368 Scientific Proceedings, Royal Dublin Society. 



1807 Dr. Thomas Young explained their origin as due to the 

 coalescence of rapid beats into a single resultant tone. 1 



This explanation was rejected by von Helmholtz, who suggested 

 a wholly different origin of the Tartini's tones ; and though it has 

 been subjected to criticism, Helmholtz's view is now generally ac- 

 cepted by physicists. This explanation depends on the fact that in 

 the case of any violent vibration of an elastic medium the restoring 

 force is not proportional to the displacement. This notably occurs 

 in a limited air-cavity, or in an unsymmetrical membrane like 

 that of the drum-skin of the ear. Hence a " simple harmonic force 

 acting on the membrane, or on the cavity, will produce not merely 

 a simple harmonic vibration of its own frequency, but a series of 

 harmonic vibrations giving rise to overtones, which we may term 

 self-combination tones. When two harmonic forces of different 

 frequencies act on the membrane, or cavity, they produce vibrations 

 having frequencies equal to the differences of frequencies of the 

 original vibrations, or their harmonics. These give rise to 

 difference combination-tones. There are also vibrations equal to the 

 sum of the frequencies of the original vibrations, or their harmonics. 

 These give rise to summation combination-tones. Hence two pure 

 tones of frequencies m and n, entering the ear, may give rise to 

 the following notes: — 



m n primaries. 



2m 2n self-combination tones. 



m-n first difference tone. 



m + n first summation tone." 2 



And various other combinations of less importance. 



In most cases these combination-tones arise within the ear, and 

 have no existence external to the auditory apparatus. But 

 Helmholtz showed that when the primaries were powerful, and 

 issued from a common and confined air-space, as in the case of 

 the double siren and harmonium he used, the combination-tones do 

 have an existence in the external air. To prove this Helmholtz 



1 Young says : " "When the beats of two sounds are too frequent to be heard as 

 distinct augmentations of their force, they have the same effect as any other impulses 

 which recur in regular succession, and produce a musical note which has been called 

 the grave harmonic." — Young's Lectures, 1845 edition, p. 306. 



2 Poynting and Thomson, " Sound," new edition, p. 156. 



