Chapter III.). You then had frequent occasion to notice that, even
before perfect synchronism had been established between the string and the tuning-fork to which it was attached, the string began to respond to the fork. But you also noticed how rapidly the vibrating amplitude of the string increased, as it came close to perfect synchronism with the vibrating fork. On approaching unison the string would open out, say to an amplitude of an inch; and then a slight tightening or slackening, as the case might be, would bring it up to unison, and cause it to open out suddenly to an amplitude of six inches.
So also in reference to the experiment made a moment ago with the sonometer; you noticed that the unhorsing of the paper riders was preceded by a fluttering of the bits of paper; showing that the sympathetic response of the second string had begun, though feebly, prior to perfect synchronism. Instead of two strings, conceive three strings, all nearly of the same pitch, to be stretched upon the sonometer; and suppose the vibrating period of the middle string to lie midway between the periods of its two neighbors, being a little higher than the one and a little lower than the other. Each of the side strings, sounded singly, would cause the middle string to respond. Sounding the two side strings together they would produce beats; the corresponding intermittence would be propagated to the central string, which would beat in synchronism with the beats of its neighbors. In this way we make plain to our minds how a Corti’s fibre may, to some extent, take up the vibrations of a note, nearly, but not exactly, in unison with its own; and that when two notes close to the pitch of the fibre act upon it together, their beats are responded to by an intermittent motion on the part of the fibre. This power of sympathetic vibration would fall rapidly on both sides of the perfect unison, so that on increasing the interval between the two notes, a time would soon arrive when the same fibre would refuse to be acted on simultaneously by both. Here the condition of the organ, necessary for the perception of audible beats, would cease.
In the middle region of the pianoforte, with the interval of a semitone, the beats are sharp and distinct, falling indeed upon the ear as a grating dissonance. Extending the interval to a whole tone, the beats become more rapid, but less distinct. With the interval of a minor third between the two notes, the beats in the middle region of the scale cease to be sensible. But this smoothening of the sound is not wholly due to the augmented rapidity of the beats. It is due in part to the fact, for which the foregoing considerations have prepared us, that the two notes here sounded are too far removed from that of the intermediate Corti’s fibre to affect it powerfully. By ascending to the higher regions of the scale we can produce, with a narrower interval than the minor third, the same, or even a greater, number of beats, which are sharply distinguishable because of the closeness of their component notes. In the very highest regions of the scale, however, the beats, when they become very rapid, cease to appeal as roughness to the ear.
Hence both the rapidity of the beats, and the width of the interval, enter into the question of consonance. Helmholtz judges that in the middle and higher regions of the musical scale, when the beats reach 33 per second, the dissonance reaches its maximum. Both slower and quicker beats have a less grating or dissonant effect. When the beats are very slow, they may be of advantage to the music; and, when they reach 132 per second, their roughness is no longer discernible.
Thanks to Helmholtz, whose views I have here sought to express in the briefest possible language, we are now in a condition to grapple with the question of musical intervals, and to give the reason why some are consonant and some dissonant to the ear. Circumstanced as we are upon earth, all our feelings and emotions, from the lowest sensation to the highest æsthetic consciousness, have a mechanical cause: though it may be forever denied to us to take the step from cause to effect; or to understand why the agitations of nervous matter can awaken the delights which music imparts. Take, then, the case of a violin. The fundamental tone of every string of this instrument is demonstrably accompanied by a crowd of overtones; so that, when two violins are sounded, we have not only to take into account the consonance or dissonance of the fundamental tones, but also those of the higher tones of both. Supposing two strings sounded whose fundamental tones, and all of whose partial tones, coincide, we have then absolute unison; and this we actually have when the ratio of vibration is 1:1. So also when the ratio of vibration is accurately 1:2, each overtone of the fundamental finds itself in absolute coincidence with either the fundamental tone or some higher tone of the octave. There is no room for beats or dissonance. When we examine the interval of a fifth, with a ratio of 2:3, we find the coincidence of the partial tones of the two so perfect as almost, though not wholly, to exclude every trace of dissonance. Passing on to the other intervals, we find the coincidence of the partial tones less perfect, as the numbers expressing the ratio of the vibrations become more large. Thus, the dissonance of intervals whose rates of vibration can only be expressed by large numbers, is not to be ascribed to any mystic quality of the numbers themselves, but to the fact that the fundamental tones which require such numbers are inexorably accompanied by partial tones whose coalescence produces beats, these producing the grating effect known as dissonance.
§ 6. _Graphic Representation of Consonance and Dissonance_
Helmholtz has attempted to represent this result graphically, and from his work I copy, with some modification, the next two diagrams. He assumes, as already stated, the maximum dissonance to correspond to 33 beats per second; and he seeks to express different degrees of dissonance by lines of different lengths. The horizontal line _c′ c″_, Fig. 164, represents a range of the musical scale in which _c″_ is our middle C, with 528 vibrations, and _c′_ the lower octave of _c″_. The distance from any point of this line to the curve above it represents the dissonance corresponding to that point. The pitch here is supposed to ascend continuously, and not by jumps. Supposing, for example, two performers on the violin to start with the same note _c′_, and that, while one of them continues to sound that note, the other gradually and continuously shortens his string, thus gradually raising its pitch up to the octave _c″_. The effect upon the ear would be represented by the irregular curved line in Fig. 164. Soon after the unison, which is represented by contact at _c′_, is departed from, the curve suddenly rises, showing the dissonance here to be the sharpest of all. At _c′_, the curve approaches the straight line _c′ c″_, and this point corresponds to the major third. At _f′_ the approach, is still nearer, and this point corresponds to the fourth. At _g′_ the curve almost touches the straight line, indicating that at this point, which corresponds to the fifth, the dissonance almost vanishes. At _a′_ we have the major sixth; while at _c″_, where the one note is an octave above the other, the dissonance entirely vanishes. The _e s′_ and the _a s′_, of this diagram are the German names of a third and a flat sixth.
Maintaining the same fundamental note _c′_, and passing through the octave above _c″_, the various degrees of consonance and dissonance are those shown in Fig. 165. That is to say, beginning with the octave _c′-c″_, and gradually elevating the pitch of one of the strings till it reaches _c″′_, the octave of _c″_, the curved line represents the effect upon the ear. We see, from both these curves, that dissonance is the general rule, and that only at certain definite points does the dissonance vanish, or become so decidedly enfeebled as not to destroy the harmony. These points correspond to the places where the numbers expressing the ratio of the two rates of vibration are small whole numbers. It must be remembered that these curves are constructed on the supposition that the beats are the cause of the dissonance; and the agreement between calculation and experience sufficiently demonstrates the truth of the assumption.[77]
You have thus accompanied me to the verge of the Physical portion of the science of Acoustics, and through the æsthetic portion I have not the knowledge of music necessary to lead you. I will only add that, in comparing three or more sounds together, that is to say, in choosing them for _chords_, we are guided by the principles just mentioned. We choose sounds which are in harmony with the fundamental sound and with each other. In choosing a series of sounds for combination two by two, the simplicity alone of the ratios would lead us to fix on those expressed by the numbers 1, 5/4, 4/3, 3/2, 5/3, 2; these being the simplest ratios that we can have within an octave. But, when the notes represented by these ratios are sounded in succession, it is found that the intervals between 1 and 5/4, and between 5/3 and 2, are wider than the others, and require the interpolation of a note in each case. The notes chosen are such as form chords, not with the fundamental tone, but with the note _3/2_ regarded as a fundamental tone. The ratios of these two notes with the fundamental are 9/8 and 15/8. Interpolating these, we have the eight notes of the natural or diatonic scale, expressed by the following names and ratios:
Names C. D. E. F. G. A. B. C′. Intervals 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. Rates of vibration 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2.
Multiplying these ratios by 24, to avoid fractions, we obtain the following series of whole numbers, which express the relative rates of vibration of the notes of the diatonic scale:
24, 27, 30, 32, 36, 40, 45, 48.
The meaning of the terms third, fourth, fifth, etc., which we have so often applied to the musical intervals, is now apparent; the term has reference to the position of the note in the scale.
§ 7. _Composition of Vibrations_
In our second lecture I referred to, and in part illustrated, a method devised by M. Lissajous for studying musical vibrations. By means of a beam of light reflected from a mirror attached to a tuning-fork, the fork was made to write the story of its own motion. In our last lecture the same method was employed to illustrate optically the phenomenon of beats. I now propose to apply it to the study of the composition of the vibrations which constitute the principal intervals of the diatonic scale. We must, however, prepare ourselves for the thorough comprehension of this subject by a brief preliminary examination of the vibrations of a common pendulum.
Such a pendulum hangs before you. It consists of a wire carefully fastened to a plate of iron at the roof of the house, and bearing a copper ball weighing 10 lbs. I draw the pendulum aside and let it go; it oscillates to and fro almost in the same plane.
I say “almost,” because it is practically impossible to suspend a pendulum without some little departure from perfect symmetry around its point of attachment. In consequence of this, the weight deviates sooner or later from a straight line, and describes an oval more or less elongated. Some years ago this circumstance presented a serious difficulty to those who wished to repeat M. Foucault’s celebrated experiment, demonstrating the rotation of the earth.
Nevertheless, in the case now before us, the pendulum is so carefully suspended that its deviation from a straight line is not at first perceptible. Let us suppose the amplitude of its oscillation to be represented by the dotted line _a b_, Fig. 166. The point _d_, midway between _a_ and _b_, is the pendulum’s point of rest. When drawn aside from this point to _b_, and let go, it will return to _d_, and in virtue of its momentum will pass on to _a_. There it comes momentarily to rest, and returns through _d_ to _b_. And thus it will continue to oscillate until its motion is expended.
The pendulum having first reached the limit of its swing at _b_, let us suppose a push in a direction perpendicular to _a b_ imparted to it; that is to say, in the direction _b c_. Supposing the time required by the pendulum to swing from _b_ to _a_ to be one second,[78] then the time required to swing from _b_ to _d_ will be half a second. Suppose, further, the force applied at _b_ to be such as would carry the bob, if free to move in that direction alone, to _c_ in half a second, and that the distance _b c_ is equal to _b d_, the question then occurs, where will the bob really find itself at the end of half a second? It is perfectly manifest that both forces are satisfied by the pendulum reaching the point _e_, exactly opposite the centre _d_, in half a second. To reach this point, it can be shown that it must describe the circular arc _b e_, and it will pursue its way along the continuation of the same arc, to _a_, and then pass round to _b_. Thus, by the rectangular impulse the rectilinear oscillation is converted into a rotation, the pendulum describing a circle, as shown in Fig. 167.
If the force applied at _b_ be sufficient to urge the weight in half a second through a greater distance than _b c_, the pendulum will describe an ellipse, with the lines _a b_ for its smaller axis; if, on the contrary, the force applied at _b_ urge the pendulum in half a second through a distance less than _b c_, the weight will describe an ellipse, with the line _a b_ for its greater axis.
Let us now inquire what occurs when the rectangular impulse is applied at the moment the ball is passing through its position of rest at _d_.
Supposing the pendulum to be moving from _a_ to _b_, Fig. 168, and that at _d_ a shock is imparted to it sufficient of itself to carry it in half a second to _c_; it is here manifest that the resultant motion will be along the straight line _d g_ lying between _b d_ and _d c_. The pendulum will return along this line to _d_, and pass on to _h_. In this case, therefore, the pendulum will describe a straight line, _g h_, oblique to its original direction of oscillation.
