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LXI. On the Simultaneous Sounding of two Notes. 

 By Dr. Rudolph Konig, Paris. 



[Continued from p. 446.] 



III. Difference-notes and Summationrnotes. 



IT is well known that Helmholtz has proved theoretically 

 that " whenever the vibration of the air, or of any other 

 elastic body set in motion by both the primary notes at 

 the same moment, becomes so strong that the vibrations can 

 no longer be considered as infinitesimal, vibrations of the air 

 must arise whose pitches are equal to the difference and to 

 the sum of the number of vibrations of the primary notes." 

 These combination-notes, both those of the difference and 

 those of the sum, are quite distinct from the beats, and are 

 much weaker than the original note. 



If we turn our attention first to the difference-notes, we find 

 that, in all intervals n : n + m, when m is not much greater 



than - they coincide with the original note, and therefore can- 

 not be proved by it. We have seen, however, that, in all in- 

 tervals n : n -f m, when m is much greater than - the beat- 

 notes are m / = n — m. In the intervals n : hn •+- m, when m is less 

 than - rnf=m; and when m is much greater than ^, m' '= n — m, 



and therefore not the same as the difference of the vibrations of 

 the primary notes. We must therefore try to observe the dif- 

 ference-notes in these intervals. 



As I have already stated, these intervals, composed of high 

 notes, allow the beat-notes to be heard quite loud, while no 

 trace of the difference-notes is to be perceived. c" f : V" 

 (8 : 15) allows only c (1) and no trace of 7 to be heard ; 

 f : d lv (4 : 9) only </ (1) and nothing of e'" (5) ; c"' :/ IV 

 (3 : 8) only/' and/", and absolutely no a!" (5) ; and it fol- 

 lows, therefore, that the difference-notes in any case must 

 be immeasurably weaker than the beat-notes. I was able, 

 however, to prove their existence beyond a doubt by forming 

 the above quoted intervals from deeper notes, which, by their 

 longer duration, enabled me to make use of auxiliary tuning- 

 forks which gave a certain number of beats with the required 

 notes. 



If I allowed the great forks c' and V (8 : 15) to sound 

 in front of the sounding-boxes, the first thing that fell on the 

 ear was the loud rattle of the 32 beats m' =n—m ; if, how- 

 ever, I held a tuning-fork of 440 v. s. at a greater distance 



