loO On the Stanhope Ternpcrament. 



Mr. Farey does here distinctly admit, that mi/ calcula- 

 tions agree exactly with his', so far as I use my method of 

 geometrical mean proportlonuh, to determine correctly the 

 hi-equal \\\\x([?,, and like^nise the ^ri-eg^wr/Z quints. But he 

 imagines that he has found out a very radical defect (as he 

 calls it) in my other mode of obtaining those same major 

 thirds and quints; viz. bv means of the " eqiiuLity of the 

 leQth/gs," between such thirds, when compared to such 

 equal thirds ; and also between the equal quints, respectively. 

 Mr. Farey's radical mistake seems to consist in this; 

 namely, that he does not underiiand the scientific terms 

 which I have been obliged to use ; and that he does not 

 even know the clear and wide difference which there is be- 

 tween " beats" and " beatings." 



Beats, as applicable to this subject, are nearly synony- 

 mous to vibrations. For ea^h vibration produces one heat 

 against the air. So that, when, fcr example, a proper mu- 

 sical string is shortened, the vibrations increase in number 

 inversely a.5 the length of the string; and the beats in- 

 crease, of course, in the very same proportion. If the 

 longer string be, for instance, the note C, that shorter 

 string, which will yield the precise sound of the perfect 

 octave C next above, will produce, in any given time, ex- 

 actly tifice the number of beats that were produced by the 

 longer string. So that, in this case, we have two distinct 

 sets of BEATS, but there are no beatings whatsoeiver. 

 For there are never any beatings in any case, when either 

 an octave, a quint, a fourth, or a major third, \is quite per- 

 fect. And it is precisely from the absence oi X\\& beatings 

 (as they are technically called), that we know that the par- 

 ticular musical interval, then under examination, is perfect. 

 In like manner, in a perfect u9iiso?i there are no beatings. 

 But, in this case, as in the former, there are two sets of 

 BEATS 5 although, from the isochronism of the vibrations of 

 the two strings in' unison (that is to say, from the equality 

 of tlie number of their respective vibrations in equal times), 

 those /wo sets of beats appear, to the ear, nearly as if they 

 were hut on;:. 



But, if a unison^ an octave^ a quint, a fifth, a fourth, or 



a major 



