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Different Systems of Temperament.
made to reduce the subject to arithmetical principles; but it remains a mere matter of individual taste. It has generally been considered preferable, however, to preserve some keys more free from error than others; partly for variety, and partly because keys with five or six sharps or flats are comparatively little used, so that these may safely be left more imperfect, (which is called by some, throwing the wolf into these keys.)
a. A variety of systems of temperament have been devised for producing the best harmony by a system of 21 fixed sounds, viz. each note of the seven in the diatonic scale with its sharp and flat. The first and most celebrated is that of Huygens; and an exceedingly refined system has also been proposed by Dr. Smith. Either system, no doubt, will give very good harmony; but as on the piano-forte only 12 keys can be admitted, and as this instrument is now become an essential element in all concerts, and indeed the chief of all, a temperament must be devised which will accomodate itself to that condition.
b. The system called by Dr. Smith that of mean tones, or the vulgar tempérament, though the most inartificial, is probably as good as any which the nature of music admits, holding a sort of mean between the advantages and defects of all the rest. In this system, the third (3) of the diatonic scale is perfect, and the fifth (5) is tempered a little flat, flatter than a perfect fifth by a quarter of a comma. Note (2) is half-way between (1) and (3); (6) is the fifth of (2) and the third of (4); and (7) is the third of (5). The sharps and flats of the chromatic scale are inserted by bisecting the larger intervals.
c. Mr. Logier endeavoured to place the interpolation of the intermediate notes between those of the natural scale on à priori grounds, by assuming the flat seventh (7) as the seventh harmonic of the fundamental note (1); that is to say, the note produced by subdividing into seven equal parts the length of a string whose
Ways in which Solids may Vibrate.
fundamental tone is (1), or at least one of the octaves of that note. But sevenths, tuned on his principle, will require a much more violent temperament than either fifths or thirds, either of which might be used as a means of introducing the intermediate notes; and the system must in consequence be abandoned, as must every system which professes to render musical arithmetic any thing more than a matter of convention and approximation.
OF THE SONOROUS VIBRATIONS OF BARS, RODS, AND PLATES.
117. THE vibrations of all bodies, if of a proper degree of frequency, and of sufficient force to be communicated through the air, or any other intermedium, to our organs of hearing, produce sounds whose pitch depends on their frequency; and their force and quality on the extent, or other mechanical circumstances of the vibrations, and the nature of the vibrating body. The mathematical investigation of these vibratory motions is altogether foreign to our purpose; it is one of the most intricate and least manageable branches of Dynamics.
118. A solid body may vibrate, either in consequence of its inherent elasticity, by which it tends to return to its own proper figure and state, when forcibly deranged, or in consequence of an external tension.
Ways in which Solids may Vibrate.
To the former sort of vibrations belong those of rods, tuningforks, plates, rings, bells, gongs, and vessels of all shapes, or generally, of all solid masses which ring when struck. To the latter, those of vibrating strings and membranes, such as the parchment of a drum or a tambourine, &c.
119. But, further, a solid may vibrate by its own proper elasticity in two very different ways.
First, an undulation may be propagated through it, as through an elastic compressible medium; and in this case, the waves will consist of alternate strata of condensed and rarefied solid matter, precisely similar to those of an elastic fluid, the laws of motion in different directions varying with the varying elasticity of the solid.
If the solid be homogeneous, such as the metals, glass, &c., the elasticity being the same in all directions, the waves will be propagated from the centre of disturbance, according to exactly the same laws as in a mass of air of the same shape. But, if crystallized, this may not be the case, or the vibrations, instead of being in the direction of the propagated wave, may be transverse or oblique to it, or may even not be confined to one plane, but may be performed in circles or ellipses.
120. If a straight rod of glass, or a metal, be struck at the end in the direction of its length, or rubbed lengthways with a moistened finger, it will yield a musical sound, which, unless its length be very great, will be of an extremely acute pitch; much more so than in the case of a column of air of the same length.
The reason of this is the greater velocity with which sound is propagated in solids than in air. Thus the velocity of propagation in cast-iron being 10 times that in air, a rod of cast-iron so excited
Longitudinal Vibrations of a straight Rod.
will yield for its fundamental note a sound identical with that of an
1 of its length stopped at both ends, or of its 10 length if open at one end. See Chapter III. Part II., all the details of which are applicable to the present case.
To such vibrations Chladni, who first noticed them in long wires, has applied the term longitudinal.
121. To produce the harmonics of such a rod or wire, it must be held lightly at the place of one of its intended nodes between the finger and the thumb ; and the friction must be applied in the middle of one of the vibrating segments.
If the rod be of metal, the friction, which Chladni found to succeed, was that of a bit of cloth sprinkled with powdered rosin; if of glass, the cloth or the finger may be moistened and touched with some very fine sand or pumice powder.
122. It may be observed here, that, generally speaking, a fiddle-bow well rosined is the readiest and most convenient means of setting solid bodies in vibration.
To educe their gravest or fundamental tones, the bow must be pressed hard and drawn slowly; but for the higher harmonics, a short swift stroke with light pressure is most proper.
In all cases, the point intended to be a node must be lightly touched with the finger; and the vibration must be excited in the middle of a ventral segment.
The vibrations of a cylindrical rod or tube so excited are, in general, more complex than in the above case which was analyzed by Chladni.
Transverse Vibrations of a Rigid Rod,
123. Secondly. By far the most usual species of vibration, executed by solid bodies, is that in which their external form is forcibly changed, and recovered again by their spring.
124. The simplest case is that of a rod, executing vibrations to and fro in a direction transverse to its length.
a. This case has been investigated mathematically by D. Bernouilli and Euler, as also by Riccati; and their results have been compared with those of experiment by Chladni, and found correct.
b. The cases enumerated by Chladni are six in number.
I. When one end of the rod is firmly fixed in a vice or let into a wall, the other quite free. In this case, the curvature assumed by the rod in its vibrations must of necessity have its axis or position of rest for a tangent, as fig. 35.
II. One end applied or pressed perpendicularly against an obstacle, the other free. In this case, the excursions of the applied end to and fro are prevented by the friction and adhesion to the obstacle, but the axis is not of necessity a tangent. See fig. 36. III. Both ends free. Fig. 37.
IV. Both ends applied. Fig. 38.
V. Both ends fixed. Fig. 39.
VI. One end fixed, the other applied. Fig. 40.
c. All these cases have been examined by Chladni at length. We shall, however, select only the fourth case where both ends are applied, because it will afford room for an important remark. In this, then, the several modes of vibration corresponding to 1, 2, 3 vibrating or ventral segments of the rod will be as in figs. 38, 41, 42.
Now these are similar to the curves which would be assumed by a vibrating string under the same circumstances of subdivision. But the notes produced are very different. For, whereas in the case of a string the vibrations of the successive harmonics are