Accordingly, the principal design of the following propositions will be to investigate the actual frequency of occurrence of different chords in practice; and from this and the two other above-mentioned considerations united, to deduce the best system of temperament for a scale, containing any given number of sounds to the octave, and particularly for the common Douzeave, or scale of twelve degrees.
Proposition I.
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, 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:
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 bB, cC, &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, Bb, Cc, &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.
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 Cc, Ee, &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 Cc, Ee, &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.