On Musical Temperament. 199 
Which have not been hitherto noticed, as not being susceptible 
of mathematical computation. 
Ist. 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 svstems, what would be 
actually heard, if the 1600 scores, whose signatures were ex- 
amined, 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 reason- 
able 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 sys- 
tems incommon use. Their merits, when every considera- 
tion is taken into view, are nearly in the inverse ratio of the 
sums denoting their aggregate disssonance. That of Mr. 
wkes is the best, and, in many respects, has a remarkable 
inalogy to the one derived from the preceding investigations. 
_ Cor. 2. As the aggregate dissonance of the changeable scale 
's calculated on the same principles, in Prop. VIIL., as that of 
 Douzeave in this, a comparison of the results in Table VITL- 
with those in Tablé 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 
* this essay, is nearly as 17:27. Hence it appears, that by 
inserting eight new sounds between those of the common 
sctave, 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, 