Supposing the direction of motion at the moment the push is applied to be from _b_ to _a_, instead of from _a_ to _b_, it is manifest that the resultant here will also be a straight line oblique to the primitive direction of oscillation; but its obliquity will be that shown in Fig. 169.
When the impulse is imparted to the pendulum neither at the centre nor at the limit of its swing, but at some point between both, we obtain neither a circle nor a straight line, but something between both. We have, in fact, a more or less elongated ellipse with its axis oblique to _a b_, the original direction of vibration. If, for example, the impulse be imparted at _d′_, Fig. 170, while the pendulum is moving toward _b_, the position of the ellipse will be that shown in Fig. 170; but if the push at _d′_ be given when the motion is toward _a_, then the position of the ellipse will be that represented in Fig. 171.
By the method of M. Lissajous we can combine the rectangular vibrations of two tuning-forks, a subject which I now wish to illustrate before you. In front of an electric lamp, L, Fig. 172, is placed a large tuning-fork, T′, fixed in a stand horizontally, and provided with a mirror, on which a narrow beam of light, L T′, is permitted to fall. The beam is thrown back, by reflection. In the path of the reflected beam is placed a second upright tuning-fork, T, also furnished with a mirror. By the horizontal fork, when it vibrates, the beam is tilted laterally; by the vertical fork, vertically. At the present moment both forks are motionless, the beam of light being reflected from the mirror of the horizontal to that of the vertical fork, and from the latter to the screen, on which it prints a brilliant disk. I now agitate the upright fork, leaving the other motionless. The disk is drawn out into a fine luminous band, 3 feet long. On sounding the second fork, the straight band is instantly transformed into a white ring _o p_, Fig. 172, 36 inches in diameter. What have we done here? Exactly what we did in our first experiment with the pendulum. We have caused a beam of light to vibrate simultaneously in two directions, and have accidentally hit upon the phase when one fork has just reached the limit of its swing and come momentarily to rest, while the beam is receiving the maximum impulse from the other fork.
That the _circle_ was obtained is, as stated, a mere accident; but it was a fortunate accident, as it enables us to see the exact similarity between the motion of the beam and that of the pendulum. I stop both forks, and, agitating them afresh, obtain an ellipse with its axis oblique. After a few trials we obtain the straight line, indicating that both the forks then pass simultaneously through their positions of equilibrium. In this way, by combining the vibrations of the two forks, we reproduce all the figures obtained with the pendulum.
When the vibrations of the two forks are, in all respects, absolutely alike, whatever the figure may be which is first traced upon the screen, it remains unchanged in form, diminishing only in size as the motion is expended. But the slightest difference in the rates of vibration destroys this fixity of the image. I endeavored before the lecture to reader the unison between these two forks as perfect as possible, and hence you have observed very little alteration in the shape of the figure. But by moving a small weight along the prong of either fork, or by attaching to either of them a bit of wax, the unison is impaired. The figure then obtained by the combination of both passes slowly from a straight line into an oblique ellipse, thence into a circle; after which it narrows again to an ellipse with an opposed obliquity, it then passes again into a straight line, the direction of which is at right angles to the first direction. Finally, it passes, in the reverse order, through the same series of figures to the straight line with which we began. The interval between two successive identical figures is the time in which one of the forks succeeds in executing one complete vibration more than the other. Loading the fork still more heavily, we have more rapid changes; the straight line, ellipse, and circle being passed through in quick succession. At times the luminous curve exhibits a stereoscopic depth, which renders it difficult to believe that we are not looking at a solid ring of white-hot metal.
By causing the mirror of the fork, T, to rotate through a small arc, the steady circle first obtained is drawn out into a luminous scroll stretching right across the screen, Fig. 173. The same experiment made with the changing figure, obtained by throwing the forks out of unison, gives us a scroll of irregular amplitude, Fig. 174.[79]
We have next to combine the vibrations of two forks, one of which oscillates with twice the rapidity of the other; in other words, to determine the figure corresponding to the combination of a note and its octave. To prepare ourselves for the mechanics of the problem, we must resort once more to our pendulum; for it also can be caused to oscillate in one direction twice as rapidly as in another. By a complicated mechanical arrangement this might be done in a very perfect manner, but at present simplicity is preferable to completeness. The wire of our pendulum is therefore permitted to descend from its point of suspension, A, Fig. 175, midway between two horizontal glass rods, _a b_, _a′ b′_, supported firmly at their ends, and about an inch asunder. The rods cross the wire at a height of 7 feet above the bob of the pendulum. The whole length of the pendulum being 28 feet, the glass rods intercept one-fourth of this length. On drawing the pendulum aside in the direction of the rods, _a b_, _a′ b′_, and letting it go, it oscillates freely between them. I bring it to rest and draw it aside in a direction perpendicular to the last; a length of 7 feet only can now oscillate, and by the laws of oscillation a pendulum 7 feet long vibrates with twice the rapidity of a pendulum 28 feet long.
I wish to show you the figure described by the combination of these two rates of vibration. Attached to the copper ball, _p_, is a camel’s-hair pencil, intended to rub lightly upon a glass plate placed on black paper and over which is strewed white sand. Allowing the pendulum to oscillate as a whole, the sand is rubbed away along a straight line which represents the amplitude of the vibration. Let _a b_, Fig. 176, represent this line, which, as before, we will assume to be described in one second. When the pendulum is at the limit, _b_, of its swing, let a rectangular impulse be imparted to it sufficient to carry it to _c_ in one-fourth of a second. If this were the only impulse acting on the pendulum, the bob would reach _c_ and return to _b_ in half a second. But under the actual circumstances it is also urged toward _d_, which point, through the vibration of the whole pendulum, it ought also to reach in half a second. Both vibrations, therefore, require that the bob shall reach _d_ at the same moment; and to do this it will have to describe the curve _b c′ d_. Again, in the time required by the long pendulum to pass from _d_ to _a_, the short pendulum will pass _to and fro_ over the half of its excursion; both vibrations must therefore reach _a_ at the same moment, and to accomplish this the pendulum describes the lower curve between _d_ and _a_. It is manifest that these two curves will repeat themselves at the opposite sides of _a b_, the combination of both vibrations producing finally a figure of 8, which you now see fairly drawn upon the sand before you.
The same figure is obtained if the rectangular impulse be imparted when the pendulum is passing its position of rest, _d_.
I have here supposed the time occupied by the pendulum in describing the line _a b_ to be one second. Let us suppose three-fourths of the second exhausted, and the pendulum at _d′_, Fig. 177, in its excursion toward _b_; let the rectangular impulse then be imparted to it, sufficient to carry it to _c_ in one-fourth of a second. Now the long pendulum requires that it should move from _d′_ to _b_ in one-fourth of a second; both impulses are therefore satisfied by the pendulum taking up the position _c′_ at the end of a quarter of a second. To reach this position it must describe the curve _d′ c′_. It will manifestly return along the same curve, and at the end of another quarter of a second find itself again at _d′_. From _d′_ to _d_ the long pendulum requires a quarter of a second. But at the end of this time the short pendulum must be at the lower limit of its swing: both requirements are satisfied by the pendulum being at _e_. We thus obtain one arm, _c′ e_, of a curve, which repeats itself to the left of _e_; so that the entire curve, due to the combination of the two vibrations, is that represented in Fig. 165. This figure is a parabola, whereas the figure of 8 before obtained is a lemniscata.
We have here supposed that, at the moment when the rectangular impulse was applied, the motion of the pendulum was _toward_ _b_: if it were toward _a_ we should obtain the inverted parabola, as shown in Fig. 178.
Supposing, finally, the impulse to be applied, not when the pendulum is passing through its position of equilibrium, nor when it is passing a point corresponding to three-fourths or one-fourth of the time of its excursion, but at some other point in the line, _a b_, between its end and centre. Under these circumstances we should have neither the parabola nor the perfectly symmetrical figure of 8, but a distorted 8.
And now we are prepared to witness with profit the combined vibration of our two tuning-forks, one of which sounds the octave of the other. Permitting the vertical fork, T, Fig. 172, to remain undisturbed in front of the lamp, we can oppose to it a horizontal fork, which vibrates with twice the rapidity. The first passage of the bow across the two forks reveals the exact similarity of this combination, and that of our pendulum. A very perfect figure of 8 is described upon the screen. Before the lecture the vibrations of these two forks were fixed as nearly as possible to the ratio of 1:2, and the steadiness of the figure indicates the perfection of the tuning. Stopping both forks, and again agitating them, we have the distorted 8 upon the screen. A few trials enable me to bring out the parabola. In all these cases the figure remains fixed upon the screen. But if a morsel of wax be attached to one of the forks, the figure is steady no longer, but passes from the perfect 8 into the distorted one, thence into the parabola, from which it afterward opens out to an 8 once more. By augmenting the discord, we can render those changes as rapid as we please.
When the 8 is steady on the screen, a rotation of the mirror of the fork, T, produces the scroll shown in Fig. 179.
Our next combination will be that of two forks vibrating in the ratio of 2:3. Observe the admirable steadiness of the figure produced by the compounding of these two rates of vibration. On attaching a four-penny-piece with wax to one of the forks the steadiness ceases, and we have an apparent rocking to and fro of the luminous figure. Passing on to intervals of 3:4, 4:5, and 5:6, the figures become more intricate as we proceed. The last combination, 5:6, is so entangled that to see the figure plainly a very narrow band of light must be employed. The distance existing between the forks and the screen also helps us to unravel the complication.
And here it is worth noting that, when the figure is fully developed, the loops along the vertical and horizontal edges express the ratio of the combined vibrations. In the octave, for example, we have two loops in one direction, and one in another; in the fifth, two loops in one direction, and three in another. When the combination is as 1:3, the luminous loops are also as 1:3. The changes which some of these figures undergo, when the tuning is not perfect, are extremely remarkable. In the case of 1:3, for example, it is difficult at times not to believe that you are looking at a solid link of white-hot metal. The figure exhibits a depth, apparently incompatible with its being traced upon a plane surface.
Fig. 180 (page 445) is a diagram of these beautiful figures, including combinations from 1:1 to 5:6. In each case, the characteristic phases of the vibration are shown; and through all of these each figure passes when the interval between the two forks is not pure. I also add here, Fig. 181, two phases of the combination 8:9.
To these illustrations of rectangular vibrations I add two others, Figs. 182 and 183, from a very beautiful series obtained by Mr. Herbert Airy with a compound pendulum. The experiments are described in “Nature” for August 17 and September 7, 1871. As their loops indicate, the figures are those of an octave and a twelfth.
But the most instructive apparatus for the compounding of rectangular vibrations is that of Mr. Tisley. Figs. 184 and 185 are copies of figures obtained by him through the joint action of two distinct pendulums; the rates of vibration corresponding to these particular figures being 2:3 and 3:4 respectively. The pen which traces the figures is moved simultaneously by two rods attached to the pendulums above their places of suspension. These two rods lie in the two planes of vibration, being at right angles to the pendulums, and to each other. At their place of intersection is the pen. By means of a ball and socket, of a special kind, the rods are enabled to move with a minimum of friction in all directions, while the rates of vibration are altered, in a moment, by the shifting of movable weights. The figures are drawn either with ink on paper, or, when projection on a screen is desired, by a sharp point on smoked glass. When the pendulums, having gone through the entire figure, return to their starting-point, they have lost a little in amplitude. The second excursion will, therefore, be smaller than the first, and the third smaller than the second. Hence the series of fine lines, inclosing gradually-diminishing areas, shown in these exquisite figures.[80] Mr. Tisley’s apparatus reflects the highest credit upon its able constructor.
