tuned to a quite low A (426 Hz) 135 octaves above
middle C, according to contemporary equal temperament ( ♪ sound example 17.7).^34 Alter-
natively, we can understand Planck ’ s equation E = h ν as translating any energy E into a
frequency ν = E/h , so that energy corresponds to pitch , with Planck ’ s constant h as the
conversion factor.
Planck ’ s term “ natural ” brings to mind his foregoing struggle with the problem of
natural versus equal-tempered tuning. There, he had to yield to conventionality and the
human feeling (which he shared) for the greater sweetness (as well as familiarity) of
tempered versus natural intervals. But when he transferred the problem of tuning to the
black body, Planck was able both to have complete equality of “ temperament ” through
the equally spaced “ notes ” of the quantum harmonium as well as naturalness, through the
fundamental “ pitch ” implied by the very temperament itself, the Planck frequency. The
tuning dilemmas he had faced while playing the Eitz harmonium turned out to be solvable
in the case of the black-body resonators.