HELMHOLTZ IN BONN 



the same vibrational numbers as the combination 

 tones.' * Recently Riicker and Edser have demon- 

 strated the objective existence of such tones. 2 Com- 

 binational tones may, however, as Helmholtz clearly 

 showed, be produced in the ear itself, because of the 

 elastic asymmetry of the drum 



The importance of these combinational tones in the 

 theory of hearing is obvious. If the ear can only 

 analyse compound waves into simple pendular vibra- 

 tions, in a certain order, how can it detect combina- 

 tional tones, which no doubt can be heard, and yet do 

 not belong to that order ? For example, when the in- 

 terval is harmonic (400, 500) the combined wave-forms 

 make a wave of 100 of frequency, so that the ear, like 

 Fourier's theorem, may easily pick out this tone ; but 

 how will it deal with intervals of incommensurate 

 numbers, such as 407, 483 ? Yet this combination 

 tone of a frequency of 76 will be heard as distinctly as 

 one arising from two tones having frequencies of 400 

 and 500. This is still a great difficulty, as it cannot 

 be said that experiments made by many others since 

 the time of Helmholtz have removed it. Experiment 

 shows that combinational tones are produced when the 

 notes of intervals are sounded strongly on instruments 

 like tuning-forks, whose notes are nearly simple tones, 

 free from upper partials ; and that these combinational 

 tones may produce beats with any of the generators, or 

 among themselves ; and these beats, feeble as they 



1 Sensations of Tone, trans, by Ellis, p. 235. * PAH. Mag., April 1895. 



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