Smith is accordingly clear that other "roughnesses" exist besides the beats which Sauveur considered, and if the investigations had been continued on the basis of Sauveur's idea, these additional roughnesses would have turned out to be the beats of the overtones, and the theory thus have attained the point of view of Helmholtz.
Reviewing the differences between Sauveur's and Helmholtz's theories, we find the following:
1. The theory according to which consonance depends on the frequent and regular coincidence of vibrations and their ease of enumeration, appears from the new point of view inadmissible. The simplicity of the ratios obtaining between the rates of vibration is indeed a mathematical characteristic of consonance as well as a physical condition thereof, for the reason that the coincidence of the overtones as also their further physical and physiological consequences is connected with this fact. But no physiological or psychological explanation of consonance is given by this fact, for the simple reason that in the acoustic nerve-process nothing corresponding to the periodicity of the sonant stimulus is discoverable.
2. In the recognition of beats as a disturbance of consonance, both theories agree. Sauveur's theory, however, does not take into account the fact that clangs, or musical sounds generally, are composite and that the disturbance in the consonances of distant intervals principally arise from the beats of the overtones. Furthermore, Sauveur was wrong in asserting that the number of beats must be less than six in a second in order to produce disturbances. Even Smith knows that very slow beats are not a cause of disturbance, and Helmholtz found a much higher number (33) for the maximum of disturbance. Finally, Sauveur did not consider that although the number of beats increases with the recession from unison, yet their strength is diminished. On the basis of the principle of specific energies and of the laws of sympathetic vibration the new theory finds that two atmospheric motions of like amplitude but different periods, a sin(rt) and a sin[(r + ρ)(t + τ)], cannot be communicated with the same amplitude to the same nervous end-organ. On the contrary, an end-organ that reacts best to the period r responds more weakly to the period r + ρ, the two amplitudes bearing to each other the proportion a: φa. Here φ decreases when ρ increases, and when ρ = 0 it becomes equal to 1, so that only the portion of the stimulus φa is subject to beats, and the portion (1-φ)a continues smoothly onward without disturbance.
If there is any moral to be drawn from the history of this theory, it is that considering how near Sauveur's errors were to the truth, it behooves us to exercise some caution also with regard to the new theory. And in reality there seems to be reason for doing so.
The fact that a musician will never confound a more perfectly consonant chord on a poorly tuned piano with a less perfectly consonant chord on a well tuned piano, although the roughness in the two cases may be the same, is sufficient indication that the degree of roughness is not the only characteristic of a harmony. As the musician knows, even the harmonic beauties of a Beethoven sonata are not easily effaced on a poorly tuned piano; they scarcely suffer more than a Raphael drawing executed in rough unfinished strokes. The positive physiologico-psychological characteristic which distinguishes one harmony from another is not given by the beats. Nor is this characteristic to be found in the fact that, for example, in sounding a major third the fifth partial tone of the lower note coincides with the fourth of the higher note. This characteristic comes into consideration only for the investigating and abstracting reason. If we should regard it also as characteristic of the sensation, we should lapse into a fundamental error which would be quite analogous to that cited in (1).
The positive physiological characteristics of the intervals would doubtless be speedily revealed if it were possible to conduct aperiodic, for example galvanic, stimuli to the single sound-sensing organs, in which case the beats would be totally eliminated. Unfortunately such an experiment can hardly be regarded as practicable. The employment of acoustic stimuli of short duration and consequently also free from beats, involves the additional difficulty of a pitch not precisely determinable.
II.