Our next question is, what is the length of the column of air which resounds to this fork? By measurement with a two-foot rule it is found to be 13 inches. But the length of the wave emitted by the fork is 52 inches; hence the length of the column of air which resounds to the fork is equal to one-fourth of the length of the sound-wave produced by the fork. This rule is general, and might be illustrated by any other of the forks instead of this one.
Let the prong, vibrating between the limits a and b, be placed over its resonant jar, A B, Fig. 93. In the time required by the prong to move from a to b, the condensation it produces runs down to the bottom of the jar, is there reflected, and, as the distance to the bottom and back is 26 inches, the reflected wave will reach the fork at the moment when it is on the point of returning from b to a. The rarefaction of the wave is produced by the retreat of the prong from b to a. This rarefaction will also run to the bottom of the jar and back, overtaking the prong just as it reaches the limit, a, of its excursion. It is plain from this analysis that the vibrations of the fork are perfectly synchronous with the vibrations of the aërial column A B; and in virtue of this synchronism the motion accumulates in the jar, spreads abroad in the room, and produces this vast augmentation of the sound.
When we substitute for the air in one of these jars a gas of different elasticity, we find the length of the resounding column to be different. The velocity of sound through coal-gas is to its velocity in air about as 8:5. Hence, to synchronize with our fork, a jar filled with coal-gas must be deeper than one filled with air. I turn this jar, 18 inches long, upside down, and hold close to its open mouth our agitated tuning-fork. It is scarcely audible. The jar, with air in it, is 5 inches too deep for this fork. Let coal-gas now enter the jar. As it ascends the note at a certain point swells out, proving that for the more elastic gas a depth of 18 inches is not too great. In fact, it is not great enough; for if too much gas be allowed to enter the jar the resonance is weakened. By suddenly turning the jar upright, still holding the fork close to its mouth, the gas escapes, and at the point of proper admixture of gas and air the note swells out again.45
§ 9. Reinforcement of Bell by Resonance
This fine, sonorous bell, Fig. 94, is thrown into intense vibration by the passage of a resined bow across its edge. You hear its sound, pure, but not very forcible. When, however, the open mouth of this large tube, which is closed at one end, is brought close to one of the vibrating segments of the bell, the tone swells into a musical roar. As the tube is alternately withdrawn and advanced, the sound sinks and swells in this extraordinary manner.
The second tube, open at both ends, is capable of being lengthened and shortened by a telescopic slider. When brought near the vibrating bell, the resonance is feeble. On lengthening the tube by drawing out the slider at a certain point, the tone swells out as before. If the tube be made longer, the resonance is again enfeebled. Note the fact, which shall be explained presently, that the open tube which gives the maximum resonance is exactly twice the length of the closed one. For these fine experiments we are indebted to Savart.
§ 10. Expenditure of Motion in Resonance
With the India-rubber tube employed in our third chapter it was found necessary to time the impulses properly, so as to produce the various ventral segments. I could then feel that the muscular work performed, when the impulses were properly timed, was greater than when they were irregular. The same truth may be illustrated by a claret-glass half filled with water. Endeavor to move your hand to and fro, in accordance with the oscillating period of the water: when you have thoroughly established synchronism, the work thrown upon the hand apparently augments the weight of the water. So likewise with our tuning-fork; when its impulses are timed to the vibrations of the column of air contained in this jar, its work is greater than when they are not so timed. As a consequence of this the tuning-fork comes sooner to rest when it is placed over the jar than when it is permitted to vibrate either in free air, or over a jar of a depth unsuited to its periods of vibration.46
Reflecting on what we have now learned, you would have little difficulty in solving the following beautiful problem: You are provided with a tuning-fork and a siren, and are required by means of these two instruments to determine the velocity of sound in air. To solve this problem you lack, if anything, the mere power of manipulation which practice imparts. You would first determine, by means of the siren, the number of vibrations executed by the tuning-fork in a second; you would then determine the length of the column of air which resounds to the fork. This length multiplied by 4 would give you, approximately, the wave-length of the fork, and the wave-length multiplied by the number of vibrations in a second would give you the velocity in a second. Without quitting your private room, therefore, you could solve this important problem. We will go on, if you please, in this fashion, making our footing sure as we advance.
§ 11. Resonators of Helmholtz
Helmholtz has availed himself of the principle of resonance in analyzing composite sounds. He employs little hollow spheres, called resonators, one of which is shown in Fig. 94a. The small projection b, which has an orifice, is placed in the ear, while the sound-waves enter the hollow sphere through the wide aperture at a. Reinforced by the resonance of such a cavity, and rendered thereby more powerful than its companions, a particular note of a composite clang may be in a measure isolated and studied alone.
