Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/128

be obtained independently of each other, but they are also to be heard simultaneously, particularly, for the reason just given, those that are tower in the scale. A practised ear easily discerns the coexistence of these various tones when a pianoforte or violin string is thrown into vibration.

Fig. 19.

It is evident that, in such case, the string, while vibrating as a whole between its fixed extremities, is at the same time executing subsidiary oscillations about its middle point, its points of trisection, &c., as shown in fig. 19, for the fundamental and the first harmonic.

59.The easiest means for bringing out the harmonics of a string consists in drawing a violin-bow across it near to one end, while the feathered end of a quill or a hair-pencil is held lightly against the string at the point which it is intended shall form a node, and is removed just after the bow is withdrawn. Thus, if a node is made in this way, at ${\tfrac {1}{3}}$ of ${\text{AB}}$ from ${\text{A}}$, the note heard will be the twelfth. If light paper rings be strung on the cord, they will be driven by the vibrations to the nodes or points of rest, which will thus be clearly indicated to the eye.

60.The formula $n_{1}={\tfrac {\text{V}}{2l}}$ shows that the pitch of the fundamental note of a wire of given length rises with the velocity of propagation of sound through it. Now we have learned (§ 28) that this velocity, in ordinary circumstances, is enormously greater for a wire vibrating longitudinally than for the same wire vibrating transversely. The fundamental note, therefore, is far higher in pitch in the former than in the latter case.

As, however, the quantity ${\text{V}}$ depends, for longitudinal vibrations, solely on the nature of the medium, the pitch of the fundamental note of a wire rubbed along its length depends—the material being the same, brass for instance—on its length, not at all on its thickness, &c.

But as regards strings vibrating transversely, such as are met with in our instrumental music, ${\text{V}}$, as we have seen (§ 27), depends not only on the nature of the substance used, but also on its thickness and tension, and hence the pitch of the fundamental, even with the same length of string, will depend on all those various circumstances.

61.If we put for ${\text{V}}$ its equivalent expressions before given, we have for the fundamental note note of transversely vibrating strings:

$$n={\tfrac {\sqrt {gc}}{2l}}{\text{ or }}n={\tfrac {1}{2l}}{\sqrt {\tfrac {g{\text{P}}}{w}}},$$

whence the following inferences may be easily drawn:

If a string, its tension being kept invariable, have its length altered, the fundamental note will rise in pitch in exact proportion with its diminished length, that is, $n$ varies then inversely as $l$.

Hence, on the violin, by placing a finger successively on any one ot the strings at ${\tfrac {8}{9}},{\tfrac {4}{5}},{\tfrac {3}{4}},{\tfrac {2}{3}},{\tfrac {3}{5}},{\tfrac {8}{15}},{\tfrac {1}{2}}$, we shall obtain notes corresponding to numbers of vibrations bearing to the fundamental the ratios to unity of the following, viz., ${\tfrac {9}{8}}$, ${\tfrac {5}{4}}$, ${\tfrac {4}{3}}$, ${\tfrac {3}{2}}$, ${\tfrac {15}{8}}$, $2$, which notes form, therefore, with the fundamental, the complete scale.

62.By tightening a musical string, its length remaining unchanged, its fundamental is rendered higher. In fact, then, $n$ is proportional to the square root of the tension. Thus, by quadrupling the tension, the note is raised an octave. Hence, the use of keys in tuning the violin, the, pianoforte, &c.

63.Equal lengths of strings of the same density and . equally stretched, but of different thicknesses, give fundamentals which are higher in pitch in proportion to diminution of thickness (i.e., $n$ varies inversely as the thickness). Thus, of two strings of same kind of gut, same length and same tension, if one be twice as thick as the other, its fundamental will be an octave lower. Hence, three of the strings of the violin, though all of gut, have different fundamentals, because unequally thick.

64.Equally long and equally stretched strings or wires of different thickness and different material, have fundamentals higher in pitch the less the weights of the strings; $n$ here varies inversely as the square root of the weight $w$ of a given length of the string.

65.If, in last case, the thicknesses of the strings which are to be compared together are equal, then $n$ varies inversely as the square root of the density.

Hence, in the violin and in the pianoforte, the lower notes are obtained from wires formed of denser material. Thus, the fourth string of the violin is formed of gut covered with silver wire.

66.A highly ingenious and instructive method for illustrating the above laws of musical strings, has been recently contrived by M. Melde, and consists simply in attaching to the ventral segment of a vibrating body, such as a tuning-fork or a bell-glass, a silk or cotton thread, the other extremity being either fixed or passing over a pulley and supporting weights by which the thread may be stretched to any degree required. The vibrations of the larger mass are communicated to the thread which, by proper adjustment of its length and tension, vibrates in unison and divides itself into one or more ventral segments easily discernible by a spectator. If the length of the thread be kept invariable, a certain tension will give but one ventral segment; the fundamental note of the thread is then of same pitch as the note of the body to which it is attached. By reducing the tension to ${\tfrac {1}{4}}$of its previous amount, the number of ventral segments will be seen to be increased to two, indicating that the first harmonic of the thread is now in unison with the solid, and consequently that its fundamental is an octave lower than it was with the former tension; thus confirming the law that $n$ varies as ${\sqrt {\text{P}}}$. In like manner, on further lowering the tension to ${\tfrac {1}{9}}$, three ventral segments will be formed, and so on.

The law that, cœt. par., $n$ varies inversely as the thickness may be tested by forming a string of four lengths of the single thread used before, and consequently of double the thickness of the latter, when, for the same length and tension, the compound thread will exhibit double the number of ventral segments presented by the single thread.

The other laws admit of similar illustration.

Part VII.
Stiff Rods, Plates, &c.

Fig. 20.

67.If, instead of a string or thin wire, we make use of a rod or narrow plate, sufficiently stiff to resist flexure, we may cause it to vibrate transversely when fixed at one end only. In this case the number of vibrations corresponding to the fundamental note varies as the thickness directly, and as the square of the length inversely. The annexed figures re present the modes of vibration corresponding to the fundamental and the first two overtones, the rod passing to and fro between the positions ${\text{AGKC}}$ and ${\text{AHLD}}$. In all cases ${\text{A}}$