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Scan 20
A dictionary of musical terms, 1889
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1vibrations whose periods are aliquot parts of that of the former; and we have conclusive reasons for supposing that, in the same way as a compound periodic vibration gives rise to a compound sound (§ 9), so the simple tones into which the ear resolves the latter are respectively due to the simple harmonic vibrations which, as the above mentioned theorem proves, make up the former.*
312. The theorem of Fourier referred to in the preceding article is of such great importance in all questions connected with acoustics that a few words illustrative of it may not be out of place.†
5If a peg is fixed into the rim of a wheel capable of revolving about a fixed centre, and at right angles to the plane of the wheel, and if the latter is caused to rotate uniformly and is looked at edgways the peg will appear to move up and down in a straight line, its velocity being the greatest at the middle of its course, and diminishing as it approaches each end. Under these circumstances the peg appears to perform harmonic vibrations.
7Now suppose a second wheel, also furnished with a peg in its rim, is made to revolve about the peg of the first as an axis. If the latter is at rest the peg of the second will appear, looked at as above, to perform harmonic vibrations; but if the former is also caused to revolve these vibrations are no longer harmonic, but are the result of adding together the separate harmonic vibrations of the two pegs, in other words of superposing the harmonic vibrations which the second peg performs if the first wheel is at rest, upon those which the first peg performs when it is itself in motion. Now it is evident that by continuing this process indefinitely, and by giving the wheels different radii, and different uniform velocities of rotation, the final motion of the last peg looked at sideways as before, would be an exceedingly complicated one, and that an infinite number of different vibrations could be produced by varying the number, position at starting, radii, and velocities of the wheels, though it could not be assumed without proof that every possible variety could be so produced. This however is what Fourier's theorem asserts, provided that the velocities of rotation of the several wheels of the series are in the proportion of 1, 2, 3, 4, &c.‡ In other words, every periodic vibration is the resultant of a certain number of harmonic vibrations whose periods are one-half, one-third, one-fourth, &c., &c., that of the former.
913. A harmonic scale is formed by taking a series of notes produced by vibrations whose numbers in a given time are respectively as 1, 2, 3, 4, &c.
11If we take as fundamental tone the open C string of the violoncello, the series of tones which with it form a harmonic scale will be as follows:—
13The notes marked with an asterisk do not exactly represent the corresponding tones; but are the nearest representatives which the modern notation supplies. All the notes of the harmonic scale can theoretically be produced by either a single string, or by a simple tube used as a trumpet. If we lightly touch the string of a violin, without causing it to come in contact with the finger board, at any one of a series of points dividing it into a number of equal parts, and excite it by means of a bow, it no longer vibrates as a whole, but separates into the number of equal vibrating segments which is the least possible consistent with that point forming one of their points of division; the latter remain stationary, or very nearly so, and are called nodes, their number being evidently just one less than that of the segments. It is plain that if the point of application of the bow be one of a series of nodes, no sound will be produced, provided, of course, the finger remains on any other of the same series, and this may serve to explain why it is sometimes difficult to bring out the higher harmonics of a violin, as the bow may, unconsciously to the performer, be passing exactly over one of the corresponding nodes. The first harmonic, as it is called, of the open string is produced by touching it while in a state of vibration at its middle point, and thereby dividing it into two equal portions, both of which vibrate twice as fast as the whole, and accordingly give the octave. The second harmonic, or the twelfth of the fundamental, corresponds to a division of the string into three equal portions, and so on. And generally, in order to produce the nth harmonic the finger should touch the string at any one of the series of points which divide it into n equal portions.§
15* A periodic vibration is any movement which recurs after equal intervals or periods of time, such as that of a uniformly working punching machine, or of the hammer of a clock bell when it is striking, and so on. It should be observed that though all harmonic vibrations are periodic, it is by no means the case that all periodic vibrations are harmonic. See foot note to § 3.
17† For a complete discussion and demonstration of the theorem, the reader is referred to the work on Acoustics by the late Professor Donkin, published in the Clarendon Press series.
19‡ The order in which the wheels are arranged with respect to their velocities is quite arbitrary.
21§ That is supposing n to be a prime number (i.e. having no divisors). If such is not the case, it is plain that some points of the series when touched would give harmonics of lower pitch.
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