August 1 8, 1881] 



NATURE 



359 



certain speed both can be heard going on simultaneously. 

 Helmholtz gives to the grave harmonic the name of 

 ^^difference-tone'' because its number of vibrations ex- 

 actly corresponds to the difference between the number 

 of vibrations of the primaries. Two notes whose fre- 

 quencies are respectively m per second and n per second 

 will give rise to a difference-tone whose frequence is;«- n 

 per second, which is, in fact, just the same number as the 

 number of beats between the two. Konig uses a different 

 name, and agreeably to his (and Young's) theory, calls 

 these notes ^^ beat-notes," and classifies them into two 

 sets, lower and tipper, the lower beat-note being that 

 corresponding to the beats between the lower note and 

 the one that is sharper than it, the higher beat-note being 

 that corresponding to the beats between the higher note 

 and the octave of the lower. For example, if the notes c" 

 and d' are sounded together, their frequencies being in the 

 ratio 8 : 9, there will be heard a beat-note whose frequency 

 is relatively i, or three octaves below the lower note. If 



c' and b' (a seventh) are sounded, their frequencies being 

 in the ratio 8:15, there will be heard a beat-note of the 

 upper series of relative frequency i (being the difierence 

 between 15 and 16), or also three octaves below the c'. 

 So also the interval between c' and %f" (the twelfth-tone 

 flattened by about a semitone, so as to make the ratio 

 8 : 23) will also give a beat-tone of relative frequency I, 

 being the difference between 23 and 24. 



Now on Helmholtz' s theory' beats can only arise 

 between vibrations so near together on the scale as to 

 act on the same fibre of Corti in the ear (provided the 

 vibrations be pure and free from upper partial tones), 

 and they should therefore be audible not as tii'o tones but 

 as fluctuations in loudness of one tone. But when e' and 

 d' arc sounded we certainly hear two separate tones pbts 

 the low note which we call the grave harmonic. Helm- 

 holtz has therefore concluded that another explanation 

 must be sought, and this he finds in a mathematical 

 investigation of the resultant displacements due to super- 





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posing two tones, on the supposition that the vibrations 

 of the primaries are so large that the moving forces are 

 no longer simply proportional to the displacement, but 

 are influenced by the squares or higher powers of them. 

 He has shown that when this is the case combinational 

 tones must arise whose frequencies correspond to the 

 difference in the number of vibrations, and he further 

 conjectures that to the dissymmetry of the drumskin and 

 other vibrating parts of the ear is due the fact that the 

 squares of the displacements can thus affect the resultant 

 vibration. If so, all the combinational tones other than 

 those of mistuned unisons must really arise in the ear 

 itself and be sid>jective in character, as indeed Mr. 

 Bosanquet, who has lately studied the matter most care- 

 fully, roundly declares. 



Dr. Konig, however, undaunted by Helmholtz's rea- 

 sonings, has returned to the contest with new weapons. 

 He_ has repeated all his former experiments with new 

 tuning-forks specially made of massive form, so as to be 

 yet more perfect in tone, and finds his observations on 



beat-tones confirmed. He has further constructed a new 

 instrument, the wave-siren, with which to establish his 

 doctrine that beats, v/hen too rapid to be heard separately, 

 blend into a beat-note. In this instrument vibrations are 

 set up in the air by blowing through a slit against the 

 edges of a notched disk or rim which rotates rapidly upon 

 an axis. In 1872 Dr. Konig constructed sirens on this 

 principle, the indentations at the edges of the disks being 

 simple harmonic curves, or "wave-forms," which there- 

 fore gave rise to simple tones. In the new wave-siren 

 (Fig. i) the indentations are determined in the following 

 manner : — Two simple vibrations whose ratio is known 

 are mechanically compounded together by machinery, and 

 a resultant curve is obtained which exactly corresponds 

 on a large scale to the resultant motion of the air when 

 the two notes having this interval are sounded together. 

 This compound curve is then set off very exactly round 

 the periphery of a metal disk, and cut out in the metal 

 with the utmost nicety. Fig. 2 shows the form of the 

 curve (set out on the edge of a.Jlat disk) for the interval of 



