DIFFERENT SOUNDS OF THE SAME STRING. 351 



in a query. " Whence comes it, that if we tune 

 a fifth 2 : 3, a third sound is faintly heard, the 

 octave below the lower of the two notes ? Nature 

 shows that with 2 : 3, she still requires the unity 

 to perfect the order 1, 2, 3." The truth is, that 

 these numbers express the frequency of the vibra- 

 tions, and thus there will be coincidences of the 

 notes 2 and 3, which are of the frequency 1, and 

 consequently give the octave below the sound 2. 

 This is the explanation given by Lagrange 7 , and 

 is indeed obvious. 



7 Mem. Tur. i. p. 104. 



