"For nothing gives greater offence to the hearer, though ignorant of the cause of it, than those rapid, piercing beats of high and loud sounds, which make imperfect consonances with one another. And yet a few slow beats, like the slow undulations of a close shake now and then introduced, are far from being disagreeable." 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_ + [rho])(_t_ + [tau])], 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_ + [rho], the two amplitudes bearing to each other the proportion _a_: [phi]_a_. Here [phi] decreases when [rho] increases, and when [rho] = 0 it becomes equal to 1, so that only the portion of the stimulus [phi]_a_ is subject to beats, and the portion (1-[phi])_a_ continues smoothly onward without disturbance.