Sir Charles Wheatstone devised, many years ago, a small and very efficient apparatus for the compounding of rectangular vibrations. A drawing, Fig. 186, and a description of this beautiful little instrument, for both of which I am indebted to its eminent inventor, may find a place here: _a_ is a steel rod polished at its upper end so as to reflect a point of light; this rod moves in a ball-and-socket joint at _b_, so that it may assume any position. Its lower end is connected with two arms _c_ and _d_, placed at right angles to each other, the other ends of which are respectively attached to the circumferences of the two circular disks _e_ and _f_. The axis of the disk _e_ carries at its opposite end another large disk _g_, which gives motion to the small disk _h_, placed on the axis which carries the disk _f_; and, according as this small disk _h_ is placed nearer to or further from the centre of the disk _g_, it communicates a different relative motion to the disk _f_. The nut and screw _i_ enable the disk _h_ to be placed in any position between the centre, and circumference of the larger disk _g_; and by means of the fork _j_ the disk _f_ is caused to revolve, whatever may be the position of the disk _h_. By this arrangement, while the wheel _k_ is turned regularly, the rod _a_ is moved backward and forward by the disk _e_ in one direction, and by the disk _f_, with any relative oscillatory motion, in the rectangular direction. The end of the rod is thus made to describe and to exhibit optically all the beautiful acoustical figures produced by the composition of vibrations of different periods in directions rectangular to each other. A lever _l_, bearing against the nut _i_, indicates, on a scale _m_, the numerical ratio of the two vibrations.[81]
I close these remarks on the combination of rectangular vibrations with a brief reference to an apparatus constructed by Mr. A. E. Donkin, of Exeter College, Oxford, and described in the “Proceedings of the Royal Society,” vol. xxii., p. 196. In its construction great mechanical knowledge is associated with consummate skill. I saw the apparatus as a wooden model, before it quitted the hands of its inventor, and was charmed with its performance. It is now constructed by Messrs. Tisley and Spiller.
SUMMARY OF CHAPTER IX
By the division of a string Pythagoras determined the consonant intervals in music, proving that, the simpler the ratio of the two parts into which the string was divided, the more perfect is the harmony of the sounds emitted by the two parts of the string. Subsequent investigators showed that the strings act thus because of the relation of their lengths to their rates of vibration.
With the double siren this law of consonance is readily illustrated. Here the most perfect harmony is the unison, where the vibrations are in the ratio of 1:1. Next comes the octave, where the vibrations are in the ratio of 1:2. Afterward follow in succession the fifth, with a ratio of 2:3; the fourth, with a ratio of 3:4; the major third, with a ratio of 4:5; and the minor third, with a ratio of 5:6. The interval of a tone, represented by the ratio 8:9, is dissonant, while that of a semitone, with a ratio of 15:16, is a harsh and grating dissonance.
The musical interval is independent of the absolute number of the vibrations of the two notes, depending only on the _ratio_ of the two rates of vibration.
The Pythagoreans referred the pleasing effect of the consonant intervals to number and harmony, and connected them with “the music of the spheres.” Euler explained the consonant intervals by reference to the constitution of the mind, which, he affirmed, took pleasure in simple calculations. The mind was fond of order, but of such order as involved no weariness in its contemplation. This pleasure was afforded by the simpler ratios in the case of music.
The researches of Helmholtz prove the rapid succession of beats to be the real cause of dissonance in music.
By means of two singing-flames, the pitch of one of them being changeable by the telescopic lengthening of its tube, beats of any degree of slowness or rapidity may be produced. Commencing with beats slow enough to be counted, and gradually increasing their rapidity, we reach, without breach of continuity, downright dissonance.
But, to grasp this theory in all its completeness, we must refer to the constitution of the human ear. We have first the tympanic membrane, which is the anterior boundary of the drum of the ear. Across the drum stretches a series of little bones, called respectively the _hammer_, the _anvil_, and the _stirrup-bone_; the latter abutting against a second membrane, which forms part of the posterior boundary of the drum. Beyond this membrane is the labyrinth filled with water, and having its lining membrane covered with the filaments of the auditory nerve.
Every shock received by the tympanic membrane is transmitted through the series of bones to the opposite membrane; thence to the water of the labyrinth, and thence to the auditory nerve.
The transmission is not direct. The vibrations are in the first place taken up by certain bodies, which can swing sympathetically with them. These bodies are of three kinds: the otolites, which are little crystalline particles; the bristles of Max Schultze; and the fibres of Corti’s organ. This latter is to all intents and purposes a stringed instrument, of extraordinary complexity and perfection, placed within the ear.
As regards our present subject, the strings of Corti’s organ probably play an especially important part. That one string should respond, in some measure, to another, it is not necessary that the unison should be perfect; a certain degree of response occurs in the immediate neighborhood of unison.
Hence each of two strings, not far removed from each other in pitch, can cause a third string, of intermediate pitch, to respond sympathetically. And if the two strings be sounded together, the beats which they produce are propagated to the intermediate string.
So, as regards Corti’s organ, when single sounds of various pitches, or rather when vibrations of various rapidities, fall upon its strings, the vibrations are responded to by the particular string whose period coincides with theirs. And when two sounds, close to each other in pitch, produce beats, the intermediate Corti’s fibre is acted on by both, and responds to the beats.
In the middle and upper portions of the musical scale the beats are most grating and harsh when they succeed each other at the rate of 33 per second. When they occur at the rate of 132 per second, they cease to be sensible.
The perfect consonance of certain musical intervals is due to the absence of beats. The imperfect consonance of other intervals is due to their existence. And here the overtones play a part of the utmost importance. For, though the primaries may sound together without any perceptible roughness, the overtones may be so related to each other as to produce harsh and grating beats. A strict analysis of the subject proves that intervals which require large numbers to express them are invariably accompanied by overtones which produce beats; while in intervals expressed by small numbers the beats are practically absent.
The graphic representation of the consonances and dissonances of the musical scale, by Helmholtz, furnishes a striking proof of this explanation.
The optical illustration of the musical intervals has been effected in a very beautiful manner by Lissajous. Corresponding to each interval is a definite figure, produced by the combination of its vibrations.
The compounding of vibrations has, of late years, been beautifully illustrated by apparatus constructed by Sir C. Wheatstone, Mr. Herbert Airy, and Mr. A. E. Donkin; and by the beautiful pendulum apparatus of Mr. Tisley, of the firm of Tisley and Spiller.
The pressure which, on a former occasion, prevented me from adding a “summary” to this chapter, was also the cause of hastiness, and partial inaccuracy, in its sketch of the theory of Helmholtz. That the sketch needed emendation I have long known, but I did not think it worth while to anticipate the correction here made; as the chapter, imperfect as it was, had been published, without comment, in Germany, by Helmholtz himself.
APPENDICES
APPENDIX I
ON THE INFLUENCE OF MUSICAL SOUNDS ON THE FLAME OF A JET OF COAL-GAS. BY JOHN LE CONTE, M.D.[82]
A short time after reading Prof. John Tyndall’s excellent article “On the Sounds produced by the Combustion of Gases in Tubes,”[83] I happened to be one of a party of eight persons assembled after tea for the purpose of enjoying a private musical entertainment. Three instruments were employed in the performance of several of the grand trios of Beethoven, namely, the piano, violin, and violoncello. Two “_fish-tail_” gas-burners projected from the brick wall near the piano. Both of them burned with remarkable steadiness, the windows being closed and the air of the room being very calm. Nevertheless, it was evident that _one_ of them was under a pressure nearly sufficient to make it _flare_.
Soon after the music commenced, I observed that the flame of the last-mentioned burner exhibited pulsations in height which were _exactly synchronous_ with the audible beats. This phenomenon was very striking to every one in the room, and especially so when the strong notes of the violoncello came in. It was exceedingly interesting to observe how perfectly even the _trills_ of this instrument were reflected on the sheet of flame. _A deaf man might have seen the harmony_. As the evening advanced, and the diminished consumption of gas in the city _increased the pressure_, the phenomenon became more conspicuous. The _jumping_ of the flame gradually increased, became somewhat irregular, and finally it began to flare continuously, emitting the characteristic sound indicating the escape of a greater amount of gas than could be properly consumed. I then ascertained by experiment that the phenomenon _did not_ take place unless the discharge of gas was so regulated that the flame approximated to the condition of _flaring_. I likewise determined by experiment that the effects _were not_ produced by jarring or shaking the floor and walls of the room by means of repeated concussions. Hence it is obvious that the pulsations of the flame _were not_ owing to _indirect_ vibrations propagated through the medium of the walls of the room to the burning apparatus, but must have been produced by the _direct_ influence of the aërial sonorous pulses on the burning jet.
In the experiments of M. Schaffgotsch and Prof. J. Tyndall, it is evident that “the shaking of the singing-flame within the glass tube,” produced by the voice or the siren, was a phenomenon perfectly analogous to what took place under my observation _without the intervention of a tube_. In my case the discharge of gas was so regulated that there was a tendency in the flame to flare, or to emit a “_singing-sound_.” Under these circumstances, strong aërial pulsations occurring at _regular intervals_ were sufficient to develop synchronous fluctuations in the height of the flame. It is probable that the effects would be more striking when the tones of the musical instrument are _nearly_ in unison with the sounds which would be produced by the flame under the slight increase in the rapidity of discharge of gas required to manifest the phenomenon of flaring. This point might be submitted to an experimental test.
As in Prof. Tyndall’s experiments on the jet of gas burning within a tube, clapping of the hands, shouting, etc., were ineffectual in converting the “silent” into the “singing-flame,” so, in the case under consideration, _irregular_ sounds did not produce any perceptible influence. It seems to be necessary that the impulses should _accumulate_, in order to exercise an appreciable effect.
With regard to the mode in which the sounds are produced by the combustion of gases in tubes, it is universally admitted that the explanation given by Prof. Faraday in 1818 is essentially correct. It is well known that he referred these sounds to the successive explosions produced by the periodic combination of the atmospheric oxygen with the issuing jet of gas. While reading Prof. J. Plateau’s admirable researches (third series) on the “Theory of the Modifications experienced by Jets of Liquid issuing from Circular Orifices when exposed to the Influence of Vibratory Motions,”[84] the idea flashed across my mind that the phenomenon which had fallen under my observation was nothing more than a _particular case_ of the effects of sounds on _all kinds of fluid jets_. Subsequent reflection has only served to fortify this first impression.
The beautiful investigations of Felix Savart, on the influence of sounds on jets of water, afford results presenting so many points of analogy with their effects on the jet of burning gas, that it may be well to inquire whether both of them may be referred to a common cause. In order to place this in a striking light, I shall subjoin some of the results of Savart’s experiments. Vertically-descending jets of water receive the following modifications under the influence of vibrations:
1. The continuous portions become shortened; the vein resolves itself into separate drops nearer the orifice than when _not_ under the influence of vibrations.
2. Each of the masses, as they detach themselves from the extremity of the continuous part, becomes flattened alternately in a vertical and horizontal direction, presenting to the eye, under the influence of their translatory motion, regularly-disposed series of maxima and minima of thickness, or ventral segments and nodes.
3. The foregoing modifications become much more developed and regular when a note, in unison with that which would be produced by the shock of the discontinuous part of the jet against a stretched membrane, is sounded in its neighborhood. The continuous part becomes considerably shortened, and the ventral segments are enlarged.
4. When the note of the instrument is _almost_ in unison, the continuous part of the jet is alternately lengthened and shortened and the beats which coincide with these variations in length _can be recognized by the ear_.
