SOUND AND MUSIC 89 



beautiful and characteristic sound. In all music the over- 

 tones are evidently more numerous than the fundamental, 

 and if they had been out of harmony with it and with 

 each other, the effect would have been disastrous. A 

 musical string, a musical instrument, could not have been 

 made. That none such appear is due to the perfect order 

 that reigns in nature, to the perfect adjustment of its 

 particles in aggregation for harmonic action. 



The vibrations of rods. When they are fixed at both 

 ends, the laws of their action are, in some respects, the 

 same as in strings. Those of vibrating parts, when divisions 

 are made by damping, are different on account of the differ- 

 ent forces brought into play, namely, the elasticity of 

 the rod itself, and not external causes of tension. The 

 number of vibrations, when the whole rod oscillates, and 

 when there are two or three divisions, are as 9 to 25, to 

 49, to 81, or as the squares of the uneven numbers. 

 Reasons might be assigned why the proportions should be 

 as the squares, but the facts are very striking. 



The vibrations of a rod, fixed at one end and free at 

 the other, are also produced through the elasticity of its 

 substance. The number of vibrations in a unit of time is 

 inversely proportional to the square of the length of the 

 rod. If a rod 36 inches long vibrate once in a second, 

 one 12 inches long will vibrate nine times, one 6 inches 

 long thirty-six times, one 3 inches 144 times, one a 

 single inch 1296 times. Plates and bells follow the same 

 laws. The celebrated Chladin first employed fine sand 

 sprinkled on plates to show to the eye the motions in 

 them when vibrating. A square plate of metal is fixed 

 at its centre. The middle of one of the sides is damped 

 by the touch of the finger nail, and a violin bow is drawn 

 across near to one of the corners. The sand is at once set 

 in motion, and arranges itself in lines from side to side 



