SOUND AND MUSIC 87 



the same, illustrating the exactness with which their 

 elasticity and density have been determined and meted 

 out to them. 



Vibration of strings. Pluck aside a string at its middle 

 point, and the whole vibrates to and fro, yielding what 

 is called its fundamental note. When a bridge is placed 

 at the middle and either half is plucked aside, a note is 

 produced which is the octave of the fundamental. A third 

 part yields a note which is a fifth above the octave. 

 Half the string creates twice as many vibrations as the 

 whole; one-third three times as many; one-fourth four 

 times the number. The law is that the number of vibra- 

 tions is inversely proportional to the length of the string. 



The number also depends on the tension of the string, 

 or on the force by which it is stretched. For experiment a 

 string is fixed at a point, passes at once over one bridge, 

 and after an interval over another, then over a wheel 

 which moves freely. It is then stretched by a weight. 

 If the weight be a pound, and the vibrations so many, in 

 order to double the number, or obtain the octave, it is 

 necessary to stretch by a weight of four pounds ; to treble 

 the number, the weight must be nine pounds ; to quad- 

 ruple, it must be sixteen. If other things be equal, the 

 number of vibrations is in proportion to the square roots 

 of the stretching weights. 



The thickness of the string also plays its part, produces 

 its appropriate and measured effect. If two strings be 

 alike in every respect except in thickness, and the dia- 

 meter of the one be twice that of the other, the thinner 

 string will create twice the number of vibrations of the 

 thicker. If the diameter of the one be three or four or 

 five times that of the other, the number of vibrations will 

 be lessened three or four or five times, showing the law 

 to be that it is inversely proportional to the thickness. 