5. Other tones act with less energy on the jet, and some produce no sensible effect.
When a jet is made to ascend _obliquely_, so that the discontinuous part appears scattered into a kind of _sheaf_ in the same vertical plane, M. Savart found:
_a._ That, under the influence of vibrations of a determinate period, this sheaf may form itself into _two_ distinct jets, each possessing regularly-disposed ventral segments and nodes; sometimes with a different node the sheaf becomes replaced by _three_ jets.
_b._ The note which produces the greatest shortening of the continuous part always reduces the whole to a _single_ jet, presenting a perfectly regular system of ventral segments and nodes.
In the last memoir of M. Savart—a posthumous one, presented to the Academy of Sciences of Paris, by M. Arago, in 1853[85]—several remarkable acoustic phenomena are noticed in relation to the musical tones produced by the efflux of liquids through short tubes. When certain precautions and conditions are observed (which are minutely detailed by this able experimentalist), the discharge of the liquid gives rise to a succession of musical tones of great intensity and of a peculiar quality, somewhat analogous to that of the human voice. That these notes were not produced by the descending drops of the liquid vein was proved by permitting it to discharge itself into a vessel of water, while the orifice was below the surface of the latter. In this case the jet of liquid must have been _continuous_, but nevertheless the notes were produced. These unexpected results have been entirely confirmed by the more recent experiments of Prof. Tyndall.[86]
According to the researches of M. Plateau, all the phenomena of the influence of vibrations on jets of liquid are referable to the conflict between the vibrations and the _forces of figure_ (“_forces figuratrices_”). If the physical fact is admitted—and it seems to be indisputable—that a liquid cylinder attains a _limit of stability_ when the proportion between its length and its diameter is in the ratio of twenty-two to seven, it is almost a _physical necessity_ that the jet should assume the constitution indicated by the observations of Savart. It likewise seems highly probable that a liquid jet, while in a transition stage to discontinuous drops, should be exceedingly sensitive to the influence of all kinds of vibrations. It must be confessed, however, that Plateau’s beautiful and coherent theory does not appear to embrace Savart’s last experiment, in which the musical tones were produced by a jet of water issuing under the surface of the same liquid. It is rather difficult to imagine what agency the “forces of figure” could have, under such circumstances, in the production of the phenomenon. This curious experiment tends to corroborate Savart’s original idea, that the vibrations which produce the sounds must take place in the glass reservoir itself, and that the cause must be inherent in the phenomenon of the flow.
To apply the principles of Plateau’s theory to gaseous jets, we are compelled to abandon the idea of the _non-existence of molecular cohesion in gases_. But is there not abundant evidence to show that cohesion _does exist_ among the particles of gaseous masses? Does not the deviation from rigorous accuracy, both in the law of Mariotte and Gay-Lussac—especially in the case of condensable gases, as shown by the admirable experiments of M. Regnault—clearly prove that the hypothesis of the non-existence of cohesion in aëriform bodies is fallacious? Do not the expanding rings which ascend when a bubble of phosphuretted hydrogen takes fire in the air indicate the existence of some cohesive force in the gaseous product of combustion (aqueous vapor), whose outlines are marked by the opaque phosphoric acid? In short, does not the very _form_ of the flame of a “fish-tail” burner demonstrate that cohesion _must exist_ among the particles of the issuing gas? It is well known that in this burner the single jet which issues is formed by the union of _two oblique jets_ immediately before the gas is emitted. The result is a perpendicular _sheet of flame_. How is such a result produced by the mutual action of two jets, unless the force of cohesion is brought into play? Is it not obvious that such a fanlike flame must be produced by the same causes as those varied and beautiful forms of aqueous sheets, developed by the mutual action of jets of water, so strikingly exhibited in the experiments of Savart and of Magnus?
If it be granted that gases possess molecular cohesion, it seems to be physically certain that jets of gas must be subject to the same laws as those of liquid. Vibratory movements excited in the neighborhood ought, therefore, to produce modifications in them analogous to those recorded by M. Savart in relation to jets of water. Flame or incandescent gas presents gaseous matter in a _visible_ form, admirably adapted for experimental investigation; and, _when produced by a jet_, should be amenable to the principles of Plateau’s theory. According to this view, the pulsations or _beats_ which I observed in the gas-flame when under the influence of musical sounds, are produced by the conflict between the aërial vibrations and the “forces of figure” (as Plateau calls them) giving origin to periodical fluctuations of intensity, depending on the sonorous pulses.
If this view is correct, will it not be necessary for us to modify our ideas in relation to the agency of tubes in developing musical sounds by means of burning jets of gas? Must we not look upon all burning jets—as in the case of water-jets—as _musically inclined_; and that the use of tubes merely places them in a condition favorable for developing the tones? It is well known that burning jets frequently emit a _singing-sound_ when they are perfectly _free_. Are these sounds produced by successive explosions analogous to those which take place in glass tubes? It is very certain that, under the influence of molecular forces, any cause which tends to elongate the flame, without affecting the velocity of discharge, must tend to render it discontinuous, and thus bring about that mixture of gas and air which is essential to the production of the explosions. The influence of tubes, as well as of aërial vibrations, in establishing this condition of things, is sufficiently obvious. Was not the “beaded line” with its succession of “luminous stars,” which Prof. Tyndall observed when a flame of olefiant gas, burning in a tube, was examined by means of a moving mirror, an indication that the flame became _discontinuous_, precisely as the continuous part of a jet of water becomes _shortened_, and resolved into isolated drops, under the influence of sonorous pulsations? But I forbear enlarging on this very interesting subject, inasmuch as the accomplished physicist last named has promised to examine it at a future period. In the hands of so sagacious a philosopher, we may anticipate a most searching investigation of the phenomena in all their relations. In the meantime I wish to call the attention of men of science to the view presented in this article, in so far as it groups together several classes of phenomena under one head, and may be considered a partial generalization.—From Silliman’s “American Journal” for January, 1858.
APPENDIX II
ON ACOUSTIC REVERSIBILITY[87]
On the 21st and 22d of June, 1822, a commission, appointed by the Bureau des Longitudes of France, executed a celebrated series of experiments on the velocity of sound. Two stations had been chosen, the one at Villejuif, the other at Montlhéry, both lying south of Paris, and 11·6 miles distant from each other. Prony, Mathieu, and Arago were the observers at Villejuif, while Humboldt, Bouvard, and Gay-Lussac were at Montlhéry. Guns, charged sometimes with two pounds and sometimes with three pounds of powder, were fired at both stations, and the velocity was deduced from the interval between the appearance of the flash and the arrival of the sound.
On this memorable occasion an observation was made which, as far as I know, has remained a scientific enigma to the present hour. It was noticed that while every report of the cannon fired at Montlhéry was heard with the greatest distinctness at Villejuif, by far the greater number of the reports from Villejuif failed to reach Montlhéry. Had wind existed, and had it blown from Montlhéry to Villejuif, it would have been recognized as the cause of the observed difference; but the air at the time was calm, the slight motion of translation actually existing being from Villejuif toward Montlhéry, or against the direction in which the sound was best heard.
So marked was the difference in transmissive power between the two directions, that on June 22d, while every shot fired at Montlhéry was heard _à merveille_ at Villejuif, but one shot out of twelve fired at Villejuif was heard, and that feebly, at the other station.
With the caution which characterized him on other occasions, and which has been referred to admiringly by Faraday,[88] Arago made no attempt to explain this anomaly. His words are: “Quant aux différences si remarquables d’intensité que le bruit du canon a toujours présentées suivant qu’il se propageait du nord au sud entre Villejuif et Montlhéry, ou du sud au nord entre cette seconde station et la première, nous ne chercherons pas aujourd’hui à l’expliquer, parce que nous ne pourrions offrir au lecteur que des conjectures denuées de preuves.”[89]
I have tried, after much perplexity of thought, to bring this subject within the range of experiment, and have now to submit the following solution of the enigma: The first step was to ascertain whether the sensitive flame, referred to in my recent paper in the “Philosophical Transactions,” could be safely employed in experiments on the mutual reversibility of a source of sound and an object on which the sound impinges. Now, the sensitive flame usually employed by me measures from eighteen to twenty-four inches in height, while the reed employed as a source of sound is less than a square quarter of an inch in area. If, therefore, the whole flame, or the pipe which fed it, were sensitive to sonorous vibrations, strict experiments on reversibility with the reed and flame might be difficult, if not impossible. Hence my desire to learn whether the seat of sensitiveness was so localized in the flame as to render the contemplated interchange of flame and reed permissible.
The flame being placed behind a cardboard screen, the shank of a funnel passed through a hole in the cardboard was directed upon the middle of the flame. The sound-waves issuing from the vibrating reed, placed within the funnel, produced no sensible effect upon the flame. Shifting the funnel so as to direct its shank upon the root of the flame, the action was violent.
To augment the precision of the experiment, the funnel was connected with a glass tube three feet long and half an inch in diameter, the object being to weaken, by distance, the effect of the waves diffracted round the edge of the funnel, and to permit those only which passed through the glass tube to act upon the flame.
Presenting the end of the tube to the orifice of the burner (_b_, Fig. 1), or the orifice to the end of the tube, the flame was violently agitated by the sounding-reed, R. On shifting the tube, or the burner, so as to concentrate the sound on a portion of the flame about half an inch above the orifice, the action was _nil_. Concentrating the sound upon the burner itself, about half an inch below its orifice, there was no action.
These experiments demonstrate the localization of “the seat of sensitiveness,” and they prove the flame to be an appropriate instrument for the contemplated experiments on reversibility.
The experiments then proceeded thus: The sensitive flame being placed close behind a screen of cardboard 18 inches high by 12 inches wide, a vibrating reed, standing at the same height as the root of the flame, was placed at a distance of 6 feet on the other side of the screen. The sound of the reed, in this position, produced a strong agitation of the flame.
The whole upper half of the flame was here visible from the reed; hence the necessity of the foregoing experiments to prove the action of the sound on the upper portion of the flame to be _nil_, and that the waves had really to bend round the edge of the screen, so as to reach the seat of sensitiveness in the neighborhood of the burner.
The positions of the flame and reed were reversed, the latter being now close behind the screen, and the former at a distance of 6 feet from it. The sonorous vibrations were without sensible action upon the flame.
The experiment was repeated and varied in many ways. Screens of various sizes were employed; and, instead of reversing the positions of the flame and reed, the screen itself was moved, so as to bring in some experiments the flame, and in other experiments the reed, close behind it. Care was also taken that no reflected sound from the walls or ceiling of the laboratory, or from the body of the experimenter, should have anything to do with the effect. In all cases it was shown that the sound was effective when the reed was at a distance from the screen, and the flame close behind it; while the action was insensible when these positions were reversed.
Thus, let _s e_, Fig. 2, be a vertical section of the screen. When the reed was at A and the flame at B there was no action; when the reed was at B and the flame at A the action was decided. It may be added that the vibrations communicated to the screen itself, and from it to the air beyond it, were without effect; for when the reed, which at B was effectual, was shifted to C, where its action on the screen was greatly augmented, it ceased to have any action on the flame at A.
