366 OVERTONES 



by be silenced, while the overtones, which correspond 

 to a node at the point touched, will not be affected 

 since the finger was placed upon one of their nodal 

 points ; the overtones will now be distinctly recognised 

 without difficulty. Thus, if the vibrating string be 

 touched in the middle, the octave above the funda- 

 mental tone will persist, together with all the tones 

 which correspond to even multiples of the rate of vibra- 

 tion of the fundamental tone ; if the string be touched 

 at 40 or 80 cm , the twelfth continues to be heard ; if at 30 

 or 90 cm , the second octave, and so on. That the over- 

 tones are not excited by touching the string is proved 

 by plucking the string originally at a node and touching 

 it afterwards at the same point ; for example, by pluck- 

 ing it in the middle, the fundamental tone will be heard 

 very strongly, but it cannot be accompanied by the 

 octave above it, because the point which would be a node 

 for the octave, vibrates in this case most strongly, and 

 if the string be now touched in the middle, all sound 

 ceases immediately and completely. 



With the help of a piano we may very easily make observations on 

 overtones. We may then either find out the overtones of a definite 

 note, or, which is more convenient, starting with a particular tone, 

 we may investigate of what other notes it is an overtone. If it i' 

 desired to experiment in the former manner, gently press down one 

 of the keys, so as to free the corresponding string from the damper 

 and strike repeatedly in rapid succession one of the lower keys. Ii 

 the freed string belongs to an overtone of the note which is struck,! 

 it will commence to vibrate, if its key is kept down, and will con- 

 tinne to be heard even after the key which is struck has beer: 

 released and its string has ceased sounding. If we keep on striking 

 C and key after key is pressed down, the overtones of C will b( 

 found to be those given previously. For experiments by the con 

 verse method, it must be remembered that, since the harmonic 

 overtones make 2, 3, 4, 5, etc. times as many vibrations as th< 

 fundamental tone, any tone is obviously an harmonic overtone of ' 



