1 1 74 THE EAR. 



that, by combining tones that differ in phase and in amplitude, we can 

 produce a great variety of wave-forms. These wave-forms, however, can 

 always, according to Fourier's mathematical law, be resolved into the 

 constituents. This law is expressed as follows : Any given regular 

 periodic form of vibration can always be produced by the addition of 

 simple vibrations having vibrational numbers, which are once, twice, 

 thrice, four times, etc., as great as the vibrational number of the given 

 motion. Applying this to the motion of the air close to the ear, we find 

 that any such motion, corresponding to a musical tone, may be always, 

 but for each case only in a single way, shown to be the sum of simple 

 motions, corresponding to the partial tones of this musical tone. 1 



The interesting question now arises whether the ear, having to deal 

 with waves varying almost infinitely in form, is differently affected by 

 such waves, according as their form represents various modes of pressure 

 (pushes and pulls) on the drum-head and conducting mechanism. 

 Suppose we sound the harmonic series of forks from u 2 to ut 5 , we hear a 

 rich harmonious sound, and we can analyse the sensation, and pick out 

 the tone of any particular partial, more especially if it is slightly 

 strengthened by a touch of the bow. By varying the order and the 

 intensity of the partials, we can produce a very large number of wave- 

 forms, but the general character of the harmonious sound, so far as we 

 can judge, remains the same. This would appear to indicate, with an 

 experiment comparatively so rough, that the ear takes no cognisance of 

 phase. Helmholtz investigated the question in a similar way, by means 

 of his vowel tone apparatus. 2 This apparatus consists of a harmonic 

 series of forks, electrically driven, and so arranged that they can be 

 sounded in any order and with various intensities, according to changes 

 produced in the size of the orifices of their appropriate resonators. 3 

 The resonators are thus put slightly out of tune, their resonance is 

 weakened, and thus the phase is altered. Helmholtz stated the result 

 as follows : — Differences in musical quality of tone depend solely on the 

 presence and strength of partial tones, and in no respect on the differ- 

 ences in phase under which these partial tones enter into composition. 



It must be admitted that this mode of investigation is not satis- 

 factory, inasmuch as when, in successive experiments produced by 

 pressing on the keys of the instrument, we apparently have the same 

 tonal result, we cannot be sure that there is no difference in sensation, as 

 we have no criterion by which to judge. We apparently hear the same 

 compound tone, but it may be slightly different from the one we just 

 heard an instant before. Accordingly, it is better to investigate this 

 question by experiments with intervals slightly inharmonious. This 

 has been done by Lord Kelvin. 4 Suppose we take the two forks, 

 ut 2 = 256 and ut 3 = 512, and sound these together, we have a sound 

 composed of a tone and its octave. If, then, we slightly flatten ut v so 

 that its vibration number is 510, we hear, not a roughness corresponding 

 to 254 beats, but a slow beat of only 2 per second. The sensation is 



1 Ah to Fourier's theorem, see Donkin's "Acoustics," pp. 60-66 ; O'Kinealy, London, 

 Edinburgh, and Dublin, Phil. Mag., London, vol. xlviii. p. 95 ; Ohm's " System der Mathe- 

 matik," Bd. ix. S. 286 ; Airy, op. cit., p. 160 ; Rayleigh, op. cit., vol. i. pp. 118, 202. 



2 Helmholtz, op. cit, p. 606. Apparatus shown in M'Kendrick's " Physiology," vol. 

 ii. p. 691. 



3 Helmholtz, op. cit., p. 174. 



4 Proc. Roy. Soc. Edin., 1878, vol. ix. p. 602 ; also " Popular Lectures and Addresses," 

 vol. ii. p. 395. 



