PHYSIOLOGICAL ACOUSTICS 293 



the octave (for example) should be distinguished by any 

 character of smoothness from adjacent intervals on either 

 side, the two groups of sensations being in any case quite 

 independent. Since the more consonant intervals at all events 

 are as a matter of fact easily recognized by the ear, even in 

 the case of apparently pure tones, and are thoroughly well 

 defined, the difficulty is a serious one. To meet it, Helmholtz 

 developed his theory of "combination-tones," which are assumed 

 to supply the function of the missing overtones. 



In most of our investigations it has been assumed that the 

 amplitude of the vibrations may be treated as infinitely small, 

 so that disturbances due to different sources may be super- 

 posed by mere addition. In the theory now in question this 

 assumption is abandoned ; the vibrations are regarded as 

 small, but not as infinitely small, and the interaction of the 

 disturbances due to different causes is, to a certain degree of 

 approximation, investigated. 



We have already had an indication in 63 of the manner 

 in which two imposed simple-harmonic disturbing forces of 

 small but finite amplitude, with frequencies N lf N 2 respectively, 

 may generate in the air other simple-harmonic vibrations 

 whose frequencies are 



2N lt 2# 2 , Ni-N,, Ni + N*, 



and whose amplitudes involve the squares or product of the 

 amplitudes of the two primaries. If the approximation were 

 continued we should meet with further vibrations whose fre- 

 quencies are of the type p l N l piN^ where p lt p 2 are integers. 

 In acoustical language, two simple-harmonic vibrations can, if 

 of sufficient intensity, give rise not only to the pure tones 

 usually associated with them, but also to a series of other pure 

 tones of higher order. The fact that a single harmonic vibration 

 can by itself give rise to a pure tone together with its octave, &c. 

 is itself of some importance, but the most interesting result is 

 due to the interaction, viz. the " difference-tone " (N-^ N 9 ). 

 The existence of difference-tones was observed, apart from 

 all theory, by Sorge (1745) and Tartini (1754). The "sum- 

 mation-tone " (Ni + N 2 ) is more difficult to hear, and its 



