OF ARTS AND SCIENCES. 3 



the necessary adjustments, the pitch of either note could be varied 

 by drawing out or pushing in the lower half of the resonator. The 

 actual pitch of the note given by the resonator was in all cases 

 determined by comparison with a tuning-fork of known rate. As 

 the pitch of the resonator when blown is to a certain extent de- 

 pendent on the position of the embouchure and the strength of the 

 blast, it would not be safe to assume that the actual note given 

 by the resonator was that marked upon it. Neither resonator 

 when sounded sensibly affected the pitch of the note given by 

 the other. 



The determination of the point at which the harshness of the 

 dissonance produced by the tones of the two resonators reaches a 

 maximum is of course a very difficult problem to be solved exactly. 

 It is, moreover, a question still to be investigated, whether pre- 

 cisely the same results would be reached by different observers 

 possessed of equally good perception. The following figures given 

 in Table I. were ascertained from a long series of observations by 

 one of the writers (Mr. Goodwin). They are liable to an error of 

 a single beat in either direction, possibly somewhat more in one or 

 two cases. They furnish, however, the first series of direct meas- 

 urements of this kind which we know. 



It will be noticed that the result for Ut i7 32 beats per second, 

 agrees very closely with the number 33, which Helmholtz gives 

 for maximum dissonance in the middle registers. He found that 

 of the ordinary musical intervals, b' c", as he designates it, was the 

 most dissonant, and as this gave 33 beats per second he considered 

 this number as giving the harshest effect for tones of corresponding 

 pitch. 



The notes furnished by the resonators are also well adapted for 

 the direct study of the limit at which the beats cease to be audible 

 when their number is increased. As the interval between the beat- 

 ing notes is increased, the beats heard become fainter and fainter 

 until they disappear. It is all but impossible to fix upon the pre- 

 cise point of disappearance, since the change in loudness of the 

 beats is so gradual. But, as will presently appear, our results 

 clearly show that the maximum number of beats perceptible, like 



