THE HAMILTON ASSOCIATION. 49 
We will now consider the actions of vibrating strings. Ifa string 
(or wire) is stretched tightly between two points and then twanged, 
it will give a certain fundamental note dependent on the thickness, 
tightness and length of the string. If it is bisected by a fixed point 
each section will give a sound one octave higher ; if divided at one- 
third of its length, it will give a note a twelfth higher (¢ e, an 
octave and four notes) ; and it can be proved that a sound an octave 
higher vibrates with twice the speed of the fundamental note, that 
the twelfth has three times the speed, and so on, giving us the rule 
that the number of vibrations is in inverse proportion to the length 
of string. Again, the number of vibrations will vary inversely as 
the thickness of the string. So it is evident that in stringing an 
instrument like a piano, which has strings ranging over seven octaves 
of pitch, it is a matter of the nicest calculation to grade the wires in 
thickness, from the heavy covered wires of the bass to the thin 
wires of the treble, so that the tone may be evenly graded 
throughout. 
If a stretched string is lightly touched at some section, half, third, 
quarter, or some definite fractional part of its length the rest of the 
string divides itself into sections of similar length, and this is proved 
by the fact that small pieces of paper perched on the string at these 
sections (or nodes as they are called) retain their places when the 
string is vibrated by a bow, while others placed midway are thrown 
off immediately. 
The observance of nodes, or partial- vibrations, found in vibrat- 
ing strings bring us to a most interesting part of the study of sound, 
namely, harmonic sounds or overtones. We are apt to think of 
sounds as just of a certain pitch and certain quality. Now it is 
ascertained that with every fundamental tone there coexist a number 
of higher tones called overtones or upper partials. 
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