ORGAN-PIPES
§ 12. Principles of Resonance applied to Organ-Pipes
Thus disciplined we are prepared to consider the subject of organ-pipes, which is one of great importance. Before me on the table are two resonant jars, and in my right hand and my left are held two tuning-forks. I agitate both, and hold them over this jar. One of them only is heard. Held over the other jar, the other fork alone is heard. Each jar selects that fork whose periods of vibration synchronize with its own. And instead of two forks suppose several of them to be held over the jar; from the confused assemblage of pulses thus generated, the jar would select and reinforce that one which corresponds to its own period of vibration.
When I blow across the open mouth of the jar, or, better still, for the jar is too wide for this experiment, when I blow across the open end of a glass tube, t u, Fig. 95, of the same length as the jar, a fluttering of the air is thereby produced, an assemblage of pulses at the open mouth of the tube being generated. And what is the consequence? The tube selects that pulse of the flutter which is in synchronism with itself, and raises it to a musical sound. The sound, you perceive, is precisely that obtained when the proper tuning-fork is placed over the tube. The column of air within the tube has, in this case, virtually created its own tuning-fork; for by the reaction of its pulses upon the sheet of air issuing from the lips it has compelled that sheet to vibrate in synchronism with itself, and made it thus act the part of the tuning-fork.
Selecting for each of the other tuning-forks a resonant tube, in every case, on blowing across the open end of the tube, a tone is produced identical in pitch with that obtained through resonance.
When different tubes are compared, the rate of vibration is found to be inversely proportional to the length of the tube. These three tubes are 24, 12, and 6 inches long, respectively. I blow gently across the 24-inch tube, and bring out its fundamental note; similarly treated, the 12-inch tube yields the octave of the note of the 24-inch. In like manner the 6-inch tube yields the octave of the 12-inch. It is plain that this must be the case; for, the rate of vibration depending on the distance which the pulse has to travel to complete a vibration, if in one case this distance be twice what it is in another, the rate of vibration must be twice as slow. In general terms, the rate of vibration is inversely proportional to the length of the tube through which the pulse passes.
§ 13. Vibrations of Stopped Pipes: Modes of Division: Overtones
But that the current of air should be thus able to accommodate itself to the requirements of the tube, it must enjoy a certain amount of flexibility. A little reflection will show you that the power of the reflected pulse over the current must depend to some extent on the force of the current. A stronger current, like a more powerfully stretched string, requires a great force to deflect it, and when deflected vibrates more quickly. Accordingly, to obtain the fundamental note of this 24-inch tube, we must blow very gently across its open end; a rich, full, and forcible musical tone is then produced. With a little stronger blast the sound approaches a mere rustle; blowing stronger still, a tone is obtained of much higher pitch than the fundamental one. This is the first overtone of the tube, to produce which the column of air within it has divided itself into two vibrating parts, with a node between them. With a still stronger blast a still higher note is obtained. The tube is now divided into three vibrating parts, separated from each other by two nodes. Once more I blow with sudden strength; a higher note than any before obtained is the consequence.
In Fig. 96 are represented the divisions of the column of air corresponding to the first three notes of a tube stopped at one end. At a and b, which correspond to the fundamental note, the column is undivided; the bottom of the tube is the only node, and the pulse simply moves up and down from top to bottom, as denoted by the arrows. In c and d, which correspond to the first overtone of the tube, we have one nodal surface shown by dots at x, against which the pulses abut, and from which they are reflected as from a fixed surface. This nodal surface is situated at one-third of the length of the tube from its open end. In e and f, which correspond to the second overtone, we have two nodal surfaces, the upper one, x′, of which is at one-fifth of the length of the tube from its open end, the remaining four-fifths being divided into two equal parts by the second nodal surface. The arrows, as before, mark the directions of the pulses.
We have now to inquire into the relation of these successive notes to each other. The space from node to node has been called all through “a ventral segment”; hence the space between the middle of a ventral segment and a node is a semi-ventral segment. You will readily bear in mind the law that the number of vibrations is directly proportional to the number of semi-ventral segments into which the column of air within the tube is divided. Thus, when the fundamental note is sounded, we have but a single semi-ventral segment, as at a and b. The bottom here is a node, and the open end of the tube, where the air is agitated, is the middle of a ventral segment. The mode of division represented in c and d yields three semi-ventral segments; in e and f we have five. The vibrations, therefore, corresponding to this series of notes, augment in the proportion of the series of odd numbers 1:3:5. Could we obtain still higher notes, their relative rates of vibration would continue to be represented by the odd numbers 7, 9, 11, 13, etc.