We are now, I think, prepared to consider the failure of reversibility in the larger experiments of 1822. Happily an incidental observation of great significance comes here to our aid. It was observed and recorded at the time that, while the reports of the guns at Villejuif were without echoes, a roll of echoes lasting from 20 to 25 seconds accompanied every shot at Montlhéry, being heard by the observers there. Arago, the writer of the report, referred these echoes to reflection from the clouds, an explanation which I think we are now entitled to regard as problematical. The report says that “tous les coups tirés à Montlhéry y étaient _accompagnés_ d’un roulement semblable à celui du tonnerre.” I have italicized a very significant word—a word which fairly applies to our experiments on gun-sounds at the South Foreland, where there was no sensible interval between explosion and echo, but which could hardly apply to echoes coming from the clouds. For supposing the clouds to be only a mile distant, the sound and its echo would have been separated by an interval of nearly ten seconds. But there is no mention of any interval; and, had such existed, surely the word “followed,” instead of “accompanied,” would have been the one employed. The echoes, moreover, appear to have been _continuous_, while the clouds observed seem to have been _separate_. “Ces phénomènes,” says Arago, “n’ont jamais eu lieu qu’au moment de l’apparition de quelques nuages.” But from separate clouds a continuous roll of echoes could hardly come. When to this is added the experimental fact that clouds far denser than any ever formed in the atmosphere are demonstrably incapable of sensibly reflecting sound, while cloudless air, which Arago pronounced echoless, has been proved capable of powerfully reflecting it, I think we have strong reason to question the hypothesis of the illustrious French philosopher.[90]
And, considering the hundreds of shots fired at the South Foreland, with the attention especially directed to the aërial echoes, when no single case occurred in which echoes of measurable duration did not accompany the report of the gun, I think Arago’s statement, that at Villejuif no echoes were heard when the sky was clear, must simply mean that they vanished with great rapidity. Unless the attention was specially directed to the point, a slight prolongation of the cannon-sound might well escape observation; and it would be all the more likely to do so if the echoes were so loud and prompt as to form apparently part and parcel of the direct sound.
I should be very loth to transgress here the limits of fair criticism, or to throw doubt, without good reason, on the recorded observations of illustrious men. Still, taking into account what has been just stated, and remembering that the minds of Arago and his colleagues were occupied by a totally different problem (that the echoes were an incident rather than an object of observation), I think we may justly consider the sound which he called “instantaneous” as one whose aërial echoes did not differentiate themselves from the direct sound by any noticeable fall of intensity, and which rapidly died into silence.
Turning now to the observations at Montlhéry, we are struck by the extraordinary duration of the echoes heard at that station. At the South Foreland the charge habitually fired was equal to the largest of those employed by the French philosophers; but on no occasion did the gun-sounds produce echoes approaching to 20 or 25 seconds’ duration. The time rarely reached half this amount. Even the siren-echoes, which were more remarkable and more long-continued than those of the gun, never reached the duration of the Montlhéry echoes. The nearest approach to it was on October 17, 1873, when the siren-echoes required 15 seconds to subside into silence.
On this same day, moreover (and this is a point of marked significance), the transmitted sound reached its maximum range, the gun-sounds being heard at the Quenocs buoy, 16-1/2 nautical miles from the South Foreland. I have stated in another place that the duration of the air-echoes indicates “the atmospheric depths” from which they came. An optical analogy may help us here. Let light fall upon chalk, the light is wholly scattered by the superficial particles; let the chalk be powdered and mixed with water, light reaches the observer from a far greater depth of the turbid liquid. The solid chalk typifies the action of exceedingly dense acoustic clouds; the chalk and water that of clouds of more moderate density. In the one case we have echoes of short, in the other echoes of long, duration. These considerations prepare us for the inference that Montlhéry, on the occasion referred to, must have been surrounded by a highly-diacoustic atmosphere; while the shortness of the echoes at Villejuif shows that the atmosphere surrounding that station must have been, in a high degree, acoustically opaque.
Have we any clew to the cause of the opacity? I think we have. Villejuif is close to Paris, and over it, with the observed light wind, was slowly wafted the air from the city. Thousands of chimneys to windward of Villejuif were discharging their heated currents; so that an exceeding non-homogeneous atmosphere must have surrounded that station.[91] At no great height in the atmosphere the equilibrium of temperature would be established. This non-homogeneous air surrounding Villejuif is experimentally typified by our screen, with the source of sound close behind it, the upper edge of the screen representing the place where equilibrium of temperature was established in the atmosphere above the station. In virtue of its proximity to the screen, the echoes from our sounding-reed would, in the case here supposed, so blend with the direct sound as to be practically indistinguishable from it, as the echoes at Villejuif followed the direct sound so hotly, and vanished so rapidly, that they escaped observation. And as our sensitive flame, at a distance, failed to be affected by the sounding body placed close behind the cardboard screen, so, I take it, did the observers at Montlhéry fail to hear the sounds of the Villejuif gun.
Something further may be done toward the experimental elucidation of this subject. The facility with which sounds pass through textile fabrics has been already illustrated,[92] a layer of cambric or calico, or even of thick flannel or baize, being found competent to intercept but a small fraction of the sound from a vibrating reed. Such a layer of calico may be taken to represent a layer of air, differentiated from its neighbors by temperature or moisture; while a succession of such sheets of calico may be taken to represent successive layers of non-homogeneous air.
Two tin tubes (M N and O P, Fig. 3) with open ends were placed so as to form an acute angle with each other. At the end of one was the vibrating reed _r_; opposite the end of the other, and in the prolongation of P O, the sensitive flame _f_, a second sensitive flame (_f′_) being placed in the continuation of the axis of M N. On sounding the reed, the direct sound through M N agitated the flame _f′_. Introducing the square of calico _a b_ at the proper angle, a slight decrease of the action on _f′_ was noticed, and the feeble echo from _a b_ produced a barely perceptible agitation of the flame f. Adding another square, _c d_, the sound transmitted by _a b_ impinged on _c d_; it was partially echoed, returned through _a b_, passed along P O, and still further agitated the flame _f_. Adding a third square, _e f_, the reflected sound was still further augmented, every accession to the echo being accompanied by a corresponding withdrawal of the vibrations from _f′_, and a consequent stilling of that flame.
With thinner calico or cambric it would require a greater number of layers to intercept the entire sound; hence with such cambric we should have echoes returned from a greater distance, and therefore of greater duration. Eight layers of the calico employed in these experiments, stretched on a wire frame and placed close together as a kind of pad, may be taken to represent a dense acoustic cloud. Such a pad, placed at the proper angle beyond N, cuts off the sound, which in its absence reaches _f′_, to such an extent that the flame _f′_, when not too sensitive, is thereby stilled, while _f_ is far more powerfully agitated than by the reflection from a single layer. With the source of sound close at hand, the echoes from such a pad would be of insensible duration. Thus close at hand do I suppose the acoustic clouds surrounding Villejuif to have been, a similar shortness of echo being the consequence.
A further step is here taken in the illustration of the analogy between light and sound. Our pad acts chiefly by internal reflection. The sound from the reed is a composite one, made up of partial sounds differing in pitch. If these sounds be ejected from the pad in their pristine proportions, the pad is acoustically _white_; if they return with their proportions altered, the pad is acoustically _colored_.
In these experiments my assistant, Mr. Cottrell, has rendered me material assistance.[93]
NOTE, _June 3d_.—I annex here a sketch of an apparatus[94] devised by my assistant, Mr. Cottrell, and constructed by Tisley and Spiller, for the demonstration of the law of reflection of sound. It consists of two tubes (A F, B R) with a source of sound at the end R of one of them, and a sensitive flame at the end F of the other. The axes of the tube converge upon the mirror, M, and they are capable of being placed so as to inclose any required angle. The angles of incidence and reflection are read off on the graduated semicircle. The mirror M is also movable round a vertical axis.
FOOTNOTES:
[1] It will be borne in mind that the Washington Appendix was published nearly a year after my Report to the Trinity House.
[2] That is to say, homogeneous air with an opposing wind is frequently more favorable to sound than non-homogeneous air with a favoring wind. We had the same experience at the South Foreland.—J. T.
[3] Had this observation been published, it could only have given me pleasure to refer to it in my recent writings. It is a striking confirmation of my observations on the Mer de Glace in 1859.
[4] Had I been aware of its existence I might have used the language of General Duane to express my views on the point here adverted to. See Chap. VII., pp. 340-341.
[5] This does not seem more surprising than the passage of light, or radiant heat, through rock salt.
[6] Also “Proceedings of the Royal Society,” vol. xxiii., p. 159, and “Proceedings of the Royal Institution,” vol. vii., p. 344.
[7] See page 372 of this volume.
[8] The rapidity with which an impression is transmitted through the nerves, as first determined by Helmholtz, and confirmed by Du Bois-Reymond, is 93 feet a second.
[9] And long previously by Robert Boyle.
[10] A very effective instrument, presented to the Royal Institution by Mr. Warren De La Rue.
[11] By directing the beam of an electric lamp on glass bulbs filled with a mixture of equal volumes of chlorine and hydrogen, I have caused the bulbs to explode in vacuo and in air. The difference, though not so striking as I at first expected, was perfectly distinct.
[12] It may be that the gas fails to throw the vocal chords into sufficiently strong vibration. The _laryngoscope_ might decide this question.
[13] Poisson, “Mécanique,” vol. ii., p. 707.
[14] To converge the pulse upon the flame, the tube was caused to end in a cone.
[15] It is recorded that a bell placed on an eminence in Heligoland failed, on account of its distance, to be heard in the town. A parabolic reflector placed behind the bell, so as to reflect the sound-waves in the direction of the long, sloping street, caused the strokes of the bell to be distinctly heard at all times. This observation needs verification.
[16] “Encyclopædia Metropolitana,” art. “Sound.”
[17] Placing himself close to the upper part of the wall of the London Colosseum, a circular building one hundred and thirty feet in diameter, Mr. Wheatstone found a word pronounced to be repeated a great many times. A single exclamation appeared like a peal of laughter, while the tearing of a piece of paper was like the patter of hail.
[18] “Poggendorff’s Annalen,” vol. lxxxv., p. 378; “Philosophical Magazine,” vol. v., p. 73.
[19] Thin India-rubber balloons also form excellent sound lenses.
[20] For the sake of simplicity, the wave is shown broken at _o′_ and its two halves straight. The surface of the wave, however, is really a curve, with its concavity turned in the direction of its propagation.
[21] See “Heat as a Mode of Motion,” chap. iii.
[22] In fact, the prompt abstraction of the motion of heat from the condensation, and its prompt communication to the rarefaction by the contiguous luminiferous ether, would prevent the former from ever rising so high, or the latter from ever falling so low, in temperature as it would do if the power of radiation was absent.
[23] “Heat a Mode of Motion,” chap. x.
[24] According to Burmeister, through the injection and ejection of air into and from the cavity of the chest.
[25] On July 27, 1681, “Mr. Hooke showed an experiment of making musical and other sounds by the help of teeth of brass wheels; which teeth were made of equal bigness for musical sounds, but of unequal for vocal sounds.”—Birch’s “History of the Royal Society,” p. 96, published in 1757.
The following extract is taken from the “Life of Hooke,” which precedes his “Posthumous Works,” published in 1705, by Richard Waller, Secretary of the Royal Society: “In July the same year he (Dr. Hooke) showed a way of making musical and other sounds by the striking of the teeth of several brass wheels, proportionally cut as to their numbers, and turned very fast round, in which it was observable that the equal or proportional strokes of the teeth, that is, 2 to 1, 4 to 3, etc., made the musical notes, but the unequal strokes of the teeth more answered the sound of the voice in speaking.”
[26] Galileo, finding the number of notches on his metal to be great when the pitch of the note was high, inferred that the pitch depended on the rapidity of the impulses.
