If the vibrating elastic solid is in the form of a plate the system of overtones bears a com plicated relationship to the fundamental, no longer being integer multiples in vibration fre quency. The manner in which the plate vi brates may be shown by sprinkling sand on the plate, the latter being horizontal. When the plate vibrates the sand dances away from the parts of the plate in motion and settles in ridges along the nodes. When the plate is square and emitting its lowest tone the nodal lines traced by the sand form a cross reaching from the centres of the sides. By bowing the plate at different points the plate may be made to vi brate in very complicated forms, the sand figures thus traced often making attractive de signs. The production of these various pat terns is much guided by touching the plate at various points on the edge with the fingers, thus determining the ends of nodal lines. This ex periment was first performed by Chladni, and the sand figures are called after their inventor Chladni's figures. Similar experiments can be carried out on stretched membranes, and one may investigate in this way the vibration of drum heads. The result of such an experiment shows that the quality of sound from a drum depends on the point at which it is struck, and that the upper partials are inharmonics of the fundamental.
Next to the stretched string the most inter esting case of a vibrating body is that of a col umn of air. To a first approximation the prob lem of the vibration of an air column is as simple as that of a stretched string, but in its practical forms and more accurate solution it is by no means so simple. The vibration of a column of air, according to the theory ad vanced by Bernoulli, is identical with the longi tudinal vibration of a straight bar of metal. If the column of air is in a tube open at both ends, the simplest form of vibration and that in which it emits the lowest possible note is such that the air moves to and fro at both ends having a node at the middle. The first overtone, hav ing a vibration frequency twice that of the fundamental, is produced by the column of air vibrating freely at both ends, vibrating freely at the middle, and having nodes at points one quarter of the total length of the pipe from either end. The second overtone has three times, the third overtone four times, etc., the vibration frequency of the fundamental. If the column of air is closed at one end the low est tone which it can emit is an octave below the lowest tone emitted by the same pipe open at both ends. The overtones in this case are three, five, seven, etc., times the fundamental in frequency. The analogy of this with the bar of metal is obvious. It might be added that according to Bernoulli's theory the note emit ted by the column of air is such that the sound could travel from the open end to the first node during one-quarter of a vibration.
This, only approximately true in the case of the column of air, is very strictly true in the case of the metal rod. It follows from this that the pitch of the note emitted by a column of air can be varied either by varying the length of the column, the pitch being inversely pro portional to the length, or by so exciting the air that it vibrates according to the higher forms with nodes nearer the ends. The appli cation of this to musical instruments is very simple. Take, for example, organ pipes of what are called flue stops as distinguished from reed stops. All such organ pipes are, obviously, open at the end at which they are blown. Ac cording as they are open or closed at the other end they are called open or closed pipes. Open pipes have nodes at their middle when sound ing the fundamental note, while the closed pipes have their nodes at the closed end. A closed pipe is therefore an octave lower in pitch than an open pipe of the same length, ac curately according to the theory of Bernoulli, hut as a matter of fact only approximately so. In a pipe organ the variation in pitch is accom plished not merely by using open and closed pipes, but principally by using pipes of different lengths. The pipes not uncommonly vary in length from 32 feet to half an inch. In the military trumpet we have an exceedingly simple instrument whose whole available scale consists in the overtones, the particular note being determined by lip tension and wind pres sure. In the slide trombone the scale is pro duced not merely as in the trumpet, but by varying the length by means of the slide. In the cornet the variation in length is accom plished by means of keys turning valves which throw into the length of the pipe or cut out from it short auxiliary convolutions. In the French horn the scale is played not merely by the means adopted in the cornet, but by means of the hand thrust into the bell or flared end, thus partially closing it and so lowering the pitch. In the flute, clarionet and wood wind instruments generally variation in pitch is accomplished by opening and closing ports on the side of the tube.
A little more might be said in regard to stringed instruments. The strings are, in gen eral, so narrow that when vibrating they cut through the air, communicating, practically, no motion to the air and therefore emitting, prac tically, no sound directly. The sound which we hear therefore comes not from the string, but from the sounding boards with which they are always placed in contact. It is thus be cause the sound which we bear comes from the body of the violin and scarcely at all from the strings directly that its quality depends so much more on the instrument than on the strings with which it is stretched.