It is evident that this must be the law of succession. For the time of vibration in c or d is that of a stopped tube of the length x y; but this length is one-third of the length of the whole tube, consequently its vibrations must be three times as rapid. The time of vibration in e or f is that of a stopped tube of the length x′ y′, and inasmuch as this length is one-fifth that of the whole tube, its vibrations must be five times as rapid. We thus obtain the succession 1, 3, 5; if we pushed matters further we should obtain the continuation of the series of odd numbers.
And here it is once more in your power to subject my statements to an experimental test. Here are two tubes, one of which is three times the length of the other. I sound the fundamental note of the longest tube, and then the next note above the fundamental. The vibrations of these two notes are stated to be in the ratio of 1:3. This latter note, therefore, ought to be of precisely the same pitch as the fundamental note of the shorter of the two tubes. When both tubes are sounded their notes are identical.
It is only necessary to place a series of such tubes of different lengths thus together to form that ancient instrument, Pan’s pipes, p p′, Fig. 97 (page 223), with which we are so well acquainted.
The successive divisions, and the relation of the overtones of a rod fixed at one end (described in page 205), are plainly identical with those of a column of air in a tube stopped at one end, which we have been here considering.
§ 14. Vibrations of Open Pipes: Modes of Division: Overtones
From tubes closed at one end, and which, for the sake of brevity, may be called stopped tubes, we now pass to tubes open at both ends, which we shall call open tubes. Comparing, in the first instance, a stopped tube with an open one of the same length, we find the note of the latter an octave higher than that of the former. This result is general. To make an open tube yield the same note as a closed one, it must be twice the length of the latter. And, since the length of a closed tube sounding its fundamental note is one-fourth of the length of its sonorous wave, the length of an open tube is one-half that of the sonorous wave that it produces.
It is not easy to obtain a sustained musical note by blowing across the end of an open glass tube; but a mere puff of breath across the end reveals the pitch to the disciplined ear. In each case it is that of a closed tube half the length of the open one.
There are various ways of agitating the air at the ends of pipes and tubes, so as to throw the air-columns within them into vibration. In organ-pipes this is done by blowing a thin sheet of air against a sharp edge. You will have no difficulty in understanding the construction of an open organ-pipe, from this model, Fig. 98, one side of which has been removed so that you may see its inner parts. Through the tube t the air passes from the wind-chest into the chamber, C, which is closed at the top, save a narrow slit, e d, through which the compressed air of the chamber issues. This thin air-current breaks against the sharp edge, a b, and there produces a fluttering noise, and the proper pulse of this flutter is converted by the resonance of the pipe above into a musical sound. The open space between the edge, a b, and the slit below it is called the embouchure. Fig. 99 represents a stopped pipe of the same length as that shown in Fig. 98, and hence producing a note an octave lower.
Instead of a fluttering sheet of air, a tuning-fork whose rate of vibration synchronizes with that of the organ-pipe may be placed at the embouchure, as at a b, Fig. 100. The pipe will resound. Here, for example, are four open pipes of different lengths, and four tuning-forks of different rates of vibration. Striking the most slowly vibrating fork, and bringing it near the embouchure of the longest pipe, the pipe speaks powerfully. When blown into, the same pipe yields a tone identical with that of the tuning-fork. Going through all the pipes in succession, we find in each case that the note obtained Fig. 100. by blowing into the pipe is exactly that produced when the proper tuning-fork is placed at the embouchure. Conceive now the four forks placed together near the same embouchure; we should have pulses of four different periods there excited; but out of the four the pipe would select only one. And if four hundred vibrating forks could be placed there instead of four, the pipe would still make the proper selection. This it also does when for the pulses of tuning-forks we substitute the assemblage of pulses created by the current of air when it strikes against the sharp upper edge of the embouchure.
The heavy vibrating mass of the tuning-fork is practically uninfluenced by the motion of the air within the pipe. But this is not the case when air itself is the vibrating body. Here, as before explained, the pipe creates, as it were, its own tuning-fork, by compelling the fluttering stream at its embouchure to vibrate in periods answering to its own.
