On Musical Temperament, I9f 



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 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 plaj'^ed 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 r 



Cor. We may hence draw a comparison between the sys- 

 tems in common use. Their merits, when every considera- 

 tion 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 diflferent 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 

 win never be perceireil. 



