LAWS OF SONOROUS VIBRATIONS. 743 



the quality depends largely upon the form of the vibrations. To quote the 

 words of Helmholtz, " the more uniformly rounded the form of the wave, 

 the softer and milder is the quality of the sound. The more jerking and 

 angular the wave-form, the more piercing the quality. Tuning-forks, with 

 their rounded forms of wave, have an* extraordinarily soft quality ; and the 

 qualities of sound generated by the zither and violin resemble in harshness 

 the angularity of their wave-forms." 



Harmonics, or Overtones. As before stated, nearly all sounds are compos- 

 ite, but some contain many more aliquot, or secondary vibrations than others. 

 The notes of vibrating strings are peculiarly rich in harmonics, and these 

 may be used for illustration, remembering that the phenomena here observed 

 have their analogies in nearly all varieties of musical sounds. If a stretched 

 string be made to vibrate, the secondary tones, which qualify the funda- 

 mental, are called harmonics, or overtones. 



While it is difficult at all times to distinguish by the ear the individual 

 overtones of vibrating strings, their existence can be demonstrated by certain 

 simple experiments. Take, for example, a string, the fundamental tone of 

 which is C. If this string be damped with a feather at one-fourth of its 

 length and a violin-bow be drawn across the smaller section, not only the 

 fourth part of the string across which the bow is drawn is made to vibrate, 

 but the remaining three-fourths ; and if little riders of paper be placed upon 

 the longer segment at distances equal to one-fourth of the entire string, they 

 will remain undisturbed, while riders placed at any other points on the string 

 will be thrown off. This experiment shows that the three-fourths of the 

 string have been divided. This may be illustrated by connecting one end of 

 the string with a tuning-fork. When this is done and the string is brought 

 to the proper degree of tension, it will first vibrate as a whole, then, when a 

 little tighter, will spontaneously divide into two equal parts, and under in- 

 creased tension, into three, four, and go on. By damping a string with the 

 light touch of a feather, it is possible to suppress the fundamental tone and 

 bring out the overtones, which exist in all vibrating strings but are usually 

 concealed by the fundamental. The points which mark the subdivisions of 

 the string into segments of secondary vibrations are called nodes. When the 

 string is damped at its centre, the fundamental tone is, quenched and there 

 are overtones an octave above ; damping it at a distance of one-fourth, there 

 is the second octave above, and so on. When the string is damped at ajlis- 

 tance of one-fifth from the end, the four-fifths sound the 3d of the funda- 

 mental, with the second octave of the 3d. If it be damped at a distance of 

 two-thirds, there is the 5th of the fundamental, with the octave of the 5th. 

 Every vibrating string thus possesses a fundamental tone and overtones. 

 Qualifying the fundamental there is first, as the most simple, a series of 

 octaves; next, a series of 5ths of the fundamental and their octaves; and 

 next, a series of 3ds. These are the most powerful overtones, and they form 

 the common chord of the fundamental ; but they are so far concealed by the 

 greater intensity of the fundamental, that they can not easily be distinguished 

 by the unaided ear, unless the fundamental be quenched in some way. In 