The condition of the air within an open organ-pipe, when its fundamental note is sounded, is that of a rod free at both ends, held at its centre, and caused to vibrate longitudinally. The two ends are places of vibration, the centre is a node. Is there any way of feeling the vibrating air-column so as to determine its nodes and its places of vibration? The late excellent William Hopkins has taught us the following mode of solving this problem: Over a little hoop is stretched a thin membrane, forming a little tambourine. The front of this organ-pipe, P P′, Fig. 101, is of glass, through which you can see the position of any body within it. By means of a string, the little tambourine, m, can be raised or lowered at pleasure through the entire length of the pipe. When held above the upper end of the pipe, you hear the loud buzzing of the membrane. When lowered into the pipe, it continues to buzz for a time; the sound becoming gradually feebler, and finally ceasing totally. It is now in the middle of the pipe, where it cannot vibrate, because the air around it is at rest. On lowering it still further, the buzzing sound instantly recommences, and continues down to the bottom of the pipe. Thus, as the membrane is raised and lowered in quick succession, during every descent and ascent, we have two periods of sound separated from each other by one of silence. If sand were strewed upon the membrane, it would dance above and below, but it would be quiescent at the centre. We thus prove experimentally that, when an organ-pipe sounds its fundamental note, it divides itself into two semi-ventral segments separated by a node.
What is the condition of the air at this node? Again, that of the middle of a rod, free at both ends, and yielding the fundamental note of its longitudinal vibration. The pulses reflected from both ends of the rod, or of the column of air, meet in the middle, and produce compression; they then retreat and produce rarefaction. Thus, while there is no vibration in the centre of an organ-pipe, the air there undergoes the greatest changes of density. At the two ends of the pipe, on the other hand, the air-particles merely swing up and down without sensible compression or rarefaction.
If the sounding pipe were pierced at the centre, and the orifice stopped by a membrane, the air, when condensed, would press the membrane outward, and, when rarefied, the external air would press the membrane inward. The membrane would therefore vibrate in unison with the column of air. The organ-pipe, Fig. 102, is so arranged that a small jet of gas, b, can be lighted opposite the centre of the pipe, and there acted upon by the vibrations of a membrane. Two other gas-jets, a and c, are placed nearly midway between the centre and the two ends of the pipe. The three burners, a, b, c, are fed in the following manner: through the tube, t, the gas enters the hollow chamber, e d, from which issue three little bent tubes, shown in the figure, each communicating with a capsule closed underneath by the membrane. This is in direct contact with the air of the organ-pipe. From the three capsules issue the three little burners, with their flames, a, b, c.
Blowing into the pipe so as to sound its fundamental note, the three flames are agitated, but the central one is most so. Lowering the flames so as to render them very small, and blowing again, the central flame, b, is extinguished, while the others remain lighted. The experiment may be performed half a dozen times in succession; the sounding of the fundamental note always quenches the middle flame.
By blowing more sharply into the pipe, it is caused to yield its first overtone. The middle node no longer exists. The centre of the pipe is now a place of maximum vibration, while two nodes are formed midway between the centre and the two ends. But if this be the case, and if the flame opposite the node be always blown out, then, when the first overtone of this pipe is sounded, the two flames a and c ought to be extinguished, while the central flame remains lighted. This is the case. When the first harmonic is sounded the two nodal flames are infallibly extinguished, while the flame b in the middle of the ventral segment is not sensibly disturbed.
There is no theoretic limit to the subdivision of an organ-pipe, either stopped or open. In stopped pipes we begin with 1 semi-ventral segment, and pass on to 3, 5, 7, etc., semi-ventral segments, the number of vibrations of the successive notes augmenting in the same ratio. In open pipes we begin with 2 semi-ventral segments, and pass on to 4, 6, 8, 10, etc., the number of vibrations of the successive notes augmenting in the same ratio; that is to say, in the ratio 1:2:3:4:5, etc. When, therefore, we pass from the fundamental tone to the first overtone in an open pipe, we obtain the octave of the fundamental. Fig. 103. When we make the same passage in a stopped pipe, we obtain a note a fifth above the octave. No intermediate modes of vibration are in either case possible. If the fundamental tone of a stopped pipe be produced by 100 vibrations a second, the first overtone will be produced by 300 vibrations, the second by 500, and so on. Such a pipe, for example, cannot execute 200 or 400 vibrations in a second. In like manner the open pipe, whose fundamental note is produced by 100 vibrations a second, cannot vibrate 150 times in a second, but passes, at a jump, to 200, 300, 400, and so on.
In open pipes, as in stopped ones, the number of vibrations executed in the unit of time is inversely proportional to the length of the pipe. This follows from the fact, already dwelt upon so often, that the time of a vibration is determined by the distance which the sonorous pulse has to travel to complete a vibration.
