594 Original Articles. [Oct,. 



point and the string be plucked at b, and the finger immediately 

 removed from a, the string will divide itself into two equal portions, 

 each vibrating twice as fast as before, producing therefore the higher 

 octave c, as in Fig. 3. If it be touched at b and plucked at d it 

 divides itself into three portions, each moving three times as fast, and 

 giving the twelfth of C, viz. g, Fig. 4, and so on for Fig. 5, which 

 represents the second octave of C, c". But the movement of a string 

 is not limited to any single one of these forms of vibrations. It may 

 partake of all or any ; thus, a string may be divided into two, as in 

 Fig. 2, and likewise vibrate as a whole, so that a phenomenon would 

 be presented something like that shown in Fig. 6, where the string 

 gives both the sound of C its fundamental tone, and also that of c, 

 which is one of its upper tones. 



In this case the node or fixed point a of the upper octave c is 

 itself vibrating so as to produce C. It is obvious that the portions 

 representing the higher octave may be themselves divided so as to 

 produce the second octave c', and the string would then simultaneously 

 emit three sounds, C c and c'. 



These secondary vibrations may vary much among themselves in 

 intensity, that is, in their amplitude of oscillation, so that a string may 

 give its fundamental note supplemented by upper tones, some of 

 which will enter into the sound in greater strength than others, but 

 as a rule those upper tones which lie nearest in the scale to the funda- 

 mental will be the most powerful, as they require a greater length of 

 string and are therefore capable of oscillations of greater amplitude. 

 But this is not always the case, the second, third, or fourth tone being 

 sometimes more powerful than the fundamental itself. 



This series of harmonic upper tones is always the same for all 

 musical instruments, and is as follows : — 



1st. The higher octave which makes twice as many vibrations as 

 the fundamental note. If we suppose the key-note to be c, this higher 

 octave would be c'. 



2nd. The fifth of this higher octave g', which makes nine times 

 as many vibrations as c. 



3rd. The second higher octave c", which makes four times as 

 many vibrations. 



4th. The major third of the octave e", with five times as many 

 vibrations. 



5th. The fifth of this octave, with six times as many, and so on 

 to tones which require seven, eight, or nine times as many vibrations. 

 Expressed in the musical scale, we have 



Fundamental. Yt^--, -&- f^- ~\~ 



-JZt 



¥ 



^£ 



Tones 1 SC/3 4 5 6 7 8 9 10 



Vibrations-66 132 19S 264 330 396 462 528 594 660 



It is not every ear that can, without great attention and practice, 

 distinguish these upper tones in the sounds of musical instruments. 



