RESONANCE.
When a tuning-fork is struck, and held out in the air, the vibrations can be felt for a time by the fingers; but the sound is hardly audible unless the fork be placed close to the ear. Let the stem of the fork rest upon the table, a chair, or any solid body of considerable size, and the sound is so much increased in loudness as to be heard in every part of a large room. The reason appears to be, that in the first case the vibrations are so slight that the air is not much affected. Most of the force of the vibration is absorbed by the hand that holds it; but when the stem rests upon a hard body of considerable extent, the vibrations are given up to it, and every part of its surface is giving off the vibrations to the air. In other words, it is a much larger body that is now vibrating, and consequently the air is receiving the amplified sound-waves.
If the stem of the fork had been made to rest upon a bit of rubber, the sound would not only not have been re-enforced in such a way, but the fork would very soon have been brought to rest; for India rubber _absorbs_ sound vibrations, and converts them into heat vibrations, as is proved by placing such a combination upon the face of a thermo-pile.
If one will but put his hand upon a table or a chair-back in any room where a piano or an organ is being played, or where voices are singing, especially in church, he cannot fail to feel the sound; and if he notices carefully he will perceive that some sounds make such table or seat to shake much more vigorously than others,--a genuine case of sympathetic vibrations.
It is for this reason that special materials and shapes are given to parts of musical instruments, so that they may respond to the various vibrations of the strings or reeds. For instance, the piano has an extensive thin board of spruce underneath all the strings, which is called the sounding-board. This board takes up the vibrations of the strings; but, unlike the rubber, gives them all out to the air, greatly re-enforcing their strength, and changing somewhat their quality. But the air itself may act in like manner. In almost any room or hall not more than fifteen or twenty feet long, a person can find some tone of the voice that will seem to meet some response from the room. Some short tunnels will from certain positions yield very powerful, responsive, resonant tones. There is certainly one such in Central Park, New York.
It is forty or fifty feet long. To a person standing in the middle of this, and speaking or making any kind of a noise on a certain pitch, the resonance is almost deafening. It is easy to understand. When a column of air enclosed in a tube is made to vibrate by any sound whose wave-length is twice the length of the tube, we have such column of air now filled with the condensed part of the wave, and now with the rarefied part; and as these motions cannot be conducted laterally, but must move in the direction of the length of the tube, the air has a very great amplitude of motion, and the sound is very loud. If one end of the tube be closed, then the length must be but one-fourth of the wave-length of the sound. Take a tuning-fork of any convenient pitch, say a C of 512 vibrations per second: hold it while vibrating over a vertical test-tube about eight inches long. No response will be heard; but, if a little water be carefully poured into the tube to the depth of about two inches, the tube will respond loudly, so that it might be heard over a large hall. In this case the length of the air-column that was responding, being one-fourth the wave-length, would give twenty-four inches as the wave-length of that fork.
It is easy in this way to measure approximately the number of vibrations made by a fork.
Letting _l_ = depth of tube,
_d_ = diameter of tube,
_v_ = velocity of sound reduced for temperature,
_N_ = number of vibrations,
Then _N_ = _v_ ------------ (4(_l_+_d_)).
When a vibrating tuning-fork is placed opposite the embouchure of an organ-pipe of the same pitch, the pipe will resound to it, giving quite a volume of sound. In 1872 it occurred to me, that the action of an organ-pipe might be quite like that of a vibrating reed in front of the embouchure. As the air is driven past it from the bellows, the form of the escaping air will evidently be like a thin, elastic strip; and, having considerable velocity, it will carry off by friction a little of the air in the tube: this will of course rarefy the air in the tube somewhat, and a wave of condensation will travel down the tube. At the bottom, being suddenly stopped, its re-action will be partly outwards, and so will drive the strip of air away from the tube. After this will follow, for a like reason, the other phase of the wave, the rarefaction, which will swing the strip of air towards the tube. This theory I verified by filling the bellows with smoke, and watching the motion of the escaping air and smoke with a stroboscope. This view is now advocated by an organ-builder in England, Herman Smith; but whether he discovered it before or after me, I do not know.
