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CHAPTER V.

VIBRATING STRINGS AND CORDS.

So many musical instruments are constructed on those principles involved in the theory of vibrating strings, that we should feel ourselves justified in asserting there is no branch of the science of acoustics of greater importance than that we are about to investigate. But if we consider this subject merely as a philosophical question, the interest and importance attached to it are sufficient inducements to a careful investigation. As this book is especially designed for the general reader, no strictly mathematical or rigidly philosophical investigations can be admitted into its pages; yet we shall attempt, as far as may be practicable, to explain so much of the theory of vibrating strings as shall enable the reader to understand the origin of the varieties of sound produced, and the laws by which those sounds are regulated. When this has been done, we may describe the construction and trace the history of those instruments in which strings or cords are used. The laws to which we refer evidently have relation to all vibrating strings, in what instrument soever they may be employed: the varieties of tone produced by different instruments consist in an alteration of intensity and quality, which are partly regulated by the form of the instrument, and partly by the means adopted for producing

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the excitement. In some instruments the string is made to vibrate by drawing a bundle of tense fibres, called a bow, over the stretched cords, as in the violin and violoncello; in some by the fingers, as in the harp and guitar, and in others by a small hammer, as in the pianoforte. The manner in which the vibrations are produced in these several instruments will assist in accounting for the peculiarities of their tones, so far at least as regards their quality; but the circumstances under which the vibrations are made will also have some influence.

The monochord (fig. 8) is an instrument admirably adapted to illustrate the laws which govern the production of sound in vibrating strings. A, B, C, D, is a hollow wooden box, on

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the top of which is fastened a narrow slip of wood, certain distances showing the necessary length of a string for the production of a certain note: b is a moveable bridge, p, a point to which the string is attached at one end, w, a wheel or pulley, over which the string passes, and T, a weight, by which the necessary tension is produced. By shifting the bridge, the length of the vibrating part of the string may be

either increased or decreased at pleasure, and the effects may be estimated under different circumstances.

The pitch of any note given out by a tense cord will vary according to the density, length, or degree of tension, possessed by the vibrating body. The reason of this is evident for the time required to complete a vibration will depend on these circumstances. It requires, as already observed in a former chapter, so many vibrations in a second for the production of one note, and so many, more or less, according to circumstances, for another. The mathematical theory of the vibration of stretched cords is one of great interest, and is remarkable, as Sir John Herschel has stated, " in an historical point of view, as having given rise to the first general solution of an equation of partial differences; and led geometers to the consideration of the nature and management of the arbitrary functions which enter into the integrals of these equations." But as we cannot enter into the mathematical researches which have conducted philosophers to a knowledge of the laws of vibrating, strings we shall merely state the result which has been obtained. The times of vibration in different cords are as their lengths directly, and as the square roots of the tending forces inversely; and the number of vibrations, the time being given, as the length inversely, and the square root of the tensions directly.*

NODAL POINTS.

It is a well-known but curious fact, that in every vibrating string there are certain points which always remain in a state

* Sound, art. 149-158, Ency. Metrop.

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of rest, never leaving the axis. These are called nodal points, and the distances between them are called bellies, or ventral segments. The existence of these nodal points may be readily shown on the monochord, an instrument already explained; for if a small narrow piece of paper in the form of an inverted V, be placed upon the vibrating string, it will be thrown off from every situation, except when on a node.

From this fact we are able to account for the production of harmonic sounds in vibrating strings. A delicate and practised ear can generally detect when a string is vibrating certain sounds blending with the fundamental note. This is especially the case when the string is touched lightly at particular points, and, from the concords they form with the fundamental note, they are called harmonic sounds. If the string of a violin, for instance, be lightly touched while sounding, exactly in the middle, the octave of the fundamental sound will be heard.

A cord may, when freely vibrating, have any number of nodes, and consequently be divided into any number of aliquot parts of its whole length. This fact, as well as the production of harmonic sounds as the result, was first observed by Wallis, in 1673, and was afterwards closely investigated by M. Sauveur, in a memoir read before the French Academy, in 1700. Before mathematicians commenced the investigation of this subject, musicians were probably aware that when a vibrating string is lightly touched at certain points, certain notes in concord with the fundamental tone, and consequently called harmonic sounds, were produced. In stringed instruments these attending tones would not be so perceptible as in vibrating bells and plates, and only an accurate ear could

detect them. By the use of the monochord, however, and the adoption of the method now commonly employed, the subject may be investigated experimentally by any of our readers.

VOIGT ON THE NODES OF MUSICAL STRINGS.

In the "Journal der Physick" there is an interesting and important paper by M. Voigt, of Halle, on the vibrating nodes of musical strings, to which we must call the attention of the reader. The facts which he adduces are arranged as a series of experiments, and we may follow the order in which they are placed.

A

D

E

B

Let A, B be the string of a monochord, and let it be divided into any number, four, for example, of equal parts, by the points C, D, and E, a moveable bridge being placed at the point E. Upon the points C and D, and other parts of the string, drop light pieces of paper, and touch that part of the string represented by A, E, with the bow of a violin, all the pieces of paper, except those lying on the points C and D, will be immediately thrown off by the vibration thus excited. The points C and D are called vibration nodes.

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Let A,B be now divided into five equal parts by the points C, D, E, F, and let a moveable bridge be placed at E. On the points C, D, F, and on any other parts of the string at

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