v THE SENSE OF HEAEING 233 



number of beats equal to m-n, if m and n represent the number 

 of the vibrations of the two tones. 



The diagram (Fig. 94) shmss the simplest case of two simple 

 simultaneous tones (represented by two pendular or sinusoidal 

 vibrations of different length) which, through their summation, 

 produce a beat. In the central part of the figure, where the 

 positive half-wave of one of the tones coincides with the negative 

 half- wave of the other, there should lie almost complete extinction 

 of the resulting tone. In practice, however, this is almost un- 

 realisable, because the tones emitted by the different instruments 

 are always compound, i.e. can be resolved into several partial 

 tones : so that during the beat there is never total extinction of 

 the tone, because when the two fundamentals are extinguished, 

 the first harmonics, i.e. the octaves, are reinforced. On the other 

 hand, the octaves and successive partial tones of the two different 

 fundamental tones must theoretically produce their respective 

 beats, which will be less perceptible to the ear in proportion as 

 they are less intense. 



Beats are also produced when the two simultaneous tones act 

 separately on the two ears (Dove and others). Does this depend 

 on cerebral interference between the two excitations of the 

 auditory nerve, as assumed by Scripture, Wundt, and Ewald, or 

 on the fact that the tone that impinges on one ear is transmitted 

 to the other through the bones, as held by Schafer, Bernstein, 

 and others ? This last interpretation alone seems probable. 



When two tones are simultaneously produced which differ 

 more widely from each other, and must therefore theoretically 

 give rise to so large a number of beats that they can no longer 

 be perceived distinctly as such, then in addition to the tw r o 

 primary tones a trained ear can distinguish a third, deeper, note, 

 known as Tartini's tone, because it was discovered by the eminent 

 violinist of that name (1714). This third tone was at first 

 supposed to be a subjective phenomenon, due to the beats, when 

 these attain a frequency so great that they can no longer be 

 recognised distinctly as such. But Helmholtz by calculation 

 showed the objective character of Tartini's tone, which he termed 

 the differential tone, because its vibration number is equal to the 

 difference between those of the two primary tones. If, for 

 instance, these are at an interval of a fifth, that is, are in the 

 ratio of 2:3, the differential tone is the octave below, since the 

 difference is equal to 1. 



Helmholtz referred the formation of the differential tone to 

 the fact that the transmitting medium does not react in the 

 elastic deformations with a force proportional to the displacements, 

 so that the sound-vibrations do not exactly follow the laws of 

 the pendulum, and diverge from them in proportion as their 

 amplitude of vibration is greater. 



