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upon them. This is because these instruments, unlike strings and pipes, have overtones which are not harmonics, that is, which are not multiples of the fundamental; and these overtones produce beats either among themselves or with one of the fundamentals. It is for this reason that in playing chimes the bells are struck in succession, not simultaneously.
QUESTIONS AND PROBLEMS
1. In what three ways do piano makers obtain the different pitches? 2. What did Helmholtz prove by means of his resonators?
3. If middle C is struck on a piano while the key for G in the octave above is held down, G will be distinctly heard when C is silenced. Explain.
4. At what point must the G1 string be pressed by the finger of the violinist in order to produce the note C'?
5. If one wire has twice the length of another and is stretched by four times the stretching force, how will their vibration numbers compare?
6. A wire gives out the note G. What is its fourth overtone?
7. If middle C had 300 vibrations per second, how many vibrations would F and A have?
8. What is the fourth overtone of C'? the fifth overtone?
9. There are seven octaves and two notes on an ordinary piano, the lowest note being A, and the highest one C""". If the vibration number of the lowest note is 27, find the vibration number of the highest.
10. Find the wave length of the lowest note on the piano; the wave length of the highest note. (Take the speed of sound as 1130 ft. per sec.)
11. A violin string is commonly bowed about one seventh of its length from one end. Why is this better than bowing in the middle? 12. Build up a diatonic scale on C' = 264.
411. Fundamentals of closed pipes. Let a tightly fitting rubber stopper be inserted in a glass tube a (Fig. 366), eight or ten inches long and about three fourths of an inch in diameter. Let the stopper be pushed along the tube until, when a vibrating C' fork is held before the mouth, resonance is obtained as in § 391. (The length will be six or seven inches.) Then let the fork be removed and a stream of air blown
across the mouth of the tube through a piece of tubing b, flattened at one end as in the figure.* The pipe will be found to emit strongly the note of the fork.
In every case it is found that a note which a pipe may be made to emit is always a note to which it is able to respond when used as a resonator. Since, in § 392, the best resonance was found when the wave length given out by the fork was four times the length of the pipe, we learn that when a current of air is suitably directed across the mouth of closed pipe, it will emit a note which has a wave length four times the length of the pipe. This note is called the fundamental of the pipe. It is the lowest note which the pipe can be made to produce.
FIG. 366. Musical notes from pipes
412. Fundamentals of open pipes. Since we found in § 393 that the lowest note to which a pipe open at the lower end can respond is one the wave length of which is twice the pipe length, we infer that an open pipe, when suitably blown, ought to emit a note the wave length of which is twice the pipe length. This means that if the same pipe is blown first when closed at the lower end and then when open, the first note ought to be an octave lower than the second.
Let the pipe a (Fig. 366) be closed at the bottom with the hand and blown; then let the hand be removed and the operation repeated. The second note will indeed be found to be an octave higher than the first.
We learn, therefore, that the fundamental of an open pipe has a wave length equal to twice the pipe length.
413. Overtones in pipes. It was found in § 392 that there is a whole series of pipe lengths which respond to a given
* If the arrangement of Fig. 366 is not at hand, simply blow with the lips across the edge of a piece of ordinary glass tubing within which a rubber stopper may be pushed back and forth.
fork, and that these lengths bear to the wave length of the fork the ratios 1, 4, 4, etc. This is equivalent to saying that a closed pipe of fixed length can respond to a whole series of notes whose vibration numbers have the ratios 1, 3, 5, 7, etc. Similarly, in § 393, we found that in the case of an open pipe the series of pipe lengths which will respond to a given fork bear to the wave length of the fork the ratios 1, 2, 2, 1, etc. This, again, is equivalent to saying that an open pipe can respond to a series of notes whose vibration numbers have the ratios 1, 2, 3, 4, 5, etc. Hence we infer that it ought to be possible to cause both open and closed pipes to emit notes of higher pitch than their fundamentals (that is, overtones), and that the first overtone of an open pipe should have twice the rate of vibration of the fundamental (that is, it should be do', the fundamental being considered as do); that the second overtone should vibrate three times as fast as the fundamental (that is, it should be sol'); that the third overtone should vibrate four times as fast (that is, it should be do"); that the fourth overtone should vibrate five times as fast (that is, it should be mi"); etc. In the case of the closed pipe, however, the first overtone should have a vibration rate three times that of the fundamental (that is, it should be sol'); the second overtone should vibrate five times as fast (that is, it should be mi"); etc. In other words, while an open pipe ought to give forth all the harmonics, both odd and even, a closed pipe ought to produce the odd harmonics but be entirely incapable of producing the even ones.
Let the pipe of Fig. 366 be blown so as to produce the fundamental when the lower end is open. Then let the strength of the air blast be increased. The note will be found to spring to do'. By blowing still harder it will spring to sol, and a still further increase will probably bring out do". The odd and the even harmonics are, in fact, emitted by the open pipe, as our theory predicted. When the lower end is closed, however, the first overtone will be found to be soľ, and the next one mi”, just as our theory demands for the closed pipe.
414. Mechanism of emission of notes by pipes. Blowing across the mouth of a pipe produces a musical note, because the jet of air vibrates back and forth across the lip in a period which is determined wholly by the natural resonance period of the pipe. Thus, suppose that the jet a (Fig. 367) first strikes just inside the edge, or lip, of the pipe. A condensational pulse starts down the pipe. When it returns to the mouth after reflection at the closed end, it pushes the jet outside the lip. This starts a rarefaction down the pipe, which, after return from the lower end, pulls the jet in again. There are thus sent out into the room regu
larly timed puffs, the period of which is controlled by the reflected pulses coming back from the lower end, that is, by the natural resonance period of the pipe.
By blowing more violently it is possible to create, by virtue of the friction of the walls, so great and so sudden a compression
ing air jet
in the mouth of the pipe that the jet is forced FIG. 367. Vibratout over the edge before the return of the first reflected pulse. In this case no note will be produced unless the blowing is of just the right intensity to cause the jet to swing out in the period corresponding to an overtone. In this case the reflected pulses will return from the end at just the right intervals to keep the jet swinging in this period. This shows why a current of a particular intensity is required to start any particular overtone.
Another way of looking at the matter is to think of the pipe as being filled up with air until the pressure within it is great enough to force the jet outside the lip, upon which a period of discharge follows, to be succeeded in turn by another period of charge. These periods are controlled by the length of the pipe and the violence of the blowing, precisely as described above.
With open pipes the situation is in no way different save that the reflection of a condensation as a rarefaction at the lower end makes the natural period twice as high, since the pipe length is now one-half wave length instead of one-fourth wave length (see § 393).
415. Vibrating air-jet instruments. The mechanism of the production of musical tones by the ordinary
FIG. 368. Organ pipes
organ pipe, the flute, the fife, the piccolo, and all whistles is essentially the same as in the case of the pipe of Fig. 367. In all these instruments an air jet is made to play across the edge of an opening in an air chamber, and the reflected pulses returning from the other end of the chamber cause it to vibrate back and forth, first into the chamber and then out again. In this way a series of regularly timed puffs of air is made to pass from the instrument to the ear of the observer precisely as in the case of the rotating disk of § 385. The air chamber may be either open or closed at the remote end. In the flute it is open, in whistles it is usually closed, and in organ pipes it may be either open or closed. Fig. 368 shows a cross section of two types of organ pipes. The jet of air from S vibrates across the lip L in obedience to the pressure exerted on it by waves reflected from O. Pipe organs are provided with a different pipe for each note, but the flute, piccolo,
FIG. 369. Mouthpiece of a clarinet, showing the tongue 1, which opens and closes the upper end of