186 SPECIAL SENSES, 



string at its exact centre, we sound the fundamental, but 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 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, 

 though many are probably unacquainted with its scientific 

 explanation. Performers on stringed instruments habitu- 

 ally attack the strings near their extremities. In the pi- 

 ano, 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 tones. The nodes formed at these points would pro- 

 duce the Tths and 9ths as overtones, which do not belong to 

 the perfect major chord, while the nodes for the harmonious 

 overtones are undisturbed. 1 The reason, then, why the tones 

 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. 

 The facility with which certain notes may be produced on 

 the cornet affords another illustration of the natural subdi- 

 vision of tones. In some parts of the scale, the open notes 

 are the tonic, 8th, 5th, and 3d ; that is, these notes can be 

 produced by the lips and the currents of air, without altering 

 the length of the column of air in the instrument by the 

 valves. "While some other of the notes in the scale may be 

 produced by the action of the lips alone, they are formed in 

 this way with uncertainty and difficulty. We can sound, for 

 example, the minor 3d, the 4th, or the 6th, with the lips ; but 

 these tones are more exact when we use the valves, as is al- 



1 HELMHOLTZ, Theorie physiologique de la musique, Paris, 1868, p. 109. 



