, &c. and compare the harmony of these with that of the VIII + 6ths, if he wants any farther evidence that Dr. Smith's measure of equal harmony is without foundation.
It may be thought, that even the measure of equal harmony laid down in the proposition, is more favourable to the complex consonances than the conclusions of experience will warrant. But when it is asserted by practical musicians, that the octave will bear less tempering than the Vth, the Vth less than the IIId, &c., they doubtless intend to estimate the temperament by the rate of beating, and to imply, that when different consonances to the same base are made to beat equally fast, the simpler are more offensive than the more complex consonances. This is entirely consistent with the proposition; for when equally tempered, the more complex consonances will beat more rapidly than the more simple; if on the same base, very nearly in the ratio of their major terms. (Smith's Har. Prop. XI. Cor. 4.) If, for example, an octave, a Vth, and a IIId on the same base were made to beat with a rapidity which is as the numbers 2, 3, and 5, no unprejudiced ear would probably pronounce the octave less harmonious in its kind than the IIId.
To those, on the other hand, who may incline to a measure of equal harmony between that laid down in the proposition and that of Dr. Smith, on account of the rapidity of the beats of the more complex consonances, it maybe sufficient to reply, that if the beats of a more complex consonance are more rapid than those of a simpler one, when both are equally tempered, those of the latter, cæteris paribus, are more distinct. It is the distinctness of the undulations, in tempered consonances, which is one of the principal causes of offence to the ear.
Scholium 2.
It will be proper to explain, in this place, the notation of musical intervals, which will be adopted in the following pages. It is well known that musical intervals are as the logarithms of their corresponding ratios. If, therefore, the octave be represented by .30103, the log. of 2, the value of the Vth will be expressed by .17509; that of the major tone by .05115; that of the comma by .00540, &c. But in order to avoid the prefixed ciphers, in calculations where so small intervals as the temperaments of different concords are concerned, we will multiply each of these values by 100,000, which will give a set of integral values having the same ratio. The octave will now become 30103, the comma 540, &c.; and, in general, when temperaments are hereafter expressed by numbers, they are to be considered as so many 540ths of a comma. Had more logarithmic places been taken, the intervals would have been expressed with greater accuracy; but it was supposed that the additional accuracy would not compensate for the increased labour of computation which it would occasion. This notation has been adopted by Dr. Robinson, in the article Temperament, (Encyc. Brit. Supplement;) and for every practical purpose, is as much superior to that proposed by Mr. Farey, in parts of the Schisma, lesser fraction and minute,[5] as all decimal measures necessarily are, to those which consist of different denominations.
Proposition II.
In adjusting the imperfections of the scale, so as to render all the consonances as equally harmonious as possible, only the simple consonances, such as the Vth, IIId, and 3d, with their complements to and compounds with the octave, can be regarded.
It has been generally assigned as the reason for neglecting the consonances, usually termed discords, in ascertaining the best scheme of temperament, that they are of less frequent occurrence than the concords. This, however, if it were the only reason, would lead us, not to neglect them entirely, but merely to give them a less degree of influence than the concords, in proportion as they are less used.
A consideration which seems not to have been often noticed, renders it impossible to pay them any regard in harmonical computations. All such computations must proceed on the supposition that within the limits to which the temperaments of the different consonances extend, they become harsher as their temperaments are increased. It is evident that any consonance may be tempered so much as to become better by having its temperament increased, in consequence of its approaching as near to some other perfect ratio, the terms of which are equally small; or perhaps much nearer some perfect ratio whose terms are not proportionally larger. For example, after we have sharpened the Vth more than 3 commas, it becomes more harmonious, as approaching much nearer to the perfect ratio 5/6. In this, however, and the other concords, the value of the nearest perfect ratios in small numbers, varies so much from the ratios of these concords, and the consequent limits within which the last part of Prop. I. holds true, are so wide that there is no hazard in making it a basis of calculation. And if there be a few exceptions to this, in some systems, in which the temperaments of a few of the concords become so large as to approach nearer to some other perfect ratio, whose terms are nearly as small as those of the perfect concord, although they might become more harmonious, by having their temperament increased, yet their effect in melody would be still more impaired; so that the concords may all be considered as subjected to the same rule of calculation.
But the limits within which the second part of Prop. I. holds true, with regard to the more complex consonances, are much more limited. We cannot, for instance, sharpen the 7th, whose ratio is 9 : 16 more than ½ a comma, without rendering it more harmonious, as approaching nearer another perfect ratio which is simpler; that of 5 : 9. Yet the difference between these two 7ths is so trifling that they have never received distinct names; and, indeed, their effect on the ear in melody would not be sensibly different.