MECHANICS OF THE INNER EAR 59 



Before we take up the theoretical discussion of further 

 tone combinations, the reader ought to obtain some informa- 

 tion concerning the difference tones which 

 Laws of ^^ hear in addition to the "objective" 



difference tones in the several combinations. To 



tones give such information of this kind as is 



indispensable, I shall state here the laws 

 of these phenomena in as clear and comprehensible a manner 

 as possible. These laws given below do not pretend to tell 

 all the difference tones which we might possibly hear in every 

 possible combination of objective tones. Neither do they tell 

 the relative intensities of the difference tones, although this 

 is a matter of no small importance. Laws of difference tones 

 of this scientific perfection are as yet not known and may 

 never be known. The laws below merely tell those differ- 

 ence tones which one is most likely to hear in those combi- 

 nations which correspond to relatively simple ratios of the 

 vibration rates and are therefore (musically and otherwise) 

 particularly interesting. These laws are the following four : 

 In case the ratio of the vibration rates does not differ 

 much from 1:1, let us say 11:12, or 9911 : 99Si9, a single dif- 

 ference tone is audible, whose pitch corre- 

 First law^ of sponds to the pitch of a tuning fork the 



difference tones vibration rate of which is equal to the 

 difference of the vibration rates of our 

 case. In addition to the difference tone, however, beats are 

 usually clearl}^ audible, and a mean tone may be audible too 

 which lies between the two primary tones. If the interval is 

 quite small, this mean tone is usually more pronounced than 

 either of the primary tones, particularly when we hear with 

 one ear only, having the other ear plugged. The beats just 

 mentioned seem to be the fluctuations of the intensity of the 

 mean, tone rather than of the primary tones, if we use one 

 ear only. 



