Proposition XI.

The aggregate of dissonance, heard in a given time, in the system of temperament unfolded in the last Proposition, will be less than in either of the systems generally practised.

In order to compare the foregoing system with those which have been most generally approved, the temperaments of all the concords have been calculated, in the system of equal semitones; in that of Earl Stanhope, which has had considerable celebrity; in that of Dr. T. Young; in that of Mr. Hawkes; in that of Kirnberger, which has been extensively adopted in Germany; and in that which is described by Rousseau and D'Alembert as generally practised in France. If these temperaments be multiplied into the corresponding numbers of Table IX., agreeably to what was shown under Prop. VIII., and those products which belong to the several concords of the same name be added, the sums, after the three right-hand figures are cut off, will be as follows:

TABLE XIII.

Systems.Mean Temp.Young's.Kirnber-
ger's.		French.Stan-
hope's.		Hawkes'.New Scale.
Disso-{ Vths309494681561595665810
nance{ IIIds21841541139713461175925530
of the{ 3ds2740244820192121199216761363
Total5233448340974028376232662703

From an inspection of the sums at the foot of the table, it will be seen that the amount of dissonance heard in a given time is decidedly less in the new scale than in either of the others; and that it is scarcely more than half as great as in the scale of equal semitones. On the other hand, the temperament is very unequally distributed, which must be admitted, cæteris paribus, to be a disadvantage. It is even somewhat greater than in the scheme of Mr. Hawkes, although by no means in the same ratio, as the aggregate dissonance is less. It contains one Vth, which will be somewhat harsh, and four IIIds and three 3ds, which will be quite harsh. But these, as will appear from an inspection of Table IX., are, of all others, of by far the most unfrequent occurrence; so that the unpleasant effect of a transition from a better to a much worse harmony will be very seldom felt. In the six simplest keys of the major, and in the three of most frequent occurrence in the minor mode, they are never heard, except in occasional modulations; and even then, generally no one, and rarely more than one is heard. Now these nine keys, as will appear from Table III., comprise more than five times as much of the music examined as all the rest. The same remarks might be extended to three other minor keys, were it not that the sharp seventh is so generally used, that it deserves to be considered as an essential note of the key.

But there are two important considerations, more than counterbalancing the objection to this system, derived from the greater inequality in the distribution of its temperaments, which have not been hitherto noticed, as not being susceptible of mathematical computation.

1st. We have gone on the supposition that tunes on the more difficult keys are as often performed, according to their number, as those on the simpler keys; and have taken for the measure of dissonance, in different systems, what would be actually heard, if the 1600 scores, whose signatures were examined, were all played in succession, and on the keys to which they are set. But the fact is, that those pieces which are set to the simpler keys are oftener played, and with fuller harmony, on account of the greater ease of execution, than those in which many of the short finger keys must be used.

2d. Pieces on the more difficult keys are often played on the adjacent easier keys, but the contrary is seldom or never done.

Giving to these two considerations no more than a reasonable weight, they will counterbalance the objection, and will render it evident that the sums under the several systems in the table may be taken as a true exhibition of their respective merits, without any injustice to the more equal systems at the left-hand of the table.

Cor. We may hence draw a comparison between the systems in common use. Their merits, when every consideration is taken into view, are nearly in the inverse ratio of the sums denoting their aggregate dissonance. That of Mr. Hawkes is the best, and, in many respects, has a remarkable analogy to the one derived from the preceding investigations.

Cor. 2. As the aggregate dissonance of the changeable scale is calculated on the same principles, in Prop. VIII., as that of the Douzeave in this, a comparison of the results in Table VIII. with those in Table XIII., will furnish us with the relative dissonance of different systems for these different scales. The relative dissonance of the two systems which form the object of this essay, is nearly as 17 : 27. Hence it appears, that by inserting eight new sounds between those of the common octave, the harshness of the music executed, at a medium of all the keys, may be diminished by more than one third of the whole, while the transition from a better to a worse harmony will never be perceived.