In Fig. 103, a and b (at the bottom) represent the division of an open pipe corresponding to its fundamental tone; c and d represent the division corresponding to its first, e and f the division corresponding to its second overtone, the dots marking the nodes. The distance m n is one-half, o p is one-fourth, and s t is one-sixth of the whole length of the pipe. But the pitch of a is that of a stopped pipe equal in length to m n; the pitch of c is that of a stopped pipe of the length o p; while the pitch of e is that of a stopped pipe of the length s t. Hence, as these lengths are in the ratio of 1/2:1/4:1/6, or as 1:1/2:1/3, the rates of vibration must be as the reciprocals of these, or as 3:2:1. From the mere inspection, therefore, of the respective modes of vibration, we can draw the inference that the succession of tones of an open pipe must correspond to the series of natural numbers.
The pipe a, Fig. 103, has been purposely drawn twice the length of a, Fig. 93 (p. 215). It is perfectly manifest that to complete a vibration the pulse has to pass over the same distance in both pipes, and hence that the pitch of the two pipes must be the same. The open pipe, a n, consists virtually of two stopped ones, with the central nodal surface at m as their common base. This shows the relation of a stopped pipe to an open one to be that which experiment establishes.
§ 15. Velocity of Sound in Gases, Liquids, and Solids determined by Musical Vibrations
We have already learned that the relative velocities of sound in different solid bodies may be determined from the notes which they emit when thrown into longitudinal vibration. It was remarked at the time that to draw up a table of absolute velocities we only required the accurate comparison of the velocity in any one of those solids with the velocity in air. We are now in a condition to supply this comparison. For we have learned that the vibrations of the air in an organ-pipe open at both ends are executed precisely as those of a rod free at both ends. Any difference of rapidity, therefore, between the vibrations of a rod and of an open organ-pipe of the same length must be due solely to the different velocities with which the sonorous pulses are propagated through them. Take therefore an organ-pipe of a certain length, emitting a note of a certain pitch, and find the length of a rod of pine which yields the same note. This length would be ten times that of the organ-pipe, which would prove the velocity of sound in pine to be ten times its velocity in air. But the absolute velocity in air is 1,090 feet a second; hence the absolute velocity in pine is 10,900 feet a second, which is that given in our first chapter (p. 74). To the celebrated Chladni we are indebted for this beautiful mode of determining the velocity of sound in solid bodies.
We had also in our first lecture a table of the velocities of sound in other gases than air. I am persuaded that you could tell me, after due reflection, how this table was constructed. It would only be necessary to find a series of organ-pipes which, when filled with the different gases, yield the same note; the lengths of these pipes would give the relative velocities of sound through the gases. Thus we should find the length of a pipe filled with hydrogen to be four times that of a pipe filled with oxygen, yielding the same note, and this would prove the velocity of sound in the former to be four times its velocity in the latter.
But we had also a table of velocities through various liquids. How was it constructed? By forcing the liquids through properly constructed organ-pipes, and comparing their musical tones. Thus, in water it requires a pipe a little better than four feet long to produce the note of an air-pipe one foot long; and this proves the velocity of sound in water to be somewhat more than four times its velocity in air. My object here is to give you a clear notion of the way in which scientific knowledge enables us to cope with these apparently insurmountable problems. It is not necessary to go into the niceties of these measurements. You will, however, readily comprehend that all the experiments with gases might be made with the same organ-pipe, the velocity of sound in each respective gas being immediately deduced from the pitch of its note. With a pipe of constant length the pitch, or, in other words, the number of vibrations, would be directly proportional to the velocity. Thus, comparing oxygen with hydrogen, we should find the note of the latter to be the double octave of that of the former, whence we should infer the velocity of sound in hydrogen to be four times its velocity in oxygen. The same remark applies to experiments with liquids. Here also the same pipe may be employed throughout, the velocities being inferred from the notes produced by the respective liquids.
In fact, the length of an open pipe being, as already explained, one-half the length of its sonorous wave, it is only necessary to determine, by means of the siren, the number of vibrations executed by the pipe in a second, and to multiply this number by twice the length of the pipe, in order to obtain the velocity of sound in the gas or liquid within the pipe. Thus, an open pipe 26 inches long and filled with air executes 256 vibrations in a second. The length of its sonorous wave is twice 26 inches, or 4-1/3 feet: multiplying 256 by 4-1/3 we obtain 1,120 feet per second as the velocity of sound through air of this temperature. Were the tube filled with carbonic-acid gas, its vibrations would be slower: were it filled with hydrogen, its vibrations would be quicker; and applying the same principle, we should find the velocity of sound in both these gases.
