THE ANALYTIC PROPERTIES OF THE EAR. 1173 



analysis, and therefore termed pure or simple tones ; and (2) those 

 that may be resolved into partials or compound tones. As first shown 

 by Ohm, the only form of vibration incapable of being resolved into 

 simpler constituents is that of a pendulum or tuning-fork, and the curve 

 so obtained is the curve of sines. 1 Such a curve is seen in Fig. 423, A. 

 It represents the pendular motion of the air in the external meatus 

 excited by a tuning-fork in motion. The horizontal line represents time, 

 while the ordinates (d v d 2 ) show the amount of displacement of the air. 

 Suppose now another tone, represented by curve B, an octave above the 

 first, is sounded, there will then be two vibrations of B for one of A. 

 It will be seen that by superposing B on A, so that d 1 a x in A is added to 

 l x in B, the ordinate c x in C is obtained, and by subtracting b 2 in B from 

 d 3 a 2 in A, c 2 in C is obtained ; and so on with reference to other points, 

 until the curve (C) represents the combination of A and B. It will be 

 seen that (C) is periodic, and that its periods are those of A ; or, in other 

 words, the motion of the air produced by two tones (A and B) is also 

 periodic, because one of the simple tones makes twice as many vibrations 

 as the other in the same time. Now suppose that e in B were moved to 

 the right so that e was below d t in A, then the compound curve would 

 be that shown in D. Thus, by combining A and B in various ways, many 

 curve forms might be obtained, but these could always be resolved into 

 A and B, in which the frequency of B is always double that of A. 2 

 These curves represent the motions of the air when two tuning-forks, 

 say ut :i and ut i are sounded, or when a flute is gently blown so as to 

 give a note of that pitch. But the ear can readily hear the tones of the 



Fig. 424. — Form of wave produced by combining two simple waves. 



two forks. It is more difficult to analyse the sound as produced by the 

 flute, but if we assist the ear by using two resonators, one tuned to ut a and 

 the other to ut if all difficulty disappears, and we can analyse the sound. 

 The adjoining figure (Fig. 424) shows (the thick line) the resultant wave 

 — produced by superposing a tone and its octave. Now, just as the ear 

 can pick out different musical tones when several instruments are 

 sounded, so in favourable conditions it can resolve a musical tone into a 

 series of partials. This power is succinctly expressed in Ohm's law as 

 follows : Every motion of the air which corresponds to a composite mass 

 of musical tones is capable of being analysed into a sum of simple 

 pendular vibrations, and to each such single vibration corresponds a 

 simple tone, sensible to the ear, and having a pitch determined by the 

 periodic time of the corresponding motion of the air. 3 But we have seen 



1 Helmholtz, op. cit., p. 35 ; Rayleigh, op. tit., vol. i. p. 17. 



2 Helmholtz, op. cit., p. 46. 3 Ibid., p. 51. 



