Sounding of two Notes* 515 



niaiy notes, and cannot, therefore, have been produced by them, 

 this circumstance is amongst the grounds which Helmholtz 

 puts forward in support of the view that beats cannot form 

 any note whatever (Tonempfindung , pp. 245, 263). But if, on 

 the one hand, summation-notes do not coincide with the beats, 

 yet on the other hand, as we have seen above, neither the beat- 

 notes of the intervals n : n + m, if m is much greater than ~ , 



nor the beat-notes of all intervals n : lin + m coincide 

 with the difference or the sum of the primary notes; and 

 the beat-notes are therefore as little demonstrated by the 

 cause which produces the combination-notes as the latter can 

 be demonstrated from the existence of the beats ; and we must 

 consequently suppose that each of these species of notes has its 

 peculiar origin. 



As to the question whether the nature of the beats will 

 itself admit of their forming themselves together into a note, 

 the circumstance that, when the vibrations of the primary 

 notes are not infinitesimally small, combination-notes of the 

 difference and the sum ensue, can of course prove nothing 

 either for or against this view. Helmholtz, however, gives 

 some other reasons against the older opinion of Th. Young, 

 which require a closer inquiry in order to refute them. 



The way in which the beats in the ordinary (and particularly 

 therefore in the lower) parts of the scale mostly produce very 

 weak notes is that which has principally induced Helm- 

 holtz to declare that vibrations of simpler notes, without any 

 mixture of upper notes or combination-notes, " only arise when 

 the two given notes are divided from one another by a tolerably 

 small interval," and that, " when the interval is increased by 

 only the amount of a minor third, their vibrations become in- 

 distinct" (Tonempfindung,]). 284). But if we use deep and 

 sufficiently powerful notes, the primary beats, as I have men- 

 tioned above, are audible in considerably greater intervals. In 

 the octave C.c there is no interval which does not allow them 

 to be clearly heard ; and even if we set aside the beats m', we 

 can follow the beats m alone to above the fifth; and in inter- 

 vals with the fundamental note double E they may be noticed 

 even close to the seventh. 



In the above Table I have stated that the third c : e allows 

 a rattle of 32 beats to be heard, and that this ever lessening 

 rattle may be followed as far as the fifth. These results only 

 refer to the clang of primary notes of such strength as 

 my tuning-forks, placed in front of resonators, produced. 

 When, however, I made use of the louder notes c, e, and g 

 which I produced by sounding the forks in question without 



2 M2 