When a membrane vibrates, its motion is generally perceptible to the eye; and it may have a very great amplitude of motion, as in the case of the drum; and various instruments have been devised for the study of vibrations, using membranes like rubber, gold-beater"s skin, or even tissue paper, to receive the vibrations. One of the musical instruments of a former generation of boys was the comb. A strip of paper was placed in front of it, and placed at the mouth, and sung through, the paper responding to the pitch with a loose nasal sound. Koenig fixed a membrane across a small capsule, one side of which was connected by a tube to any source of sound, and the other side to a gas-pipe and a small burner. A sound made in the tube would shake the flame, and a mirror moving in front of the flame would show a zigzag outline corresponding to the sound vibrations.
In like manner if a thin rubber be stretched over the end of a tube one or two inches in diameter and four or five inches long, and a bit of looking-gla.s.s one-fourth of an inch square be made fast to the middle of the membrane, the motions of the latter can be seen by letting a beam of sunlight fall upon the mirror so as to be reflected upon a white wall or screen a few feet away. (Fig. 8.)
[Ill.u.s.tration: FIG. 8.]
When a sound is made in this tube, the spot of light will at once a.s.sume some peculiar form,--either a straight line with some knots of light in it, or some curve simple or compound, and such as are known as Lissajous curves. If, while some of these forms are upon the screen, the instrument be moved sideways, the forms will change to undulating lines with or without loops, varying with the pitch and intensity, but being alike for the same pitch and intensity. (Fig. 9)
This instrument I called the opeidoscope.
[Ill.u.s.tration: FIG. 9.]
The vibration of a membrane and that of a solid differ chiefly in the amplitude of such vibration. The scratch of a pin at one end of a long log can be heard by an ear applied to the other end of the log; but every molecule in the log must move slightly; and there are all degrees of movement between that visible to the eye, which we call ma.s.s motion, and that called molecular simply because we cannot measure the amplitude of the motion. We may, then, roughly divide all bodies into two cla.s.ses, as to their relations to sound,--such as re-enforce it, and such as distribute it: the first depending upon the form of the body, as related to a particular sound; the second independent of form, and responding to all orders of vibrations. Air, wood, and metals belong in this latter cla.s.s. The common toy-string telegraph, or _lovers" telegraph_, is an example of this cla.s.s. Two tin boxes are connected by a string pa.s.sing through the middle of the bottom of each. When the string is stretched, and a person speaks in one box, what is said can be heard by an ear applied at the other. If the speaking-tubes be made about four inches in diameter, and about four inches deep, they are capable of doing much more service than is generally supposed to be possible. I know of two lines, one of five hundred feet and the other of a thousand feet in length, over which one can talk, and be heard with distinctness. In the line of a thousand feet, the end of the tube is made of sheepskin tightly stretched, and the line is made of No. 8 cotton thread. The greater the tension, the better is the sound transmitted. The thread is supported at intervals by running through a loop on the ends of cords not less than three feet long, attached to supports. The thread pierces the membrane, and is attached to a small b.u.t.ton which is in contact with the membrane. Wind and rain affect this line disadvantageously. The other line of five hundred feet, between a pa.s.senger and a freight depot, has the tube end covered with stretched calfskin. Instead of thread, a copper-relay wire is employed (any small uninsulated wire will do as well). This permits a good tension, and is unaffected by the weather. One may stand in front of it about three feet, and converse with ease, and in an ordinary tone. The wire is supported in loops of string, as in the other.
