Scan 52
Grove's Dictionary Of Music And Musicians Vol I A. To E., 1904
12 lines. Read by computer on 11 September 2026. Printed page 34. See the scan on archive.org.
1a gas the greater the density of a gas, and consequently the greater its inertia, the less will be the velocity of waves in that gas. If the cord is stretched more tightly the force required to draw it aside at any point is increased. Consequently the part displaced flies back more rapidly, and the wave travels with greater velocity. Progressive waves travelling along a string and reflected from the end give rise to stationary waves with nodes and segments exactly as described above for air-waves reflected from a wall; and the period of a stationary vibration is the time taken by one complete wave to travel over a given point of the string, or the time a wave takes to travel over twice the length of one of the segments. A string as used for producing musical notes is stretched between two fixed pegs, or over two bridges, and vibrates transversely. The laws connecting the period of vibration with the length, tension, and mass of the string can be determined experimentally by means of the monochord. This consists of a string stretched over a sounding box. One end of the string is fixed, and the other passes over a pulley and supports a weight that can be varied so as to give any required tension. A movable bridge is placed under the string, so that the length of the vibrating section can be altered. It will be found that if the length is reduced to one-half, the note rises an octave, if to one-third it rises a twelfth, and so on; whence it follows that the vibration number is inversely proportional to the length, or the period of vibration directly proportional to the length. In order to raise the pitch an octave by altering the tension it will be found that the weight must be made four times greater, and to raise the pitch a twelfth it must be made nine times greater, and hence the vibration number is proportional to the square root of the tension. Similarly, by using strings of the same length and tension but of different weights it will be found that the vibration number varies inversely as the square root of the density.
3All three methods of altering the pitch of a note are illustrated by the violin. It is desirable, both for mechanical and musical reasons, that the tensions of the strings should not be greatly different from each other; and hence the lower strings are thicker than the higher ones, in order to lower the pitch whilst retaining a sufficient tension. The G string is wrapped with wire so as to get sufficient density without impairing the flexibility. If all three strings were of the same density the ratio of the tensions of the G and E would be 64 to 729, as the vibration numbers are in the ratio 8 to 27, and the lower strings would then be so loosely stretched as to give a very feeble tone of a very poor quality. The tuning of a string is effected by adjusting its tension, and the production of the notes other than the open note by pressing the string down on the finger-board and so altering its length.
5When a string gives out its fundamental alone it vibrates as a whole without nodes between the fixed ends, but it is possible for it to vibrate with one, two, or more nodes, and so to produce a series of overtones. If there is only one node it will be situated at the middle of the string, and the note will be an octave higher than the fundamental. This follows from the statement made above, that the vibration number is inversely proportional to the length; for a node is a point of no motion, and might be fixed without altering the circumstances, so that a string with a node in the middle is practically equivalent to two strings of half the length and hence an octave higher in pitch. The sections of string on the opposite sides of a node are always displaced in opposite directions. If, for instance, that on the right is arched upwards that on the left will be bent downwards, and vice versa. If the two sections have the same period, this state of affairs once established will persist, and the node will remain at rest; but a moment's consideration will show that a single node cannot be elsewhere than at the middle, for if it were, the two sections of the string would have different periods, and though at one moment they might be in opposite phases of vibration they could not remain so. A time would come when they would be moving in the same direction, and the point separating them would be compelled to move with them and so could not be a node. Hence it follows that whatever the number of nodes they must be so situated as to divide the string into sections of equal length.
7Hence, since any number of nodes is possible, it is easily seen that the notes that can be produced from a string of given length will have vibration numbers in the ratio of the numbers 1, 2, 3, etc., or the harmonic series.
9Overtones are sometimes produced on the violin by touching the string gently at an aliquot part of its length from one end, so as to induce the formation of a node at the point touched, instead of pressing the string down on the finger-board. The note produced by a string is made up of the fundamental together with harmonics which vary according to the method and position of the plucking, bowing, etc. The harder the hammer and the nearer the point struck to an end of the string the more prominent will be the higher harmonics and the more tinkling will be the tone.
11It has been stated above that within certain limits the force required to produce a given displacement in an elastic body is proportional to the displacement. In such circumstances if the force acting on the body varies according to the harmonic law—that is, is proportional to the sine of a uniformly increasing angle—the displacement will also follow the harmonic law.
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