LAWS OF SONOROUS VIBRATIONS. 831 



or striking it at one of these nodal points, we abolish the overtones which vibrate from this 

 node at a fixed point. For example, if we pluck the string at its exact centre, we sound 

 the fundamental ; but we then have a dull tone, which is deficient in the overtones of the 

 octaves. We can demonstrate the fact that these overtones are absent, for, if we damp 

 the string at its centre, the fundamental is quenched, but we have no octaves, which are 

 always heard on damping the centre when the string is plucked at other points. In the 

 same way, by plucking the string at different points, we can abolish the overtones of 



FIG. 262 b. 

 FIG. 262. Resonators of Helniholtz. 



5ths, 3ds, etc. It is readily understood that, when a string is plucked at any point, it 

 will vibrate so vigorously at this point that no node can be formed. This fact has long 

 been recognized by practical musicians, although many are probably unacquainted with 

 its scientific explanation. Performers upon stringed instruments habitually attack the 

 strings near their extremities. In the piano, where the strings may be struck at almost 

 any point, the hammers are placed at from \ to \ of their extremities ; and it has been 

 ascertained by experience that this gives the richest notes. The nodes formed at these 

 points would produce the Tths and 9ths as overtones, which do not belong to the perfect 

 major chord, while the nodes for the harmonious overtones are undisturbed. The reason, 

 then, why the notes are richer and more perfect when the strings are attacked at this 

 point, is that the harmonious overtones are full and perfect, and certain of the discordant 

 overtones are suppressed. 



When two harmonious notes are produced under favorable conditions, we can 

 hear, in addition to the two sounds, a sound differing from both and much lower than 

 the lower of the two. This sound is too low for an harmonic, and it has been called a 

 resultant tone. The formation of a new sound by combining two sounds of different 

 pitch is analogous to the blending of colors in optics, except that the primary sounds are 

 not lost. The laws of the production of these resultant sounds are very simple. When 

 two notes in harmony are sounded, the resultant tone is equal to the difference between 

 the two primaries. For example, if we sound 0, with 48 vibrations, and its 5th, with 72 

 vibrations in a second, the resultant tone is equal to the difference, which is 24 vibrations, 

 and it is consequently the octave below ; or, if we sound 0, with 48 vibrations, and its 

 3d, with 60, we have a resultant tone two octaves below 0, the number of vibrations 

 being 12. 1 These resultant tones are very feeble as compared with the primary tones, and 



1 These numbers are used merely in illustration. A sound of 12 vibrations does not come within the musical scale. 



