n88 



THE EAR. 



The following table shows the pitch of several differential tones of the 

 usual intervals : l 



Differential tones, first noticed by Sorge about 1740, are usually associated 

 with the name of Tartini. 2 Summational tones were discovered by Helm- 

 holtz. 3 It is clear that there must be differential tones of several orders, 

 according as they are produced between the generating tones themselves, 

 between the differential tone and each of the generators, and so on. 4 It is not 

 difficult to detect differential tones, but this is not the case with summational 

 tones. Hehnholtz heard them first with the syren and harmonium, and after- 

 wards with organ pipes and tuning-forks. On the other hand, Hermann and 

 many others assert that they have not been able to hear these tones. 5 With 

 Appunn's apparatus, which includes a large series of reeds, which can be sounded 

 with great force, I can hear them with the use of the appropriate resonator. 

 Without the resonator, they are almost inaudible, sometimes quite inaudible, 

 because, probably, they are lost in the loud-sounding generators. It is clear 

 that a resonator can have no effect on a tone sensation that is purely 

 subjective. This proves that they have an objective existence, and that 

 they are not produced in the ear itself. Some have contended that, 

 as regards differential tones, these may be produced by a blending of beats. 

 This is open to the objections (1) That it is doubtful if any number of beats 

 can produce a sensation of tone although it is held by Hermann and other 

 critics that " the ear is capable of recognising as a tone any periodicity within 

 certain limits of frequency " ; 6 (2) that the explanation does not include the 

 existence of summational tones ; and (3) that, by using appropriate sympathetic 

 membranes, Helmholtz was able to demonstrate the existence outside of, and 

 quite independently of, the ear, of both kinds of combination tones. Further, 

 Helmholtz states that " whenever the vibrations of the air or of other elastic 

 bodies, which are set in motion at the same time by two generating simple 

 tones, are so powerful that they can no longer be considered infinitely small, 

 mathematical theory shows that vibrations of the air must arise which have 

 the same vibrational numbers as the combination tones." 7 



By other experiments, however, it may be shown that, in certain circum- 

 stances, combination tones may be produced by the mechanism of the ear 

 itself. If two tones are sounded from sources very close together, the com- 

 bination tone is strong; but if the sources are wide apart, it is weak or 

 inaudible. Helmholtz held that the unsymmetrical form of the membrana 



1 Helmholtz, op. cit., p. 231. 



2 Sorge, " Vorgemach musikalischer Composition," 1740 ; Helmholtz, op. cit., p. 229. 



3 Ann. d. Phys. u. Chem., Leipzig, Bd. xcix. S. 497 ; Monatsb. Berl. Acad., May 22, 

 1856. As to the mathematical theory of such tones, see Helmholtz, op. cit., app. xii. p. 621. 



4 Hallstrom, Ann. d. Phys. u. Chem., Leipzig, 1832, Bd. xxiv. S. 438. 



5 Arch.f. d. ges. PhysioL, Bonn, 1891, Bd. xlix. S. 499. 



6 Rayleigh, op. cit., vol. ii. p. 461 ; also Hermann, op. cit., S. 514. 



7 Helmholtz, op. cit., p. 235. Forsyth and Sowter have recently furnished evidence of the 

 objective reality of combination tones, Proc. Hoy. Soc. London, vol. Ixiii. p. 396. See also 

 Riicker and Edser, London, Edinburgh, and Dublin Phil. Mag., London, 1895, vol. xxxix. 

 p. 341. 