[27] When a rough tide rolls in upon a pebble beach, as at Blackgang Chine or Freshwater Gate in the Isle of Wight the rounded stones are carried up the slope by the impetus of the water and when the wave retreats the pebbles are dragged down. Innumerable collisions thus ensue of irregular intensity and recurrence. The union of these shocks impresses us as a kind of scream. Hence the line in Tennyson’s “Maud”
“Now to the scream of a maddened beach dragged down by the wave.”
The height of the note depends in some measure upon the size of the pebble, varying from a kind of roar—heard when the stones are large—to a scream; from a scream to a noise resembling that of frying bacon; and from this, when the pebbles are so small as to approach the state of gravel, to a mere hiss. The roar of the breaking wave itself is mainly due to the explosion of bladders of air.
[28] The error of Savart consists, according to Helmholtz, in having adopted an arrangement in which overtones (described in Chapter III.) were mistaken for the fundamental one.
[29] “The deepest tone of orchestra instruments is the E of the double-bass, with 41-1/4 vibrations. The new pianos and organs go generally as far as C^{1}, with 33 vibrations; new grand pianos may reach A^{11}, with 27-1/2 vibrations. In large organs a lower octave is introduced, reaching to C^{11}, with 16-1/2 vibrations. But the musical character of all these tones under E is imperfect, because they are near the limit where the power of the ear to unite the vibrations to a tone ceases. In height the pianoforte reaches to a^{iv}, with 3,520 vibrations, or sometimes to c^{v}, with 4,224 vibrations. The highest note of the orchestra is probably the d^{v} of the piccolo flute, with 4,752 vibrations.”—Helmholtz, “Tonempfindungen,” p. 30. In this notation we start from C, with 66 vibrations, calling the first lower octave C^{1}, and the second C^{11}; and calling the first highest octave c, the second c^{1}, the third c^{11}, the fourth c^{12}, etc. In England the deepest tone, Mr. Macfarren informs me, is not E, but A, a fourth above it.
[30] It is hardly necessary to remark that the quickest vibrations and shortest waves correspond to the extreme violet, while the slowest vibrations and longest waves correspond to the extreme red, of the spectrum.
[31] Experiments on this subject were first made by M. Buys Ballot on the Dutch railway, and subsequently by Mr. Scott Russell in this country. Doppler’s idea is now applied to determine, from changes of wave-length, motions in the sun and fixed stars.
[32] An ordinary musical box may be substituted for the piano in this experiment.
[33] To show the influence of a large vibrating surface in communicating sonorous motion to the air, Mr. Kilburn incloses a musical box within cases of thick felt. Through the cases a wooden rod, which rests upon the box, issues. When the box plays a tune, it is unheard as long as the rod only emerges; but when a thin disk of wood is fixed on the rod, the music becomes immediately audible.
[34] Chladni remarks (“Akustik,” p. 55) that it is usual to ascribe to Sauveur the discovery, in 1701, of the nodes of vibration corresponding to the higher tones of strings; but that Noble and Pigott had made the discovery in Oxford in 1676, and that Sauveur declined the honor of the discovery when he found that others had made the observation before him.
[35] The first experiment really made in the lecture was with a bar of steel 62 inches long, 1-1/2 inch wide, and 1/2 an inch thick, bent into the shape of a tuning-fork, with its prongs 2 inches apart, and supported on a heavy stand. The cord attached to it was 9 feet long and a quarter of an inch thick. The prongs were thrown into vibration by striking them briskly with two pieces of lead covered with pads and held one in each hand. The prongs vibrated transversely to the cord. The vibrations produced by a single stroke were sufficient to carry the cord through several of its subdivisions and back to a single ventral segment. That is to say, by striking the prongs and causing the cord to vibrate as a whole, it could, by relaxing the tension, be caused to divide into two, three, or four vibrating segments; and then, by increasing the tension, to pass back through four, three, and two divisions, to one, _without renewing the agitation of the prongs_. The cord was of such a character that, instead of oscillating to and fro in the same plane, each of its points described a circle. The ventral segments, therefore, instead of being flat surfaces were surfaces of revolution, and were equally well seen from all parts of the room. The tuning-forks employed in the subsequent illustrations were prepared for me by that excellent acoustic mechanician, König, of Paris, being such as are usually employed in the projection of Lissajou’s experiments.
[36] A string steeped in a solution of the sulphate of quinine, and illuminated by the violet rays of the electric lamp, exhibits brilliant fluorescence. When the fork to which it is attached vibrates, the string divides itself into a series of spindles, and separated from each other by more intensely luminous nodes, emitting a light of the most delicate greenish-blue.
[37] The subject of musical intervals will be treated in a subsequent lecture.
[38] “This quality of sound, sometimes called its register, color, or timbre.”—Thomas Young, “Essay on Music.”
[39] “Lehre von den Tonempfindungen,” p. 135.
[40] The action of such a string is substantially the same as that of the siren. The string renders intermittent the current of air. Its action also resembles that of a _reed_. See Lecture V.
[41] Chladni also observed this compounding of vibrations, and executed a series of experiments, which, in their developed form, are those of the kaleidophone. The composition of vibrations will be studied at some length in a subsequent lecture.
[42] I copy this figure from Sir C. Wheatstone’s memoir; the nodes, however, ought to be nearer the ends, and the free terminal portions of the dotted lines ought not to be bent upward or downward. The nodal lines in the next two figures are also drawn too far from the edge of the plates.
[43] Under the shoulder of the Wetterhorn I found in 1867 a pool of clear water into which a driblet fell from a brow of overhanging limestone rock. The rebounding water-drops, when they fell back, rolled in myriads over the surface. Almost any fountain, the spray of which falls into a basin, will exhibit the same effect.
[44] This experiment succeeds almost equally well with a glass tube.
[45] This experiment is more easily executed with hydrogen than with coal-gas.
[46] Only an extremely small fraction of the fork’s motion is, however, converted into sound. The remainder is expended in overcoming the internal friction of its own particles. In other words, nearly the whole of the motion is converted into heat.
[47] The clear illustrations of organ-pipes and reeds introduced here, and at page 226, have been substantially copied from the excellent work of Helmholtz. Pipes opening with hinges, so as to show their inner parts, were shown in the lecture.
[48] I owe it to this eminent artist to direct attention to his experiments communicated to the Royal Society in May, 1855, and recorded in the “Philosophical Magazine” for 1855, vol. x., page 218.
[49] The velocity in glass varies with the quality; the result of each experiment has therefore reference only to the particular kind of glass employed in the experiment.
[50] This experiment was first made with a hydrogen-flame by Sir C. Wheatstone.
[51] A gas-jet, for example, can be ignited five inches above the tip of a visible gas-flame, where platinum-leaf shows no redness.
[52] “Philosophical Magazine,” March, 1858, p. 235. In the Appendix Prof. Le Conte’s interesting paper is given _in extenso_. Some years subsequently Mr. (now Professor) Barrett, while preparing some experiments for my lectures, observed the action of a musical sound upon a flame, and by the selection of suitable burners he afterward succeeded in rendering the flame extremely sensitive. Le Conte, of whose discovery I informed Mr. Barrett, was my own starting-point.
[53] A gas-bag properly weighted also answers for these experiments.
[54] In the actions described in the case of the blow-pipe and candle-flames, it was the jet of air issuing from the blow-pipe, and not the flame itself, that was directly acted on by the external vibrations.
[55] Numerous modifications of these experiments are possible. Other inflammable gases than coal-gas may be employed. Mixtures of gases have also been found to yield beautiful and striking results. An infinitesimal amount of mechanical impurity has been found to exert a powerful influence.
[56] Referring to these effects, Helmholtz says: “Die erstaunliche Empfindlichkeit eines mit Rauch imprägnirten cylindrischen Luftstrahls gegen Schall ist von Herrn Tyndall beschrieben worden; ich habe dieselbe bestätigt gefunden. Es ist dies offenbar eine Eigenschaft der Trennungsflächen die für das Anblasen der Pfeifen von grösster Wichtigkeit ist.”—“Discontinuirliche Luftbewegung,” Monatsbericht, April, 1868.
[57] When these two tuning-forks were placed _in contact_ with a vessel from which a liquid vein issued, the visible action on the vein continued long after the forks had ceased to be heard.
[58] The experiments on sounding flames have been recently considerably extended by my assistant, Mr. Cottrell. By causing flame to rub against flame, various musical sounds can be obtained—some resembling those of a trumpet, others those of a lark. By the friction of unignited gas-jets, similar though less intense effects are produced. When the two flames of a fish-tail burner are permitted to impinge upon a plate of platinum, as in Scholl’s “perfectors,” the sounds are trumpet-like, and very loud. Two ignited gas-jets may be caused to flatten out like Savart’s water-jets. Or they may be caused to roll themselves into two hollow horns, forming a most instructive example of the _Wirbelflächen_ of Helmholtz. The carbon-particles liberated in the flame rise through the horns in continuous red-hot or white-hot spirals, which are extinguished at a height of some inches from their place of generation.
[59] “Essay on Sound,” par. 21.
[60] “Report of the British Association for 1863,” page 105.
[61] A very sagacious remark, as observation proves.
[62] Powerful electric lights have since been established and found ineffectual.
[63] This is also Sir John Herschel’s way of regarding the subject. “Essay on Sound,” par. 38.
[64] In all cases nautical miles are meant.
[65] Sir John Herschel gives the following account of Arago’s observation: “The rolling of thunder has been attributed to echoes among the clouds; and, if it is considered that a cloud is a collection of particles of water, however minute, in a liquid state, and therefore each individually capable of reflecting sound, there is no reason why very large sounds should not be reverberated confusedly (like bright lights) from a cloud. And that such is the case has been ascertained by direct observation on the sound of cannon. Messrs. Arago, Matthieu, and Prony, in their experiments on the velocity of sound, observed that under a perfectly clear sky the explosions of their guns were always single and sharp; whereas, when the sky was overcast, and even when a cloud came in sight over any considerable part of the horizon, they were frequently accompanied by a long-continued roll like thunder.”—“Essay on Sound,” par. 38. The distant clouds would imply a long interval between sound and echo, but nothing of the kind is reported.
[66] A friend informs me that he has followed a pack of hounds on a clear calm day without hearing a single yelp from the dogs; while on calm foggy days from the same distance the musical uproar of the pack was loudly audible.
[67] The horn here was temporarily suspended, but doubtless would have been well heard.
[68] Experiments so important as those of De la Roche ought not to be left without verification. I have made arrangements with a view to this object.
[69] The Elder Brethren have already had plans of a new signal-gun laid before them by the constructors of the War Department.
[70] Described in Chapter V., p. 229.
[71] The figure is but a meagre representation of the fact. The band of light was two inches wide, the depth of the sinuosities varying from three feet to zero.
[72] In his admirable experiments on tuning, Scheibler found in the beats a test of differences of temperature of exceeding delicacy.
[73] Sir John Herschel and Sir C. Wheatstone, I believe, made this experiment independently.
[74] A subject to be dealt with in Chapter IX.
[75] Nor indeed any of those tones whose rates of vibration are _even_ multiples of the rate of the fundamental.
[76] According to Kolliker, this is the number of fibres in Corti’s organ.
[77] The comparison employed by Mr. Sedley Taylor appeals with graphic truth to a mountaineer. Considering, the above curve to represent a mountain-chain, he calls the discords _peaks_, and the concords _passes_.
[78] This supposition is of course made for the sake of simplicity, the real period of oscillation of a pendulum 28 feet long being between two and three seconds.
