Scan 69
Groves Dictionary Of Music And Musicians,vol.1 Ed.3, 1904
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1ACOUSTICS
3They 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. Thus it follows that whatever the number of nodes they must be so situated as to divide the string into sections of equal length.
5Hence, 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.
7Overtones 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.
9The 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 of a pianoforte 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.
11COMBINATION OR RESULTANT TONES.—It 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 an uniformly increasing angle—the displacement will also follow the harmonic law.
13A vibrating body A communicates vibrations to an elastic body B in contact with it, by acting on it with a force that varies periodically. If A vibrates harmonically the forces will be harmonic, and if B obeys Hooke's Law, its vibrations will also be harmonic. If B does not obey Hooke's Law its vibrations will not be harmonic though they will be periodic, and will have the same period as the vibrations of A. Now small changes of volume of air are approximately proportional to the changes of pressure that cause them, and hence a harmonically vibrating body will set up approximately harmonic vibrations in the air if the vibrations are small. When, however, the periodic forces acting on air are large this is no longer the case. The condensation is no longer proportional to the increase of pressure above the normal, and therefore a body that is itself vibrating harmonically may cause vibrations in the air that are not harmonic. The vibrations will be periodic, and therefore can be expressed as the sum of a series of harmonic constituents, the overtones being more conspicuous the greater the amplitude of the vibrations. Thus we have the result that when a body that executes nearly simple harmonic vibrations communicates its vibrations to the air, the sound produced may be a complex note when it is very loud, and will gradually alter in quality as it dies away, becoming more and more nearly a pure tone. Harmonics formed in this way are very faint unless the vibrations are amplified by confining the vibrating air in a small cavity. If a strongly vibrating fork is held over the mouth of a bottle, the octave may sometimes be heard.
15When two trains of simple harmonic waves of different periods reach such a restricted cavity simultaneously the amplitude of vibration of the two, when combined, may be sufficient to exceed the limits within which force is proportional to displacement. Helmholtz has shown that in such circumstances there are formed Combination Tones, the most important of which is one whose frequency is the difference of the frequencies of the two original tones, and is called the First Difference Tone. There is formed also a Summation Tone whose frequency is the sum of the frequencies of the original tones. The difference and summation tones being real tones can form further difference and summation tones with each other and with the original notes, and hence a large number of such combination tones are theoretically possible. The drum of the ear does not obey Hooke's Law. In the normal state it is stretched inwards a little, and an increase of pressure in the air near it gives a smaller displacement than an equal diminution of pressure gives in the opposite direction. Consequently two trains of harmonic waves that are not of sufficient amplitude to give combination tones in the air may do so when they reach the ear, and it is probable that most of the combination tones heard are thus formed in the ear. If they exist outside the ear they will be strengthened by a resonator; if they are produced by the ear a resonator will have no effect on their loudness. The first difference tone is heard very plainly when two notes are sounded together on a harmonium. Here the tone has a real existence outside the ear, as is proved by its being markedly strengthened by a resonator tuned to it. The tone is probably produced in the wind chest where the air is confined to a limited space, and so vibrates with great amplitude. Two whistles blown together give a very unpleasantly loud difference tone. It is easy to find the pitch of the difference tone produced by two given notes as follows. Write down the series of harmonics for any note, say C, and number them 1, 2, 3, etc., from below upwards. These numbers will be proportional to the frequencies of the various notes. Now pick out two notes of the series that give the interval under consideration;
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