Whatever confirmation Dr Smith's position (Harmo -:ics, 2d edit. p. 21), respecting the comparative simpli city of the concords, as above, agreeing with their order as to smoothness and pleasant sensations that they excite on the car, may receive from the preceding Tables, it must be plain, from an inspection of his general Table of the order of the simplicity of consonances, including both concords and discords, those beyond, or expressed by larger primes than 5, as well as those in the received system of music that involve no such large primes, that this is an imperfect and false rule of judging of the smoothness or harmoniousness of consonances in gene ral, since in his table, the false tripled minor seventh of the trumpet .1„ or 2 VIII-F7th-13.9471 occurs before the perfect VI, and the tripled major tone 4, or 2VIII+11, a discord, before the perfect 3d! which must be quite contrary to their comparative effects on the ear.
We should, perhaps, earlier have pointed out, from our series of the fifty concords in seven octaves, given above, that the three least, or 3", III, and being consi dered as the simide concords, or concordant elements, the three next largest, V, 6, and VI, are generated by adding these simple ones in pairs, in every possible way, and the next, or VIII, by adding the three together; and that every succeeding concord in the scale, is gene rated by adding an octave (3+111+4,) or two octaves (23, 2 III, 24), or three octaves, &c. to each one of the seven original concords in the first octave. Whence it appears, that no concord but the octave will bear adding to itself once, or any greater number of times, (without other combinations), without becoming a discord ; but that the addition once, or any greater number of times, of the octave, to any concord, will produce another concord.
That the complement, or remainder, when any one of the original concords is taken from the octave next above, or VIII, or from any of the succeeding octaves, the remain ders are all concords ; and so are the complements of any of the doubled, tripled, quadrupled, &c. concords in the second, third, fourth, &c. octaves, to the next, or any succeeding octave above them ; and in like manner, in the second, third, fourth, fifth, &c. octaves, one or more VIII'"' may be taken away from any of the concords therein, and still leave remainders that are concords, &c.
Among the various attempts of philosophers to de fine the limits, or show characteristic distinctions be tween concords and discords, generally, Mersenue and Kircher maintain, that those consonances are most sim ple or agreeable which are generated in the least time, or have the smallest least terms to their ratios ; and those, on the contrary, the most compound and harsh, which are generated in the largest time, or have the larg est least term or numerators to their ratios. This rule is shewn, however, by Malcolm, to be defective; and Dr Smith has done the same thing, and thence concludes, " that the frequency of coincidences is of itself too gene ral a character of the simplicity or smoothness of a con sonance, and therefore an imperfect one." (Harmonics,
p. 23.) In another place, Dr Smith says, (p. 15.) that it is the " mixture of pulses succeeding one another in a given cycle of times, terminated at both ends by coinci dent pulses, and sufficiently repeated, which excites the sensation of a given consonance ;" and " one consonance may be considered as more or less simple than another, according as the cycle of times belonging to it is more or less simple than the cycle belonging to the other.
M. Euler says, when the ear readily discovers the re lation subsisting between the terms of the ratios of two notes, their combination is denominated consonance or concord ; and if it be very difficult, or even impossible, to catch this relation, the combination is termed disso nance, or discord." Letters, vol. i.
Mr Holder attempts, but without any success, to ac count for the pleasure derived from concords, or sounds in the more simple musical ratios, by the mind being oc cupied in parcelling out the numbers, but not by division, (which with primes is indeed impossible,) but by une qual and fanciful partitions of them into what he calls factors or parcels, as 5 into 2, 1, 2, 7 into 3, 1, 3, &c.; and, principally on this whimsical ground, he labours to show, that 7 ought to have place among musical ra tios ! &c. " No combination," says he, " ought to be esteemed concord, however simple and eligible its terms may be in every other respect, if the implied sound (that is, its grave harmonic) be three octaves or more below the lower term." Essay, p. 376. And again, 66 An interval which is concord in the upper parts, is often no concord when taken in the bass !" for " we lay it down as a rule, that the implied sound of a concord ought always to be within the limits of audible sound." The introduction of which last absurdities into his Essay, Mr Farey has shcwn to have arisen, from Mr Holder being unacquainted with the true nature of the grave harmonics, or the rule for calculating those belonging to any assigned consonances.
Dr Robison says, " a musical sound is the sensation of a certain form of the aerial undulation which agitates the auditory organ. The perception of harmonious sound, is the sensation produced by another definite form of the agitation : This is the composition of two other agitations ; but it is the compound agitation alone that affects the ear, and it is its form, or kind, which deter mines the sensation, making it pleasant or unpleasant, or in other words a concord or a discord." Our limits will not admit of enlarging further on this very curious and intricate subject, which presents yet a rich field for the successful cultivator of it. (g)