352 Prof. S. P. Thompson on the Phenomena 



2. The beats of mis tuned consonances of the form n : 1, 

 which were not explicitly mentioned in the author's papers, 

 nor at that time subjected by him to detailed examination, 

 have lately claimed attention on account of the revival of the 

 dispute between acousticians of two schools concerning the 

 origin of difference-tones, or grave harmonics. Dr. Konig 

 of Paris still consistently advocates the theory of Smith and of 

 Young — that the beats (which represent in frequency the dif- 

 ference between the vibration-frequencies of the two interfering 

 tones) pass, when occurring sufficiently rapidly, into beat- 

 tones (as he calls the differential tones). All mathematical 

 acousticians agree, on the other hand, that this is physically 

 impossible if the tones are pure and of so small an amplitude 

 that the squares and higher powers of the displacements may 

 be neglected. A very important paper on this matter by Mr. 

 Bosanquet has lately appeared*, in which the following state- 

 ments are made: — (I) That the beats and difference-tones of im- 

 perfect consonances other than unisons are subjective^ ; (2) that 

 the beats consist, as Konig discovered in 1875, of variations 

 in the intensity of the lower of the two interfering tones ; 

 (3) that if the squares and higher powers of the displacements 

 be not neglected in the equations, a term appears having a 

 period whose frequency is the difference of the frequencies of 

 the generating tones, as required by Helmholtz's theory of the 

 difference-tones, and corresponding to Konig's lower beat- 

 notes. In this last we have the apparent reconciliation of the 

 experiments of Konig with the theory of Helmholtz. To his 

 former experiments Konig has now added a fresh series with 



* Phil. Mag. June 1881, and Supplementary Number, June 1881. 



t I am not quite sure whether 1 entirely understand the sense in which 

 Mr. Bosanquet uses the word subjective. If he means by this term that 

 the phenomena of beats and difference-tones exist only in the mind, brain, 

 or nerve-structures of the ear, being generated in the sensory apparatus 

 by something which physically has no existence, being in fact only phan- 

 toms of the imagination, then I entirely differ from him. But if by sub- 

 jective Mr. Bosanquet means that the existence of these phenomena, 

 though physically and mechanically true, is limited to the receptive me- 

 chanism of the ear, then I beg in the first place to disagree with such a 

 perversion of the adjective, and in the second place to deny that any such 

 limitation exists. The beats are objective : they can be seen in a mano- 

 metric flame if the primary tones are sufficiently loud. The difference- 

 tones of the mistuned consonances in question are also objective, inasmuch 

 as a suitable asymmetrical resonator will reinforce them. What is the 

 ear but a complex resonator upon whose parts vibrations from without 

 can be forced ? And if, by reason of asymmetry in the ear, the higher 

 terms of the displacements can " by transformation " generate resultant 

 displacements whose periods are those of the tones in question, and force 

 them upon the sentient apparatus, then the same tones can, for precisely 

 the same reason, be taken up by any other suitable asymmetrical resonator. 