[79] This figure corresponds to the interval 15:16. For it and some other figures, I am indebted to that excellent mechanician, M. König, of Paris.
[80] For some beautiful figures of this description I am indebted to Prof. Lyman, of Yale College.
[81] Mr. Sang, of Edinburgh, was, I believe, the first to treat this subject analytically.
[82] This able paper was the starting-point of the experiments on sensitive flames, recorded in Chapters VI. and VII.; the researches of Thomas Young and Savart being the starting-point of the experiments on smoke-jets and water-jets.—J. T.
[83] “Philosophical Magazine,” section 4, vol. xiii., p. 413, 1857.
[84] “Philosophical Magazine,” section 4, vol. xiv., p. 1, _et seq._, July, 1857.
[85] “Comptes Rendus” for August, 1853. Also “Philosophical Magazine,” section 4, vol. vii., p. 186, 1854.
[86] “Philosophical Magazine,” section 4, vol. viii., p. 74, 1854.
[87] “Proceedings of the Royal Institution,” January 15, 1875.
[88] “Researches in Chemistry and Physics,” p. 484.
[89] “Connaissance des Temps,” 1825, p. 370.
[90] See Chapter VII., Part II.
[91] The effect of the air of London is sometimes strikingly evident.
[92] “Philosophical Transactions,” 1874, Part I., p. 208, and Chapter VII. of this volume.
[93] Since this was written I have sent the sound through fifteen layers of calico, and echoed it back through the same layers, in strength sufficient to agitate the flame. Thirty layers were here crossed by the sound. The sound was subsequently found able to penetrate two hundred layers of cotton net; a single layer of wetted calico being competent to stop it.
[94] The cut reached me too late for introduction at the proper place.
INDEX
A
Acoustic clouds, echoes from, 325 —— reversibility, 461-469 —— transparency, great change of, 323
Air, process of the propagation of sound through the, 33 —— propagation of sound through air of varying density, 41 —— elasticity and density of air, 54 —— influence of temperature on the velocity of sound, 55 —— thermal changes produced by the sonorous wave, 60 —— ratio of specific heats at constant pressure and at constant volume deduced from velocities of sound, 62 —— mechanical equivalent of heat deduced from this ratio, 64 —— inference that atmospheric air possesses no sensible power to radiate heat, 66 —— velocity of sound in, 69 —— musical sounds produced by puffs of air, 89 —— other modes of throwing the air into a state of periodic motion, 91 —— reflection from heated air, 338
Albans, St., echo in the Abbey Church of, 50
Amplitude of the vibration of a sound-wave, 42
Arago, his report on the velocity of sound, 328
Atmosphere, reflection from atmospheric air, 335
Atmosphere, its effect on sound, 365
Auditory nerve, office of the, 32 —— manner in which sonorous motion is communicated to the, 33
B
Bars, heated, musical sounds produced by, 87 —— examination of vibrating bars by polarized light, 209
Beats, theory of, 385 —— action of, on flame, 387 —— optical illustration of, 390 —— various illustrations of, 397 —— dissonance due to beats, 399, 428
Bell, experiments on a, placed _in vacuo_, 36-37
Bells, analysis of vibrations of, 190, 198 —— fluctuations of, 351-354
Bourse, at Paris, echoes of the gallery of the, 49
Burners, fish-tail and bat’s-wing, experiments with, 277
C
Carbonic acid, velocity of sound in, 65 —— reflection from, 335 —— oxide, velocity of sound in, 69
Chladni, his tonometer, 168 —— his experiments on the modes of vibration possible to rods free at both ends, 174 —— his analysis of the vibrations of a tuning-fork, 176
Chladni, his device for rendering the vibrations visible, 178 —— illustrations of his experiments, 180
Chords, musical, 432
Clang, definition of, 153
Claque-bois, formation of the, 175, 197
Clarinet, tones of the, 237
Clouds, sounds reflected from the, 49
Corti’s fibres, in the mechanism of the ear, 426
Cottrell, Mr., his experiment of an echo from flame, 339
D
Derham, Dr., on fog-signals, 306
Diatonic scale, 263
Difference-tones, 404
Diffraction of sound, 76 _note_, 78
Disks, analysis of vibrations of, 187, 198
Dissonance, cause of, 428 —— graphic representation of, 430
Doppler, his theory of the colored stars, 113
E
Ear, limits of the range of hearing of the, 106, 118 —— causes of artificial deafness, 108, 119 —— mechanism of the ear, 424 —— consonant intervals in relation to, 426
Echoes, 48 —— instances of, 48-50 —— aërial, production of, 328-329 —— from flame, 339-340 —— reputed cloud echoes, 328
Eolian harp, formation of the, 159-160
Erith, effects of the explosion of 1864 on the village and church of, 53
Eustachian tube, the, 198 —— mode of equalizing the air on each side of the tympanic membrane, 109, 119
F
Falsetto voice, causes of the, 239
Faraday, Mr., his experiment on sonorous ripples, 195
Fiddle, formation of the, 123 —— sound-board of the, 123 —— the iron fiddle, 169, 197 —— the straw-fiddle, 175, 197
Flames, sounding, 261, 302 —— rhythmic character of friction, 260, 301 —— influence of the tube surrounding the flame, 263, 302 —— singing-flames, 264, 302 —— effect of unisonant notes on singing-flames, 275 —— action of sound on naked flames, 275, 302-304 —— influence of pitch, 283 —— extraordinary delicacy of flames as acoustic reagents, 285 —— the vowel-flame, 286 —— discovery of a new sensitive flame by Philip Barry, 288 —— echo from, 339 —— action of beats on flame, 387
Flute, tones of the, 237
Fog, its want of power to obstruct sound, 348 —— observations in London, 348 —— fog-signals in, 355 —— artificial, experiments on, 357
Fog-signals, researches on the acoustic transparency of the atmosphere in relation to the question of, 305 —— station at South Foreland, 309 —— instruments and observations, 309 —— variations of range, 315-316 —— contradictory results, 317 —— solution of contradictions, 317-323 —— extraordinary case of acoustic opacity, 318 —— in fogs, 355 —— minimum range of, 371 —— its position, 370 —— disadvantages of the gun, 368
Foreland, South, fog-signal station at, 309 —— fog at, 354
G
Gaines’s Farm, account of the battle of, 324
Gases, velocity of sound in, 69
Gun, range of, for fog-signals, 312-313 —— inferiority to the siren, 369 —— its disadvantages as a signal, 368
H
Hail, doubt as to its power to obstruct sound, 342
Harmonic tones of strings, 152-154
Harmony, 410 —— notions of the Pythagoreans, 411 —— Euler’s theory, 419 —— conditions of harmony, 411 —— influence of overtones on harmony, 429 —— graphic representations of consonance and dissonance, 431
Harmonica, the glass, 176
Hawksbee, his experiment on sounding bodies placed _in vacuo_, 36
Hearing, mechanism of, 424
Heat, thermal changes in the air produced by the sonorous wave, 60 —— ratio of specific heats at constant pressure and at constant volume deduced from velocities of sound, 64
—— mechanical equivalent of heatdeduced from this ratio, 66 —— inference that atmospheric air possesses no sensible power to radiate heat, 68 —— musical sounds produced by heated bars, 87
Helmholtz, his theory of resultant tones, 405, 406 —— —— consonance, 414, 420
Herschel, Sir John, his article on “Sound” quoted, 50 —— his account of Arago’s observation on velocity of sound, 328
Hooke, Dr. Robert, his anticipation of the stethoscope, 75 —— his production of musical sounds by the teeth of a rotating wheel, 85
Horn, as an instrument for fog-signalling, 310-311
Hydrogen, action of, upon the voice, 40 —— deadening of sound by, 38 —— velocity of sound in, 55, 69
I
Inflection of sound, 53 —— case of the Erith explosion, 53
Interference of sonorous waves, 381-382, 407 —— extinction of sound by sound, 383, 408 —— theory of beats, 385, 408
Intervals, optical illustration of, 440
J
Joule’s equivalent, 67
Jungfrau, echoes of the, 49
K
Kaleidophone, Wheatstone’s, formation of, 170, 196
Kundt, M., his experiments, 344
L
Laplace, his correction of Newton’s formula for the velocity of sound, 58-59
Le Conte, Professor, his observation upon sensitive naked flames, 274-275 —— on the influence of musical sounds on the flame of a jet of coal-gas, 454-460
Lenses, refraction of sound by, 51
Light, analogy between sound and, 45-50 —— analogy of, 320
Liquids, velocity of sound in, 69 —— transmission of musical sounds through, 113 —— constitution of liquid veins, 291 —— action of sound on liquid veins, 294, 303-304 —— Plateau’s theory of the resolution of a liquid vein into drops, 295 —— delicacy of liquid veins, 300
Lissajous, M., his method of giving optical expression to the vibrations of a tuning-fork, 93 —— illustration of beats of two tuning-forks, 390
M
Mayer, his formula of the equivalent of heat, 66
Melde, M., his experiments with vibrating strings, 141, 427 —— and with sonorous ripples, 194
Metals, velocity of sound transmitted through, 72 —— determination of velocity in, 73
Molecular structure, influence of, on the velocity of sound, 73, 212
Monochord or sonometer, the, 121
Motion, conveyed to the brain by the nerves, 31 —— sonorous motion. See SOUND.
Mouth, resonance of the, 241-242
Music, physical difference between noise and, 82, 117 —— a musical tone produced by periodic, noise by unperiodic, impulses, 83, 117 —— production of musical sounds by taps, 84, 117 —— —— by puffs of air, 89, 117 —— pitch and intensity of musical sounds, 90, 92, 117 —— description of the siren, 97 —— definition of an octave, 105 —— description of the double siren, 110 —— transmission of musical sounds through liquids and solids, 113 —— musical chords, 432-433 —— the diatonic scale, 432-433 —— See also HARMONY.