So likewise the length of a solid rod free at both ends, and sounding its fundamental note, is half that of the sonorous wave in the substance of the solid. Hence we have only to determine the rate of vibration of such a rod, and multiply it by twice the length of the rod, to obtain the velocity of sound in the substance of the rod. You can hardly fail to be impressed by the power which physical science has given us over these problems; nor will you refuse your admiration to that famous old investigator, Chladni, who taught us how to master them experimentally.
REEDS AND REED-PIPES
The construction of the siren and our experiments with that instrument are, no doubt, fresh in your recollection. Its musical sounds are produced by the cutting up into puffs of a series of air-currents. The same purpose is effected by a vibrating reed, as employed in the accordion, the concertina, and the harmonica. In these instruments it is not the vibrations of the reed itself which, imparted to the air, and transmitted through it to our organs of hearing, produce the music; the function of the reed is constructive, not generative; it molds into a series of discontinuous puffs that which without it would be a continuous current of air.
Reeds, if associated with organ-pipes, sometimes command, and are sometimes commanded by, the vibrations of the column of air. When they are stiff they rule the column; when they are flexible the column rules them. In the former case, to derive any advantage from the air-column, its length ought to be so regulated that either its fundamental tone or one of its overtones shall correspond to the rate of vibration of the reed. The metal reed commonly employed in organ-pipes is shown in Fig. 104, A and b, both in perspective and in section. It consists of a long and flexible strip of metal, V V, placed in a rectangular orifice, through which the current of air enters the pipe. The manner in which the reed and pipe are associated is shown in Fig. 105. The front, b c, of the space containing the flexible tongue is of glass, so that you may see the tongue within it. A conical pipe, A B, surmounts the reed.47 The wire w i, shown pressing Fig. 105. against the reed, is employed to lengthen or shorten it, and thus to vary within certain limits its rate of vibration. At one time in the practice of music the reed closed the aperture by simply falling against its sides; every stroke of the reed produced a tap, and these linked themselves together to an unpleasant, screaming sound, which materially injured that of the associated organ-pipe. This was mitigated, but not removed, by permitting the reed to strike against soft leather; but the reed now employed is the free reed, which vibrates to and fro between the sides of the aperture, almost, but not quite, filling it. In this way the unpleasantness referred to is avoided. When reed and pipe synchronize perfectly, the sound is most pure and forcible; a certain latitude, however, is possible on both sides of perfect synchronism. But if the discordance be pushed too far, the pipe ceases to be of any use. We then obtain the sound due to the vibrations of the reed alone.
Flexible wooden reeds, which can accommodate themselves to the requirements of the pipes above them, are also employed in organ-pipes. Perhaps the simplest illustration of the action of the reed commanded by its aërial column is furnished by a common wheaten straw. At about an inch from a knot, at r, I bury my penknife in this straw, s r′, Fig. 106, to a depth of about one-fourth of the straw’s diameter, and, turning the blade flat, pass it upward toward the knot, thus raising a strip of the straw nearly an inch in length. This strip, r r′, is to be our reed, and the straw itself is to be our pipe. It is now eight inches long. When blown into, it emits this decidedly musical sound. When cut so as to make its length six inches, the pitch is higher; with a length of four inches, the pitch is higher still; and with a length of two inches, the sound is very shrill indeed. In these experiments the reed was compelled to accommodate itself throughout to the requirements of the vibrating column of air.
The clarinet is a reed-pipe. It has a single broad tongue, with which a long, cylindrical tube is associated. The reed-end of the instrument is grasped by the lips, and by their pressure the slit between the reed and its frame is narrowed to the required extent. The overtones of a clarinet are different from those of a flute. A flute is an open pipe, a clarinet a stopped one, the end at which the reed is placed answering to the closed end of the pipe. The tones of a flute follow the order of the natural numbers 1, 2, 3, 4, etc., or of the even numbers 2, 4, 6, 8, etc.; while the tones of a clarinet follow the order of the odd numbers 1, 3, 5, 7, etc. The intermediate notes are supplied by opening the lateral orifices of the instrument. Sir C. Wheatstone was the first to make known this difference between the flute and clarinet, and his results agree with the more thorough investigations of Helmholtz. In the hautboy and bassoon we have two reeds inclined to each other at a sharp angle, with a slit between them, through which the air is urged. The pipe of the hautboy is conical, and its overtones are those of an open pipe—different, therefore, from those of a clarinet. The pulpy end of a straw of green corn may be split by squeezing it, so as to form a double reed of this kind, and such a straw yields a musical tone. In the horn, trumpet, and serpent, the performer’s lips play the part of the reed. These instruments are formed of long, conical tubes, and their overtones are those of an open organ-pipe. The music of the older instruments of this class was limited to their overtones, the particular tone elicited depending on the force of the blast and the tension of the lips. It is now usual to fill the gaps between the successive overtones by means of keys, which enable the performer to vary the length of the vibrating column of air.
