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Scan 21
A dictionary of musical terms, 1889
16 lines. Read by computer on 9 September 2026. No printed page number was found. See the scan on archive.org.
1practice, however, the finger should always touch the string at the point of division adjacent to either end.
314. The harmonics of a simple tube used as a trumpet are the same as those of a vibrating string, viz., the octave, twelfth, fifteenth, &c., and are produced by modifications of the breath and lips; but there is a great difference between the nature of the vibrations which produce sound, in the case of strings and pipes. In the former case the vibrations are executed at right angles to the length of the string, that is, are lateral, while in the latter they are in the direction of the pipe, or longitudinal, and are the vibrations of the air itself within it.
515. When an open organ pipe is sounding its fundamental tone, the particles of the column of air within it are all, more or less, in a state of vibration parallel to the length of the pipe, of which the intensity is at its maximum at the two ends, growing less and less towards the middle, where there is a node, that is, a point of no disturbance. The harmonics of an open pipe follow the same law as those of a simple trumpet, or vibrating string.
7The fundamental note of a stopped organ pipe is an octave below the fundamental note of an open one of the same length. When it is sounding this note there is no node, and the first harmonic is a fifth above the octave, the second a major sixth above the first, the third a diminished fifth above the second, and so on. Or, more simply, the successive tones of the harmonic scale of an open pipe are produced by vibrations which are as 1, 2, 3, 4, &c., those of a stopped pipe by vibrations which are as 1, 3, 5, 7, &c.
916. It was stated (§ 10) that the sound of a vibrating string was in general compounded of a number of simple tones, and a well trained ear can detect a considerable number of them. If it were not for these harmonic components the tones of strings, pipes, of the human voice, or in short, of every instrument most generally used for the production of sound, would be flat and uninteresting like pure water. Each harmonic component is by itself a simple tone, and is due to the vibration of the corresponding segment of the string superposed upon that of the whole. The same statement applies, mutatis mutandis to pipes, whether open or stopped. That the harmonics of different instruments greatly influence their several characters is observable in the difference of the tones of a flute, and clarinet. A flute is an open pipe, a clarinet a stopped one; in the former, therefore, the harmonics follow the order of the natural numbers 1, 2, 3, 4, and in the latter the order 1, 3, 5, 7;—the intermediate notes being supplied by opening the lateral orifices of the instrument.
1117. When two simple tones, that is (as explained above), notes deprived of all the harmonic components which under ordinary circumstances accompany them, are sounded together very nearly in unison, there are heard what are called beats succeeding one another at regular intervals, their rapidity depending inversely on the smallness of the interval between the two tones. Their origin may be explained thus: Suppose the tones to be produced by vibrations numbering 500 and 501 per second respectively, then every 500th sound wave of the former will strike on the tympanum at exactly the same instant as every 501st of the latter and will reinforce it; while at the 250th of the first the corresponding wave of the other will be just half a period in front of it. Now a sound wave consists of a condensed and rarefied stratum of air particles, and therefore the condensed portion of one wave here coincides with the rarefied portion of the other and neutralises it. Thus there will be an alternate reinforcement and diminution of sound, every second, from the maximum intensity when both waves impinge on the tympanum at the same instant to the minimum when they counteract each other as much as possible and vice versa.
13In the above case it was supposed that the number of vibrations of one tone were only one more per second than those of the other; but if the difference of the numbers had been two, for instance, then in one second the first tone would have gained two vibrations on the other, and there would have been two beats; and in general the number of beats per second is always equal to the difference between the two rates of vibrations per second.
1518. In the preceding section, the cause of beats due to two simple tones of nearly the same pitch was explained, and it was seen that the number of beats per second was always equal to the difference of the numbers of vibrations per second of each tone; so that as the interval between them increased so would the number of beats increase in a given time. Hence it is obvious that if the interval became sufficiently large, the beats would succeed each other so rapidly as to become undistinguished. For instance, in the case of the fifth whose lower and upper tones are produced by vibrations numbering 264 and 396 per second respectively, the number of beats per second would be 132 and would therefore be undistinguishable—and still more so supposing the upper tone to have 397 or more vibrations per second; but, on the other hand it is a well-known fact, that if an imperfect fifth, octave, or any other tolerably simple interval is played on a violin or violoncello,
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