3 Recent experiments

Helmholtz was aware that the frequency of beats between two pure tones which led to the maximum perception of roughness varied with the frequency of the tones, though he used a constant value for his calculations. Plomp and Levelt determined the nature of the variation and showed that the roughest beat frequency occurred close to one quarter of the critical bandwidth, a concept introduced to characterise another area of psycho-acoustics. [2] The critical bandwidth is the frequency range within which adding more components to a complex tone does not increase its loudness. [3] The critical bandwidth is almost constant below about 200 Hz. and approximates to a constant fraction of the basic frequency above about 500 Hz. [4] This means that, in musical terms, from about the top of the treble stave upward the most dissonant interval remains constant at somewhat less than a semitone, while at middle C it is 1.5 semitones, roughly doubling for each octave downwards from there.

Pierce realised that a critical test of the extended Helmholtz theory would be its ability to predict the perception of the consonance or dissonance of pairs of synthetic tones in which the partials were at frequencies remote from the harmonic ones. His experiments, with continuous tones incorporating partials whose frequencies were irrationally related, confirmed the Helmholtz theory. [5] As noted below, natural instruments do not produce continuous tones of this character, and this experiment had to await the development of computer generated sounds.

Other workers in this field have confirmed and elaborated the extended Helmholtz theory, and it now appears to have an established position as a theory of the perception of consonance and dissonance. [6,7,8,9,10] It is not complete, since it does not show in neurological terms why beats should cause the hearer to perceive an interval as rough, nor does it explain the variation of the most dissonant beat frequency with the frequency of the fundamental tones. Helmholtz's explanation of the effect as resulting from intermittent excitement of nerve cells seems unlikely, since it would appear to predict that maximum dissonance would occur at a constant beat frequency. [11] However, lack of a satisfactory explanation for this element does not detract from the usefulness of the theory in predicting the responses of subjects who have not been conditioned by a musical education.

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