226 Chapter 14
“ equal intervals of time, ” audible as pure tones. Thus, Helmholtz uses the siren to map
visible hole spacings into audible pitches ( figure 14.9 ), bridging space and time through
the spinning disc and the concept of frequency , both as the siren ’ s rotational frequency
and the sound frequencies its disc thereby generates.
Helmholtz also advanced the technology of the siren so that it could sound two pitches
simultaneously, making possible comparisons in perception ( figure 14.10 ). Such a double
siren could produce “ combination tones, ” sounding the difference or sum of two pitches
more powerfully than any other instrument. Helmholtz himself discovered the faint sum
tones, which he could produce only with a siren or special harmonium; the stronger dif-
ference (or “ Tartini ” ) tones had long been known. Helmholtz argued that “ the greater
part of the force of the combinational tone is generated in the ear itself, ” which combines
the pure superposition of the incoming pitches, heard as two distinct tones, with their
difference or sum, as predicted by nonlinear differential equations derived from Newto-
nian mechanics.^24 Helmholtz ’ s use of mathematics shows its essential role in his argu-
ment here and in acoustics in general, as he conceives it. Helmholtz ascribed the failure
of superposition and the resultant combination tones to “ the unsymmetrical form of the
[ear] drumskin itself, ” and, more importantly, to “ the loose formation of the joint between
the hammer and anvil ” ossicles of the middle ear. “ In this case, the ossicles may click , ”
which he hears as a “ mechanical tingling in the ear ” when “ two clear and powerful
soprano voices executed passages in Thirds, in which case the combinational tone comes
out very distinctly. ”^25 Here, his musical experience impinged strongly on the formation
of his mathematical acoustics.Figure 14.8
Seebeck siren, from Helmholtz, Tonempfindungen (1863); c shows the source of the air stream that is periodically
interrupted by the holes in disc A , which the cord f rotates.