§ 16. The Voice
The most perfect of reed instruments is the organ of voice. The vocal organ in man is placed at the top of the trachea or wind-pipe, the head of which is adjusted for the attachment of certain elastic bands which almost close the aperture. When the air is forced from the lungs through the slit which separates these vocal chords, they are thrown into vibration; by varying their tension, the rate of vibration is varied, and the sound changed in pitch. The vibrations of the vocal chords are practically unaffected by the resonance of the mouth, though we shall afterward learn that this resonance, by reinforcing one or the other of the tones of the vocal chords, influences in a striking manner the quality of the voice. The sweetness and smoothness of the voice depend on the perfect closure of the slit of the glottis at regular intervals during the vibration.
The vocal chords may be illuminated and viewed in a mirror, placed suitably at the back of the mouth. Varied experiments of this kind have been executed by Sig. Garcia.48 I once sought to project the larynx of M. Czermak upon a screen in this room, but with only partial success. The organ may, however, be viewed directly in the laryngoscope; its motions, in singing, speaking, and coughing, being strikingly visible. It is represented at rest in Fig. 107. The roughness of the voice in colds is due, according to Helmholtz, to mucous flocculi, which get into the slit of the glottis, and which are seen by means of the laryngoscope. The squeaking falsetto voice, with which some persons are afflicted, Helmholtz thinks, may be produced by the drawing aside of the mucous layer which ordinarily lies under and loads the vocal chords. Their edges thus become sharper and their weight less; while, their elasticity remaining the same, they are shaken into more rapid tremors. The promptness and accuracy with which the vocal chords can change their tension, their form, and the width of the slit between them, to which must be added the elective resonance of the cavity of the mouth, render the voice the most perfect of musical instruments.
The celebrated comparative anatomist, John Müller, imitated the action of the vocal chords by means of bands of India-rubber. He closed the open end of a glass tube by two strips of this substance, leaving a slit between them. On urging air through the slit, the bands were thrown into vibration, and a musical tone produced. Helmholtz recommends the form shown in Fig. 108, where the tube, instead of ending in a section at right angles to its axis, terminates in two oblique sections, over which the bands of India-rubber are drawn. The easiest mode of obtaining sounds from reeds of this character is to roll round the end of a glass tube a strip of thin India-rubber, leaving about an inch of the substance projecting beyond the end of the tube. Taking two opposite portions of the projecting rubber in the fingers, and stretching it, a slit is formed, the blowing through which produces a musical sound, which varies in pitch, as the sides of the slit vary in tension.
§ 17. Vowel Sounds
The formation of the vowel sounds of the human voice excited long ago philosophic inquiry. We can distinguish one vowel sound from another, while assigning to both the same pitch and intensity. What, then, is the quality which renders the distinction possible? In the year 1779 this was made a prize question by the Academy of St. Petersburg, and Kratzenstein gained the prize for the successful manner in which he imitated the vowel sounds by mechanical arrangements. At the same time Von Kempelen, of Vienna, made similar and more elaborate experiments. The question was subsequently taken up by Mr. Willis, who succeeded beyond all his predecessors in the experimental treatment of the subject. The true theory of vowel sounds was first stated by Sir C. Wheatstone, and quite recently they have been made the subject of exhaustive inquiry by Helmholtz. You will find little difficulty in comprehending their origin.
Mounted on the acoustic bellows, without any pipe associated with it, when air is urged through its orifice, a free reed speaks in this forcible manner. When upon the frame of the reed a pyramidal pipe is fixed, you notice a change in the sound; and by pushing my flat hand over the open end of the pipe, the similarity between the sound produced and that of the human voice is unmistakable. Holding the palm of the hand over the end of the pipe so as to close it altogether, and then raising the hand twice in quick succession, the word “mamma” is heard as plainly as if it were uttered by an infant. For this pyramidal tube I now substitute a shorter one, and with it make the same experiment. The “mamma” now heard is exactly such as would be uttered by a child with a stopped nose. Thus, by associating with a vibrating reed a suitable pipe, we can impart to the sound the qualities of the human voice.