Musicians have in all times employed various instruments for the production of musical effects. Whistles made of bone were used by pre-historic men, some of them having finger-holes so that different tones could be produced. A stag-horn that was blown like a flageolet, and having three finger-holes, has also been found; while on the old monuments of Egypt are pictured harps, pipes with seven finger-holes, a kind of flute, drums, tambourines, cymbals, and trumpets. In later times these primeval forms have been modified into the various instruments in use in the modern orchestra. It seems as if no musician had ever been interested in the question as to why one instrument should give out a sound so different from another one, even though it was sounding upon the same pitch. No one can ever mistake the sound of a violin, or a horn, or a piano, for any other instrument; and no two persons have voices alike. This difference in tone, which enables us to identify an instrument by its sound or a friend by his voice, is called quality of tone, or _timbre_.
About twenty years ago, that great German physicist Helmholtz undertook the investigation of this subject, and succeeded in unravelling the whole mystery of the qualities of sound.
He discovered first, that a musical sound is very rarely a simple tone, but is made up of several tones, sometimes as many as ten or fifteen, having different degrees of intensity and pitch. The lowest sound, which is also the strongest, is called the _fundamental_; and it is this tone we mean when we speak of the pitch of a sound, as the pitch of middle C upon a piano, or the pitch of the _A_ string on a violin. The higher sounds that accompany the fundamental are called sometimes harmonics, sometimes upper partial tones, but generally _overtones_. The character or quality of a sound depends altogether upon the number and intensity of these overtones a.s.sociated with the fundamental. If a sound can be made upon a pipe and a violin, that consists wholly of the fundamental with no overtones, the two instruments sound absolutely alike. It is exceedingly difficult to do this; and such sound when produced is smooth, but without character, and unpleasing.
Second, Helmholtz discovered that the overtones always stand in the simplest mathematical relation to the fundamental tone,--in fact, are simple multiples of that tone, being two, three, four, and so on, times the number of vibrations of it.
This will be readily understood by considering the position of such related sounds when they are written upon the staff.
[Ill.u.s.tration]
If we start with C in the ba.s.s as indicated in the staff, calling that the fundamental, then the notes that will represent the above ratios are those indicated by smaller notes, which are the overtones up to the ninth. The first overtone, being produced by twice the number of vibrations, must be the octave; the second, the fifth of the second octave; the third will be two octaves from the first, and so on: the number of vibrations of each of these notes being the number of the fundamental multiplied by its order in the series.
Taking C with 128 vibrations, we have for this series:--
128 1 = 128 = C fundamental.
128 2 = 256 = C".
128 3 = 384 = G".
128 4 = 512 = C".
128 5 = 640 = E".
128 6 = 768 = G".
128 7 = 896 = B"[flat].
128 8 = 1,024 = C"".
128 9 = 1,152 = D"".
128 10 = 1,280 = E"".
This series is continued up to the limits of hearing. Now, it appears that all instruments do not give the complete series: indeed, it is not possible to obtain them all upon some instruments. Each of them, however, when present helps in the general effect which we call quality.
Sometimes the overtones are more prominent than the fundamental, as when a piano-wire is struck with a nail. It has always been noticed that it does not give out the sound that is wanted when it is struck in this way. Hence it is the art of an instrument-maker to so construct the instrument as to develop and re-enforce such tones as are pleasing, and to suppress the interfering and disagreeable overtones. Piano-makers learned by trial where was the proper place to strike the stretched wire in order to develop the most musical sound upon it; but no reason could be given until it was observed that striking it at a point about one-seventh or one-ninth its length from either end prevented the development of the objectionable overtones, the seventh and the ninth.
Hence they can scarcely be heard in a properly constructed instrument.
These overtones are very discordant with the lower sounds.
Organ-pipes have their specific qualities given to them by making them wide-mouthed, narrow-mouthed, conical, and so on; shapes which experience has determined give pleasing sounds with different qualities.
The violin is an instrument that seems to puzzle makers more than almost any other. Some of the old violins made two hundred years ago by the Amati family at Cremona are worth many times their weight in gold.
Recent makers have tried in vain to equal them; but, when their ingenuity and skill have failed, they declare that _age_ has much to do with such instruments, that age mellows the sounding quality of the violin. But the Cremona violins were just as extraordinary instruments when they left the hands of the makers as they are now; and the fame of the Amati family as violin-makers was over all Europe while they were living.
