If now the time of vibration in each is doubled, AC, _ac_, &c. will represent the times of vibration of imperfect unisons an octave below, and the successive dislocations will be C_c_, E_e_, &c. only half as frequent as before. But the unisons AE, _ae_, will be equally harmonious with AB, _ab_; because, although the successive dislocations are less frequent than before, yet the coincidences C′_c′_, E′_e′_ of the corresponding perfect unisons are less frequent in the same ratio. Suppose, in the second place, that the time of vibration is doubled, in only one of the unisons, _ab_; and that the times become AB and _ac_, or those of imperfect octaves. These will also be equally harmonious in their kind with the unisons AB, _ab_. For, although the dislocations C_c_, E_e_, &c. are but half as numerous as before, the coincidences of the corresponding perfect octaves will be but half as numerous. The dislocations which remain are the same as those of the imperfect unisons; and if some of the dislocations are struck out, and the increments of successive ones thus increased, no greater change is made in the nature of the imperfect than of the perfect consonance. If, thirdly, we omit two-thirds of the pulses of the lower unison, retaining the octave _ac_ of the last case, we shall have AD, _ac_, the times of vibration of imperfect Vths, to which, and to all other concords, the same reasoning may be applied as above. It may be briefly exhibited thus; since the intermission of the coincidences C′_c′_, E′_e′_ of the perfect unisons, an octave below A′B′, does not render the Vth A′D′G′ _a′c′e′g′_ less perfect than the unison A′_c′_ _a′c′_, each being perfect in its kind; so neither does the intermission of the corresponding dislocations C_c_, E_e_, of the tempered unisons, in the imperfect Vth, ADG, _aceg_, render it less harmonious in its kind than the tempered unison AB, _ab_, from which it is derived in exactly the same manner that the perfect Vth is derived from the perfect unison. The consonances thus derived, as has been shown by Dr. Smith, will have the same periods, and consequently the same beats, with the imperfect unisons. It is obvious, likewise, that they will all be equally tempered. Let _m_ AB, and _n_ _ab_, be a general expression for the times of vibration of any such consonance. The tempering ratio of an imperfect consonance is always found by dividing the ratio of the vibrations of the imperfect by that of the corresponding perfect consonance. But (mAB)/(nab) ÷ m/n = AB/ab; which is evidently the tempering ratio of the imperfect unisons. Hence, so far as any reasoning, founded on the abstract nature of coexisting pulses can be relied on, (for, in a case of this kind, rigid demonstration can scarcely be expected,) we are led to conclude that the harmoniousness of different consonances is proportionally diminished when they are equally tempered.