All consonances may be regarded, without any sensible error in practice, as equally harmonious in their kinds, when equally tempered; and when unequally tempered, within certain limits, as having their harmoniousness diminished in the direct ratio of their temperaments. As different consonances, when perfect, are not pleasing to the ear in an equal degree, some approaching nearer to the nature of discords than others, so a set of tempered consonances, _cæteris paribus_, will be best constituted when their harmoniousness is diminished _proportionally_. Suppose, for example, that the agreeable effects of the Vth, IIId, and 3d, when perfect, are as any unequal numbers, _a_, _b_, and _c_; the best arrangement of a tempered scale, other things being equal, would be, not that in which the agreeable effect of the Vth was reduced to an absolute level with that of the IIId, or 3d, but when they were so tempered that their agreeable effects on the ear might be expressed by (_m_/_n_)_a_, (_m_/_n_)_b_, and (_m_/_n_)_c_. That different consonances, in this sense, are equally harmonious in their kinds, when equally tempered, or, at least, sufficiently so for every practical purpose, may be illustrated in the following manner: [Illustration] Let the lines AB, _ab_, represent the times of vibration of two tempered unisons. Whatever be the ratio of AB to _ab_, whether rational or irrational, it is obvious that the successive vibrations will alternately recede from and approach each other, till they very nearly coincide; and, that during one of these periods, the longer vibration, AB, has gained _one_ of the shorter. Let the points, A, B, &c. represent the middle of the successive times of vibration of the lower; and _a_, _b_, &c. those of the higher of the tempered unisons. Let the arc AGN..VA be a part of a circle, representing one period of their pulses, and let the points A, _a_, be the middle points of the times of those vibrations which approach the nearest to a coincidence. It is obvious that the dislocations _b_B, _c_C, &c. of the successive pulses, increase in a ratio which is very nearly that of their distances from A, or _a_. Now if the pulses exactly coincided, the unisons would be perfect; and the same would be equally true, if the pulses of the one bisected, or divided in any other constant ratio, those of the other; as clearly appears from observation. It is, therefore, not the absolute magnitude, as asserted by Dr. Smith, but the _variableness_ of the successive dislocations, B_b_, C_c_, &c. which renders the imperfect unisons discordant; and the magnitude of the successive increments of these dislocations is the measure of the degree of discordance heard in the unisons.