356 On the Objective Reality of Combination Tones, 



We have made several attempts to detect combination- 

 tones of higher orders, such as 2p + q and 2p — q, but without 

 success. 



Conclusion. 



We may in conclusion refer to some of the suggestions 

 which have been made to account for the combination-tones 

 by theories other than those of Helmholtz. 



Konig's suggestion that they are the beat-tones of upper 

 partials has been discussed and shown to be inadequate to 

 explain the facts of observation. 



Again, it has been argued that the summation-tone is the 

 beat-note between the second partial of the higher note (the 

 octave) and the beat-tone of the two primaries. It follows as 

 a matter of algebra that such an explanation must always be 

 numerically correct, for 2a — (a — b) = a + b, and our experi- 

 ments throw no new light on the matter. It appears to us, 

 however, that since propinquity between the sources of sound, 

 causing a violent disturbance, is favourable to the production 

 of combination-tones, while it is not necessary for the pro- 

 duction of beats, the facts of experiment are in this case also 

 in favour of von Helmholtz's views. 



A still more subtle objection has been taken by Terquem 

 (Annates d'Ecole Normale, 1870, p. 356). When two rows 

 of holes are open in the siren, there may be occasions on 

 which all the holes of both rows are opened simultaneously 

 and others on which only one row is in action at one time. 

 Terquem attempts to calculate the effects of irregularities 

 such as these, but in the first place he specifically refrains 

 from attacking the theory of Helmholtz ; secondly, he does 

 not apply calculation to the siren of Helmholtz ; thirdly, he 

 points out that the relatively large size of the holes in that 

 instrument would reduce the effects he predicts ; and, lastly, 

 he admits that his results require confirmation by experiment. 

 Putting these points aside, however, his theory leads to the 

 conclusion that the two notes which we have been regarding 

 as fundamental are reinforced harmonics in a series of which 

 the fundamental note corresponds to the greatest common 

 measure of these frequencies. Both the summation and the 

 difference tone must be included in such a series ; but 

 Terquem' s theory gives no reason why they should have such 

 exceptional importance as experiment proves that they have. 

 Lastly, as he expressly repudiates the idea that partials have 

 an objective existence (loc. cit. p. 274), and includes the com- 

 bination-tones in a series of partials, the experiments described 

 by us must on this point be regarded as opposed to his views. 



