Modern psycho-acoustics and its impact on Schenkerian analysis
K C Moore
2 The Helmholtz theory
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
difference.
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Helmholtz proposed that the phenomenon of dissonance was caused by beats
between partial tones of a complex sound.[1] 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.