COMBINATION TONES. 1189 



tympani favours the production of minute vibrations superposed on those of 

 the membrane proper, and thus gives rise to weak combination tones. The 

 loose formation of the malleo-incudal joint might have a similar effect ; that 

 is to say, if acted on by two very powerful tones, minute motions might be 

 produced (corresponding to the pitch of the combinational tones) which would 

 be added to the combined motion of the two generators. This explanation 

 can only be admitted when the generators are loud ; but, as Hermann urges, 

 differential tones may be heard when the generators are very weak. Lord 

 Rayleigh supports Helmholtz in the statement that "loud generators are 

 necessary." This is consistent with the results of experiments made with 

 Appunn's apparatus. It is a reasonable objection, however, to experiments 

 with reeds, that the tone supposed to be a combination tone may really be a 

 loud partial of one or other of the reeds. A combinational tone is well brought 

 out, according to Helmholtz, " when two clear and powerful soprano voices 

 execute passages in thirds." 



The importance of these combinational tones in the theory of hear- 

 ing is obvious. If the ear can only analyse compound waves into 

 simple pendular vibrations, of a certain order, how can it detect 

 combinational tones, which no doubt can be heard, and yet do not 

 belong to that order ? Or, the question may be put, Can combinational 

 tones explain consonance and dissonance, produced by tones that are 

 simple, and are destitute of partials ? Take the octave, and sharpen the 

 upper note, so that the octave is slightly untrue. Suppose the lower 

 tone has a frequency of 100, and the upper 201 vibs. per second, a 

 combinational tone of 101 is produced, and this, with the lower 

 generator, will produce one beat per second. The beats cannot be 

 avoided, unless the octave is pure. Thus the octave, produced by simple 

 tones, is a concord bounded by discords. Again, investigate a fifth 

 slightly out of tune. Here we have, say, the generators produced by 

 200 and 301. The combinational tone of the first order is 101 ; it will 

 produce a combinational tone of the second order by the differential 

 tone of the first order, 101, beating with 200 ; the lowest generator thus 

 giving a differential tone of the second order, having a frequency of 

 99, and there will also be two beats between the differential tone of 

 the second order, 99, and the differential tone of the first order, 101. 

 Thus a fifth is a consonance less sharply cut off than the octave, owing 

 to the feebleness of the combinational tone of the second order. 1 Thus 

 we see that combinational tones are produced when the notes of intervals 

 are sounded strongly on instruments free from partials ; and that these 

 combinational tones may produce beats with either of the generators, or 

 among themselves ; and these beats, feeble as they may be, produce that 

 feeling of less and less consonance, until we come to intervals that 

 are truly dissonant. One might suppose in the case of tones that 

 abound in partials, that combinational tones might be produced by the 

 partials, and that thus a new source of beats might lead to confusion 

 and discord ; but theory shows that dissonance due to combinational 

 tones produced between partials never occurs except when it has already 

 taken place by the action of the partials among themselves. 



According to this view, combinational tones are not produced by 

 beats, but are true sounds superadded to that of the generators. The 

 ear, therefore, in dealing with them, vibrates in some part of its 

 mechanism with each generator, according to Ohm's law, while it is also 



1 For other examples, see Sedley Taylor, op. cit., p. 181. 



