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 Cc, Ee, 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 m AB n ab ÷ 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.
The remaining part of the proposition, viz. that consonances differently tempered have their harmoniousness diminished, or their harshness increased, in the direct ratio of their temperaments, will be evident, when we consider that the temperament of any consonance is the sole cause of its harshness, and that the effect ought to be proportioned to its adequate cause. We may add, that the rapidity of the beats, in a given consonance, increases very nearly in the ratio of the temperament; and universal experience shows, that increasing the rapidity of the beats of the same consonance, increases its harshness. This is on the supposition that the consonance is not varied so much as to interfere with any other whose ratio is equally simple.
Cor. We may hence infer, that in every system of temperament which preserves the octaves perfect, each consonance is equally harmonious, in its kind, with its complement to the octave, and its compounds with octaves. For the tempering ratio of the complement of any concord to the octave, is the same with that of the concord itself, differing only in its sign, which does not sensibly affect the harmony or the rate of beating; while the tempering ratio of the compounds with octaves is not only the same, but with the same sign.
Scholium 1.
There is no point in harmonics, concerning which theorists have been more divided in opinion than in regard to the true measure of equal harmony, in consonances of different kinds. Euler maintains, that the more simple a consonance is, the less temperament it will bear; and this seems to have ever been the general opinion of practical musicians.[4] Dr. Smith, on the contrary, asserts, and has attempted to demonstrate, that the simpler will bear a much greater temperament than the more complex consonances. The foregoing proposition has, at least, the merit of taking the middle ground between these discordant opinions. If admitted, it will greatly simplify the whole subject, and will reduce the labour of rendering all the concords in three octaves as equally harmonious as possible, which occupies so large a portion of Dr. Smith's volume, to a single short proposition. Dr. Smith's measure of equal harmony, viz. equal numbers of short cycles in the intervals between the successive beats, seems designed, not to render the different consonances proportionally harmonious, but to reduce the simpler to an absolute level, in point of agreeableness, with the more complex; which, as has been shown, is not the object to be aimed at in adjusting their comparative temperaments. But, in truth, his measure is far more favourable to the complex consonances than equal harmony, even in this sense, would require; and, in a great number of instances, leads to the grossest absurdities. Two consonances, according to him, are equally harmonious, when their temperaments are inversely as the products of the least numbers expressing their perfect ratio. If so, the VIII + 3d, whose ratio is 5/12, when tempered 1/20 of a comma, and the unison, whose ratio is 1/1, when tempered 3 commas, are equally harmonious. But all who have the least experience in tempered consonances will pronounce, at once, that the former could scarcely be distinguished by the nicest ear from the corresponding perfect concord, while the latter would be a most offensive discord. One instance more shall suffice. The temperaments to render the VIII + Vth, and the VIII + 6th equally harmonious, are laid down in his tables to be as 80 : 3. We will now suppose an instrument perfectly tuned in Dr. Smith's manner, and furnished with all the additional sounds which constitute his changeable scale. In this system, the IIIds, and consequently the VIII + 6ths, are tempered 1/9 of a comma; which, so far from being offensive, will be positively agreeable to the ear. This cannot be doubted by those who admit that the VIII + 6ths in the common imperfect scales, when tempered at a medium nearly seven times as much, make tolerable harmony. Yet, according to the theory which we are opposing, the VIII + Vth will be equally harmonious when tempered nearly a minor semitone. Now let any one, even with the common instruments, whenever an VIII + Vth occurs, strike the semitone next above or below: for example, instead of playing C, g, let him play C, g
; instead of A, e, let him play A, e
