; the first computation making it only 702, while the last has it 818.

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.