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FUNDAMENTALS AND OVERTONES
403. Fundamentals and overtones. If the assertion just made be correct, then a string which has a node in the middle communicates to the air twice as many pulses per second as the same string when it vibrates as a whole. This may be conclusively shown as follows:
Let the sonometer wire (Fig. 358) be plucked in the middle and the pitch of the corresponding tone carefully noted. Then let the finger be touched to the middle of the wire, and the latter plucked midway between this point and the end.* The octave of the original note will be distinctly heard. Next let the finger be touched at a point one third of the wire length from one end, and the wire again plucked. The note will be recognized as soľ. Since we learned in § 399 that soľ has three halves as many vibrations as do', it must have three times as many vibrations as the original note. Hence a wire which is vibrating in three segments sends out three times as many vibrations as when it is vibrating as a whole.
When a wire vibrates simply as a whole, it gives forth the lowest note which it is capable of producing. This note is called the fundamental or first partial of the wire. When the wire is made to vibrate in two parts, it gives forth, as has just been shown, a note an octave higher than the fundamental. This is called the first overtone or second partial. When the wire is made to vibrate in three parts it gives forth a note corresponding to three times the vibration number of the fundamental, namely, sol'. This is called the second overtone or third partial. When the wire vibrates in four parts, it gives forth the third overtone, which is two octaves above the fundamental. The overtones of wires are often called harmonics. They bear the vibration ratios 2, 3, 4, 5, 6, 7, etc. to the fundamental.†
* It is well to remove the finger almost simultaneously with the plucking. † Some instruments, such as bells, can produce higher tones whose vibration numbers are not exact multiples of the fundamental. These notes are still called overtones, but they are not called harmonics, the latter term being reserved for the multiples. Strings produee harmonics only.
404. Simultaneous production of fundamentals and overtones. Thus far we have produced overtones only by forcing the wire to remain at rest at properly chosen points during the bowing.
Now let the wire be plucked at a point one fourth of its length from one end, without being touched in the middle. The tone most distinctly heard will be the fundamental; but if the wire is now touched very lightly exactly in the middle, the sound, instead of ceasing altogether, will continue, but the note heard will be an octave higher than the fundamental, showing that in this case there was superposed upon the vibration of the wire as a whole a vibration in two segments also (Fig. 361). By touching the wire in the middle the vibration as a whole was destroyed, but that in two parts remained. Let the experiment be repeated, with this difference, that the wire is now plucked in the middle instead of one fourth its length from one end. If it is now touched in the middle, the sound will entirely cease, showing that when a wire is plucked in the middle there is no first overtone superposed upon the fundamental. Let the wire be plucked again one fourth of its length from one end and careful attention given to the compound note emitted. It will be found possible to recognize both the fundamental and the first overtone sounding at the same time. Similarly, by plucking at a point one sixth of the length of the wire from one end, and then touching it at a point one third of its length from the end, the second overtone may be made to appear distinctly, and a trained ear will detect it in the note given off by the wire, even before the fundamental is suppressed by touching at the point indicated.
FIG. 361. A wire simultaneously emitting its fundamental and first overtone
The experiments show, therefore, that in general the note emitted by a string plucked at random is a complex one, consisting of a fundamental and several overtones, and that just what overtones are present in a given case depends on where and how the wire is plucked.
405. Quality. Let the sonometer wire be plucked first in the middle and then close to one end. The two notes emitted will have exactly the same pitch, and they may have exactly the same loudness,
but they will be easily recognized as different in respect to something which we call quality. The experiment of the last paragraph shows that the real physical difference in the tones is a difference in the sorts of overtones which are mixed with the fundamental in the two cases.
Again, let a mounted C'fork be sounded simultaneously with a mounted C fork. The resultant tone will sound like a rich, full C, which will change into a hollow C when the C' is quenched with the hand.
Everyone is familiar with the fact that when notes of the same pitch and loudness are sounded upon a piano, a violin, and a cornet, the three tones can be readily distinguished. The last experiments suggest that the cause of this difference lies in the fact that it is only the fundamental which is the same in the three cases, while the overtones are different. In other words, the characteristic of a tone which we call its quality is determined simply by the number and prominence of the overtones which are present. If the overtones present are few and weak, while the fundamental is strong, the tone is, as a rule, soft and mellow, as when a sonometer wire is plucked in the middle, or a closed organ pipe is blown gently, or a tuning fork is struck with a soft mallet. The presence of comparatively strong overtones up to the fifth adds fullness and richness to the resultant tone. This is illustrated by the ordinary tone from a piano, in which several if not all of the first five overtones have a prominent place. When overtones higher than the sixth are present, a sharp metallic quality begins to appear. This is illustrated when a tuning fork is struck, or a wire plucked, with a hard body. It is in order to avoid this quality that the hammers which strike against piano wires are covered with felt.
406. Analysis of tones by the manometric flame. A very simple and beautiful way of showing the complex character of most tones is furnished by the so-called manometric flames. This device consists of the following parts: a chamber in the block B (Fig. 362), through which gas is led by way of the
tubes C and D to the flame F; a second chamber in the block A, separated from the first chamber by an elastic diaphragm made of very thin sheet rubber or paper, and communicating with the source of sound through the tube E and trumpet G; and a rotating mirror M by which the flame is observed. When a note is produced before the mouthpiece G, the vibrations of the diaphragm produce variations in the pressure of
FIG. 362. Analysis of sounds with manometric flames
the gas coming to the flame through the chamber in B, so that when condensations strike the diaphragm the height of the flame is increased, and when rarefactions strike it the height of the flame is diminished. If these up-and-down motions of the flame are viewed in a rotating mirror, the longer and shorter images of the flame, which correspond to successive intervals of time, appear side by side, as in Fig. 363. If a rotating mirror is not to be had, a piece of ordinary mirror glass held in the hand and oscillated back and forth about a vertical axis will be found to give satisfactory results.
First let the mirror be rotated when no note is sounded before the mouthpiece. There will be no fluctuations in the flame, and its image, as seen in the moving mirror, will be a straight band, as shown in 2 (Fig. 363). Next let a mounted C fork be sounded, or some other simple tone produced in front of G. The image in the mirror will be that shown in 3. Then let another fork, C', be sounded in place of the C. The image will be that shown in 4. The images of the flame are now twice as close together as before, since the blows strike the diaphragm twice as often. Next let the open ends of the resonance boxes of the tuning forks C and C' be held together in front of G. The image of the flame will be as shown in 5. If the vowel o be sung in the pitch Bb before the mouthpiece, a figure exactly similar to 5 will be produced, thus showing that this last note is a complex, consisting of a fundamental and its first overtone.
The proof that most other tones are likewise complex lies in the fact that when analyzed by the manometric flame they show figures not like 3 and 4, which correspond to simple tones, but like 5, 6, and 7, which may be produced by sounding combinations of simple tones. In the figure, 6 is produced by singing the vowel e on C"; 7 is obtained when o is sung on C". The beautiful photographs opposite page 346, taken by Prof. D. C. Miller, show the extraordinary complexity of spoken words.
407. Helmholtz's experiment. If the loud pedal on a piano is held down and the vowel sounds oo, i, ā, ah, è sung loudly into the strings, these vowels will be caught up and returned by the instrument with sufficient fidelity to make the effect almost uncanny.
It was by a method which may be considered as merely a refinement of this experiment that Helmholtz proved conclusively that quality is determined simply by the number and