Hermann Helmholtz (1821-1894) was interested in physics, physiology, acoustic
and optical perception, mathematics and the theory of knowledge. His expertise
in the first two was widely acknowledged and demonstrated by his having held,
at different times, the chairs of anatomy and physiology at Bonn and of physics
at Berlin. He became interested in acoustics in 1852 and in 1856 published a
theory which explained combination (ie summation and difference) tones as
resulting from non-linearity in the response of the ear. That this is only a
partial explanation is, of course, demonstrated by the existence of binaural
combination tones, when two different pure tones are presented, one to each ear.
By use of an ingenious double siren, of his own design, he was able to
demonstrate that beats, the variation of perceived loudness of a sound composed
of two tones near in frequency, had themselves a frequency equal to that
Helmholtz proposed that the phenomenon of dissonance was caused by beats between partial tones of a complex sound. He was able to demonstrate that beats perceived between two siren sounds nearly but not exactly in the frequency ratio 2:3 were at a frequency equal to the difference between the frequencies of the third partial of the lower sound and the second partial of the upper. Sirens produce repeating, non-sinusoidal waveforms and therefore have harmonic partial tones (see 4.1 below). Thus this demonstration strongly supports the occurrence of beats between the partials rather than the fundamentals of these sounds. Helmholtz also investigated beats between the fundamental of one sound and upper partials of another. He discovered that, over the range of frequencies of which his apparatus was capable, beats of frequency from 30 to 40 cycles per second gave maximum roughness to the ear. He used this information to guide his choice of a formula which related beat roughness to frequency. This gave maximum roughness for a frequency of 33 cycles per second and zero roughness at beat frequencies of zero and infinity but was otherwise arbitrary. Applying this formula to all the partials of the motion of the bowed violin string, which he had measured by means of the vibration microscope, an apparatus in which a lens is mounted on a vibrating tuning fork to make the string motion visible, he obtained a remarkable diagram for the predicted dissonance of pairs of notes over a continuous range of two octaves. This shows the expected zero roughness for g', c'', g'' and c''' relative to the lower note which is always c'. Other lows, most consonant first, are at points approximating to a flattened b'' flat, e'', a', f', a flattened e'' flat, a'', a' flat, f'', b'', e', a'' flat, d'', a flattened b' flat and e' flat. As would be expected, the trough of the roughness curve coincided with the frequencies of just, rather than tempered intervals.
Some of these results will have been more surprising and less convincing to Helmholtz's contemporaries than to a twentieth century musician familiar with the harmonies of impressionism and jazz. Nevertheless, while later work has improved on Helmholtz's function relating beat frequency to roughness, and has used electronics to determine the components of the sounds of the instruments rather than measuring string motion, his initial concept has been overwhelmingly confirmed by twentieth century experimenters.