his Principles for Tuning Instruments, &c. 297 



times per second ! or not above one-eighth of the rate neces- 

 sary to constitute continuous or musical sound? In the 

 case of equally tempered intervals, situate at the exact di- 

 stance of any of the concords from each other, it is gene- 

 rally true, that no " beating between the beatings" either 

 in theory or practice can happen. Suppose for instance, His 



Lordship's minor sixth C A which is flattened about — parts 



20 

 of a comma (not —• as printed by mistake p. 1 95, vol. xxvii.) ; 



this in the middle septave beats 22*6335 times per second 

 (or rather, in practice won't beat at all, but produce a con- 

 tinuous third discordant note) : if we tune another similar 

 or equally tempered sixth, on a note, a true minor sixth below 

 C the bass of the former one, that is, on His Lordship's "first 

 bass E : we shall find, that this will beat just 5-eighths as 

 fast as the above, or 14*1459 times per second, but no 

 u beating between the beatings" will take place, although 

 each are quick enough to produce them, owing to their 

 having the true relation of minor sixth between them, and 

 not because they are unisons as His Lordship would contend. 

 Let us, however, abandon the supposition of the tempered 

 sixths having basses that are exactly at concordant di- 

 stances, and tune just a similar minor sixth below C to that 

 which His Lordship has above C, that is, take two of these 

 6ths in succession; then we shall find, the lower note E 

 making 151*79 vibrations per second, and the 6th EC will 

 beat at the rate of 14*3144 times in a second: which not 

 bearing a true concordant relation to the beating of the 

 upper 6th, the sounding of the two together will be found 

 by calculation to occasion a <( beating between the beatings" 

 at the rate of 1*3477 per second : thus we see, that a " beat- 

 ing between the beatings" may happen to equally tempered 

 concords : and the same will indeed always happen, in 

 theory at least, to the tempered concords of which His 

 Lordship treats (although His Lordship asserts the contrary) ; 

 for all his tuning is to be performed by perfect intervals ex- 

 cept two successive hiequal thirds, and three successive tri* 

 equal quints, all of which will have such a " beating between 



the 



