HERMANN VON HELMHOLTZ 



tones, having a frequency which is the sum of those 

 of the tones producing them. Then, when a fifth 

 is sounded, the differential tone is an octave below the 

 low note ; with a fourth it is a twelfth j with a major 

 third, two octaves ; with a minor third, two octaves and 

 a major third ; with a major sixth, a fifth ; and with 

 a minor sixth, a major sixth. Such differential tones, 

 first heard by Sorge about 1740, are usually associated 

 with the name of Tartini. Summational tones were 

 discovered by Helmholtz. It is clear that there must 

 be differential tones of several orders, according as 

 they are produced between the generating tones them- 

 selves, then between the differential tone and each 

 of the generators, and so on. It is not difficult to 

 detect differential tones, but this is not the case with 

 summational tones. Helmholtz, who had a remark- 

 ably acute and well-trained ear, heard them first with 

 the polyphonic syren and the harmonium, and after- 

 wards with organ pipes and tuning-forks. On the 

 other hand, Hermann and others assert that they 

 cannot hear these tones. There can be little doubt, 

 however, that both kinds of combination tones may have 

 an existence outside of, and quite independent of, the 

 ear. Helmholtz states that c whenever the vibrations 

 of the air or of other elastic bodies, which are set in 

 motion at the same time by two generating simple 

 tones, which are so powerful that they can no longer 

 be considered infinitely small, mathematical theory 

 shows that vibrations of the air must arise which have 

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