HARMONY IN MUSIC. 89 



feels the tone due to each separate simple wave sepa- 

 rately, whether the compound wave originally proceeded 

 from a source capable of generating it, or became com- 

 pounded on the way. 



For example, on striking a string, it will give a tone corre- 

 sponding, as we have seen, to a wave-form widely different from 

 that of a simple tone. When the ear analyses this wave-form 

 into a sum of simple waves, it hears at the same time a series of 

 simple tones corresponding to these waves. 



Strings are peculiarly favourable for such an investigation, 

 because they are themselves capable of assuming extremely dif- 

 ferent forms in the course of their vibration, and these forms 

 may also be considered, like those of aerial undulations, as com- 

 pounded of simple waves. Fig. 4, p. 76, shows the consecutive 

 forms of a string struck by a simple rod. Fig. 11, p. 90, gives a 

 number of other forms of vibration of a string, corresponding to 

 simple tones. The continuous line shows the extreme displace- 

 ment of the string in one direction, and the dotted line in the other. 

 A.t a the string produces its fundamental tone, the deepest simple 

 tone it can produce, vibrating in its whole length, first on one 

 side and then on the other. At b it falls into two vibrating 

 sections, separated by a single stationary point /3, called a node 

 (knot). The tone is an octave higher, the same as each of the 

 two sections would separately produce, and it performs twice as 

 many vibrations in a second as the fundamental tone. At c we 

 have two nodes, yj and y 2 , and three vibrating sections, each 

 vibrating three times as fast as the fundamental tone and hence 

 giving its twelfth. At d l there are three nodes, 3 1? 3 2 , c> 3 , and 

 four vibrating sections, each vibrating four times as quickly as 

 the fundamental tone, and giving the second octave above it. 



In the same way forms of vibration may occur with 5, 6, 7 r 

 &c., vibrating sections, each performing respectively, 5, 6, 7, &c. 

 times as many vibrations in a second as the fundamental tone, 

 and all other vibratiorial forms of the string may be conceived as 

 compounded of a sum of such simple vibrational forms. 



The vibrational forms with stationary points or nodes may be 

 produced, by gently touching the string at one of these points, 

 either with the finger or a rod, and rubbing the string with a 