In the organ of voice, the reed is formed by the vocal chords, and associated with this reed is the resonant cavity of the mouth, which can so alter its shape as to resound, at will, either to the fundamental tone of the vocal chords or to any of their overtones. With the aid of the mouth, therefore, we can mix together the fundamental tone and the overtones of the voice in different proportions. Different vowel sounds are due to different admixtures of this kind. Striking one of this series of tuning-forks, and placing it before my mouth, I adjust the size of that cavity until it resounds forcibly to the fork. Then, without altering in the least the shape or size of my mouth, I urge air through the glottis. The vowel sound “U” (oo in hoop) is produced, and no other. I strike another fork, and, placing it in front of the mouth, adjust the cavity to resonance. Then removing the fork and urging air through the glottis, the vowel sound “O,” and it only, is heard. I strike a third fork, adjust my mouth to it, and then urge air through the larynx; the vowel sound ah! and no other, is heard. In all these cases the vocal chords have been in the same constant condition. They have generated throughout the same fundamental tone and the same overtones, the changes of sound which you have heard being due solely to the fact that different tones in the different cases were reinforced by the resonance of the mouth. Donders first proved that the mouth resounded differently for the different vowels.
In the formation of the different vowel sounds the resonant cavity of the mouth undergoes, according to Helmholtz, the following changes:
For the production of the sound “U” (oo in hoop), the lips must be pushed forward, so as to make the cavity of the mouth as deep as possible, and the orifice of the mouth, by the contraction of the lips, as small as possible. This arrangement corresponds to the deepest resonance of which the mouth is capable. The fundamental tone itself of the vocal chords is here reinforced, while the higher tones retreat.
The vowel “O” requires a somewhat wider opening of the mouth. The overtones which lie in the neighborhood of the middle b of the soprano come out strongly in the case of this vowel.
When “Ah” is sounded, the mouth assumes the shape of a funnel, widening outward. It is thus tuned to a note an octave higher than in the case of the vowel “O.” Hence, in sounding “Ah,” those overtones are most strengthened which lie near the higher b of the soprano. As the mouth is in this case wide open, all the other overtones are also heard, though feebly.
In sounding “A” and “E,” the hinder part of the mouth is deepened, while the front of the tongue rises against the gums and forms a tube; this yields a higher resonance-tone, rising gradually from “A” to “E,” while the hinder hollow space yields a lower resonance-tone, which is deepest when “E” is sounded.
These examples sufficiently illustrate the subject of vowel sounds. We may blend in various ways the elementary tints of the solar spectrum, producing innumerable composite colors by their admixture. Out of violet and red we produce purple, and out of yellow and blue we produce white. Thus also may elementary sounds be blended so as to produce all possible varieties of clang-tint. After having resolved the human voice into its constituent tones, Helmholtz was able to imitate these tones by tuning-forks, and, by combining them appropriately together, to produce the sounds of all the vowels.
§ 18. Kundt’s Experiments: New Modes of determining Velocity of Sound
Unwilling to interrupt the continuity of our reasonings and experiments on the sound of organ-pipes, and their relations to the vibrations of solid rods, I have reserved for the conclusion of this discourse some reflections and experiments which, in strictness, belong to an earlier portion of the chapter. You have already heard the tones, and made yourselves acquainted with the various modes of division of a glass tube, free at both ends, when thrown into longitudinal vibration. When it sounds its fundamental tone, you know that the two halves of such a tube lengthen and shorten in quick alternation. If the tube were stopped at its ends, the closed extremities would throw the air within the tube into a state of vibration; and if the velocity of sound in air were equal to its velocity in glass, the air of the tube would vibrate in synchronism with the tube itself. But the velocity of sound in air is far less than its velocity in glass, and hence, if the column of air is to synchronize with the vibrations of the tube, it can only do so by dividing itself into vibrating segments of a suitable length. In an investigation of great interest, recently published in “Poggendorff’s Annalen,” M. Kundt, of Berlin, has taught us how these segments may be rendered visible. Into this six-foot tube is introduced the light powder of lycopodium, being shaken all over the interior surface. A small quantity of the powder clings to that surface. Stopping the ends of the tube, holding its centre by a fixed clamp, and sweeping a wet cloth briskly over one of its halves, instantly the powder, which a moment ago clung to its interior surface, falls to the bottom of the tube in the forms shown in Fig. 109, the arrangement of the lycopodium marking the manner in which the column of air has been divided. Every node here is encircled by a ring of dust, while from node to node the dust arranges itself in transverse streaks along the ventral segments.