A good violin when well played gives an exquisite musical effect, and on account of its range and quality of tones it is the leading orchestral instrument, always pleasing and satisfying; but in unskilled hands even the best _Cremona_ will give forth sounds that make one grieve that it was ever invented. Overtones of all sorts and with all degrees of prominence may be easily developed upon it: therefore the skilful player draws the bow at such a place upon the strings as to develop the overtones he wants, and suppress the ones not wanted. The usual rule is to draw the bow about an inch below the bridge; but the place for the bow depends upon where the fingers are that stop the strings, and also the pressure upon it. It requires an almost incredible amount of practice to be able to play a violin very well.
In the accompanying table will be found the component parts of tones upon a few instruments in common use.
TONE COMPOSITION.
The components of the tones are indicated by lines in the column underneath the figures representing the series. Thus the narrow-stopped organ-pipe gives a sound composed of a fundamental, and overtones three, five, seven, and nine times the number of vibrations of it.
TONE COMPOSITION.
--------------------------+---+---+---+---+---+---+---+---+---+--- INSTRUMENTS. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 --------------------------+---+---+---+---+---+---+---+---+---+--- / Wide stopped | / | | | | | | | | | | +---+---+---+---+---+---+---+---+---+--- | Narrow " | / | | / | | / | | / | | / | | +---+---+---+---+---+---+---+---+---+--- | Narrow cylinder | / | / | / | / | / | / | | | | ORGAN < +---+---+---+---+---+---+---+---+---+---="" pipes.="" |="" princ.i.p.al="" }|="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" (wood)="" }|="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" |="" conically="" }|="" |="" |="" |="" |="" |="" |="" |="" |="" |="" narrow="" at="" top.="" }|="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" flute="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" violin="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" piano="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" bell="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" clarionet="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" ba.s.soon="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" +---+---+---+---+---+---+---+---+---+---="" oboe="" |="" |="" |="" |="" |="" |="" |="" |="" |="" |="" --------------------------+---+---+---+---+---+---+---+---+---+---="">
It must not be inferred that all of the overtones are of equal strength: they are very far from that; but these differ in different instruments, and it is this that const.i.tutes the difference between a good instrument and a poor one of the same name.
In a few of the s.p.a.ces very light lines are made for the purpose of indicating that such overtones are quite weak. For instance: the piano has the sixth, seventh, and eighth thus marked; these tones being suppressed by the mechanism, as described on a former page.
Only a few of the many forms of organ-pipes are given; but these are sufficient to show what a physical difference there is between the musical tones in such pipes.
As for the human voice, it is very rich in overtones; but no two voices are alike, therefore it would be impossible to tabulate the components of it in the manner they are tabulated for musical instruments.
In Helmholtz"s experiments in the a.n.a.lysis of sounds, use was made of the principle of resonance of a body of air enclosed in a vessel. In the experiment with the tuning-fork to determine the wave-length, p. 78, it is remarked that no response came until the volume of the air in the tube was reduced to a certain length, which depended upon the vibration number of the fork. If instead of a test-tube a bottle had been taken, the result would have been the same. Every kind of a vessel can respond to some tone of a definite wave-length, and a sphere has been found to give the best results. These are made with a hole on one side for the sound-wave to enter, and a projection on the opposite side, through which a hole about the one-eighth of an inch is made, this to be placed in the ear. Any sound that is made in front of the large orifice will not meet any response, unless it be that particular one which the globe can naturally re-enforce, when it will be plainly heard. Suppose, then, one has a series of twenty or more of these, graduated to the proper size for re-enforcing sounds in the ratio of one, two, three, four, and so on. Take any instrument, say a flute: have one to blow it upon the proper pitch to respond to the largest sphere, then take each of the spheres in their order, applying them to the ear while the flute is being sounded. When the overtones are present they will be heard plainly and distinct from the fundamental sound. In like manner any or all other sounds may be studied.