Musical-box, formation of the, 169, 197
N
Nerves of the human body, motion conveyed by the, to the brain, 31 —— rapidity of impressions conveyed by, 31 _note_
Newton, Sir Isaac, his calculation of the velocity of sound, 58
Nodes, 131-132 —— the nodes not points of absolute rest, 135 —— nodes of a tuning-fork, 175, 177 —— rendered visible, 177, 180 —— a node the origin of vibration, 251
Noise, physical difference between music and, 82, 117
O
Octave, definition of an, 105
Organ-pipes, 219, 256
Organ-pipes, vibrations of stopped pipes, 221, 256 —— —— Pandean pipes, 224 —— —— open pipes, 224, 256, 260 —— state of the air in sounding-pipes, 227, 257 —— reeds and reed-pipes, 234
Otolites of the ear, 425
Overtones, definition of, 153 —— relation of the point plucked to the, 155 —— corresponding to the vibrations of a rod fixed at both ends, 165 —— of a tuning-fork, 177 —— rendered visible, 177, 179 —— of rods vibrating longitudinally, 207 —— of the siren, 415 —— influence of overtones on harmony, 429
P
Pandean pipes, the, 224
Piano-wires, clang of, 158 —— curves described by vibrating, 160
Pipes. See ORGAN-PIPES
Pitch of musical sounds, 90 —— illustration of the dependence of pitch on rapidity of vibration, 100 —— relation of velocity to pitch, 211-212 —— velocity deduced from pitch, 233
Plateau, his theory of the resolution of a liquid vein into drops, 295
Pythagoreans, notions of the, regarding musical consonance, 410
R
Rain, reputed power of obstructing sound, 341-342 —— artificial, passage of sound through, 345
Reeds and reed-pipes, 234
Reeds, the clarinet and flute, 237
Reflection of sound, 45 —— from gases, 332 —— aërial, proved experimentally, 258
Refraction of sound, 51
Resonance, 213 —— of the air, 213-214, 256 —— of coal-gas, 216 —— of the mouth, 242
Resonators, 213
Resultant tones, discovery of, 399 —— conditions of their production, 400 —— experimental illustrations, 401 —— theories of Young and Helmholtz, 404, 406
Reversibility, acoustic, 461-469
Robinson, Dr., his summary of existing knowledge of fog-signals, 307 —— Professor, his production of musical sounds by puffs of air, 89
Rod, vibrations of a, fixed at both ends; its subdivisions and corresponding overtones, 165, 197 —— vibrations of a rod fixed at one end, 166, 197 —— —— of rods free at both ends, 173, 197
S
Savart’s experiments on the influence of sounds on jets of water, 457
Schultze’s bristles in the mechanism of hearing, 425
Sea-water, velocity of sound in, 70
Sensitive flames, 274
Smoke-jets, action of musical sounds on, 290
Snow, its reputed power to obstruct sound, 344
Solids, velocity of sound transmitted through, 69, 72 —— musical sounds transmitted through, 115-116, 122 —— determination of velocity in, 211
Sonometer, or monochord, the, 121
Sorge, his discovery of resultant tones, 399
Sound, production and propagation of, 32, 77 —— experiments on sounding bodies placed _in vacuo_, 36, 77 —— deadened by hydrogen, 38 —— action of hydrogen upon the voice, 40 —— propagation of sound through air of varying density, 40 —— amplitude of the vibration of a sound-wave, 42, 77 —— the action of sound compared with that of light and radiant heat, 45 —— reflection of, 45, 77 —— echoes, 48-50, 78 —— sounds reflected from the clouds, 49-50 —— refraction of sound, 51, 77 —— diffraction of sound, 53, 78 —— influence of density and elasticity on velocity, 54, 78 —— influence of temperature on velocity of sound, 55, 78 —— determination of velocity, 57, 78 —— Newton’s calculation, 58, 80 —— Laplace’s correction of Newton’s formula, 59, 80 —— thermal changes produced by the sonorous wave, 60, 80 —— velocity of sound in different gases, 69, 81 —— —— in liquids and solids, 70-73, 81 —— influence of molecular structure on the velocity of sound, 73, 81 —— velocity of sound transmitted through wood, 74, 81 —— diffraction of, 76 _note_, 78 —— physical distinction between noise and music, 82 —— musical sounds periodic, noise unperiodic, impulses, 83 —— —— produced by taps, 84 —— —— by puffs of air, 89 —— pitch and intensity o£ musical sounds, 90 —— vibrations of a tuning-fork, 91 —— M. Lissajous’s method of giving optical expression to the vibrations of a tuning-fork, 93 —— definition of the wave-length, 96 —— description of the siren, 97 —— determination of the rapidity of vibration, 101 —— and of the length of the corresponding sonorous wave, 102 —— various definitions of vibration and of sound-wave, 103 —— limits of range of hearing of the ear: highest and deepest tones, 106 —— double siren, 110 —— transmission of musical sounds through liquids and solids, 113-117 —— vibrations of strings, 120 —— the sonometer, or monochord, 121 —— influence of sound-boards, 123 —— laws of vibrating strings, 125 —— direct and reflected pulses, 129 —— stationary and progressive waves, 130 —— nodes and ventral segments, 130, 133 —— application of the results to the vibration of musical strings, 138 —— M. Melde’s experiments, 141, 427 —— longitudinal and transverse impulses, 144 —— laws of vibration thus demonstrated, 148, 162 —— harmonic tones of strings, 152, 163-164 —— definitions of timbre, or quality, of overtones and clang, 153, 164 —— relation of the point of string plucked to overtones, 155-156 —— vibrations of a rod fixed at both ends; its subdivisions and corresponding overtones, 165 —— —— of a rod fixed at one end, 166 —— Chladni’s tonometer, 168 —— Wheatstone’s kaleidophone, 170, 196 —— vibrations of rods free at both ends, 173, 197 —— nodes and overtones of a tuning-fork, 175-178, 197 —— —— rendered visible, 177-179, 197-198 —— vibrations of square plates, 184, 198 —— —— of disks and bells, 187-190, 198-199 —— sonorous ripples in water, 193 —— Faraday’s and Melde’s experiments on sonorous ripples, 194-195 —— longitudinal vibrations of a wire, 200 —— relative velocities of sound in brass and iron, 203, 206 —— examination of vibrating bars by polarized light, 209 —— determination of velocity in solids, 211 —— relation of velocity to pitch, 212 —— resonance, 213, 253, 256 —— —— of the air, 213, 256 —— —— of coal-gas, 216, 256 —— description of vowel-sounds, 240 —— Kundt’s experiments on sound-figures within tubes, 244-251, 259 —— new methods of determining velocity of sound, 247-251, 259 —— causes that obstruct the propagation of, 306 —— action of fog upon sound, 307 —— contradictory results of fog-signalling, 317 —— solution of contradictions of fog-signalling, 317-318 —— extraordinary case of acoustic opacity, 318 —— great change of acoustic transparency, 323 —— noise of battle unheard, 324 —— echoes from invisible acoustic clouds, 325, 375 —— report of Arago on the velocity of, 328 —— aërial echoes of, 330 —— demonstration of reflection from gases, 332 —— reflection from vapors, 336 —— —— heated air, 337 —— echo from flame, 340 —— investigations of the transmission of sound through the atmosphere, 341 —— action of hail and rain, 341 —— action of snow, 344 —— passage through tissues, 345 —— —— artificial showers, 346 —— action of fog, 347 —— fluctuations of bells, 351-354 —— action of wind, 361 —— atmospheric selection, 365 —— law of vibratory motions in water and air, 377, 407 —— superposition of vibrations, 381 —— interference and coincidence of sonorous waves, 382-383, 407 —— extinction of sound by sound, 384, 407 —— theory of beats, 385, 408 —— action of beats on flame, 387 —— optical illustration of beats, 390, 408 —— various illustrations of beats, 397 —— resultant tones, 399, 409 —— —— conditions of their production, 400 —— —— experimental illustrations, 401 —— —— theories of Young and Helmholtz, 405-406 —— difference-tones and summation-tones, 405 —— combination of musical sounds, 410 —— sympathetic vibrations, 421 —— mode in which sonorous motion is communicated to the auditory nerve, 426
Sound-boards, influence of, 123-124
Sound-figures within tubes, M. Kundt’s experiments with, 244-251
Stars, Doppler’s theory of the colored, 113
Steam-siren, description of, 309 —— conclusive opinion as to its power for a fog-signal, 370
Stethoscope, Dr. Hook’s anticipations of the, 74
Stokes, Professor, his explanation of the action of sound-boards, 124 —— his explanation of the effect of wind on sound, 363
Straw-fiddle, formation of the, 175, 197
Strings, vibration of, 120 —— laws of vibrating strings, 125 —— combination of direct and reflected pulses, 129 —— stationary and progressive waves, 130 —— nodes and ventral segments, 130-133 —— experiments of M. Melde, 141, 427 —— longitudinal and transverse impulses, 144 —— laws of vibration thus demonstrated, 148, 162 —— harmonic tones of strings, 152, 163-164 —— timbre, or quality, and overtones and clang, 153, 164 —— Dr. Young’s experiments on the curves described by vibrating piano-wires, 160 —— longitudinal vibrations of a wire, 200 —— —— with one end fixed, 204 —— —— with both ends free, 206
Summation-tones, 405
Siren, description of the, 97 —— sounds, description of the, 97 —— its determination of the rate of vibration, 101 —— the double siren, 110, 411-412 —— the echoes of the, 330
T
Tartini’s tones, 399. See RESULTANT TONES.
Timbre, or quality of sound, definition of, 153
Tisley, Mr., his apparatus for the compounding of rectangular vibrations, 447
Toepler, M., his experiment on the rate of vibration of the flame, 268
Tonometer, Chladni’s, 168
Trumpets, range of, for fog-signals, 313
Tuning-fork, vibrations of a, 93 —— M. Lissajous’s method of giving optical expression to the vibrations, 93 —— strings set in motion by tuning-forks, 142 —— vibrations of the tuning-forks as analyzed by Chladni, 176 —— nodes and overtones of a, 171, 197 —— interference of waves of the, 395
V
Vapors, reflection from, 336
Velocity of sound, influence of density and elasticity on, 54 —— influence of temperature on, 55 —— determination of, 57 —— Newton’s calculation, 58 —— velocity of sound in different gases, 69 —— and transmitted through various liquids and solids, 70-73 —— relative velocities of sound in brass and iron, 203, 206 —— relation of velocity to pitch, 212 —— velocity deduced from pitch, 233
Ventral segments, 132
Vertical jets, action of sound on, 297, 304
Vibrations of a tuning-fork, 93 —— method of giving optical expression to the vibrations of a tuning-fork, 93 —— illustration of the dependence of pitch on rapidity of vibration, 101 —— the rate of vibration determined by the siren, 101 —— determination of the length of the sound-wave, 102, 118 —— various definitions of vibrations, 103, 118 —— vibrations of strings, 120 —— laws of vibrating strings, 125 —— direct and reflected pulses illustrated, 129 —— application of the result to the vibration of musical strings, 138 —— M. Melde’s experiments on the vibration of strings, 141, 427 —— longitudinal and transverse impulses, 144 —— Vibrations of a red-hot wire, 147 —— laws of vibration thus demonstrated, 148, 162 —— new mode of determining the law of vibration, 148, 150 —— harmonic tones of strings, 152, 163 —— vibrations of a rod fixed at both ends; its subdivisions and corresponding overtones, 165 —— vibrations of a rod fixed at one end, 166 —— Chladni’s tonometer, 168 —— Wheatstone’s kaleidophone, 170 —— vibrations of rods free at both ends, 173 —— nodes and overtones rendered visible, 177-179 —— vibrations of square plates, 184 —— —— of disks and bells, 187-190 —— longitudinal vibrations of a wire, 200, 255 —— —— with one end fixed, 204 —— —— with both ends free, 206 —— divisions and overtones of rods, vibrating longitudinally, 207 —— examination of vibrating bars by polarized light, 209 —— vibrations of stopped pipes, 221 —— —— of open pipes, 224 —— a node the origin of vibration, 251 —— law of vibratory motions in water and air, 377 —— superposition of vibrations, 381 —— theory of beats, 385 —— sympathetic vibrations, 421 —— M. Lissajous’s method of studying musical vibrations, 433 —— apparatus for the compounding of rectangular vibrations, 447
Violin, formation of the, 123 —— sound-board of the, 123 —— the iron fiddle, 169, 197
Voice, human, action of hydrogen upon the, 40 —— sonorous waves of the, 104 —— description of the organ of voice, 238 —— causes of the roughness of the voice in colds, 239 —— causes of the squeaking falsetto voice, 239 —— Müller’s imitation of the action of the vocal chords, 240 —— formation of the vowel-sounds, 240-241 —— synthesis of vowel-sounds, 242-243
Vowel-flame, the, 286
Vowel-sounds, formation of the, 240 —— synthesis of, 242-243
W
Water-Waves, stationary, phenomena of, 136
Water, velocity of sound in, 70 —— transmission of musical sounds through, 113 —— effects of musical sounds on jets of water, 291-292 —— delicacy of liquid veins, 294 —— theory of the resolution of a liquid vein into drops, 295, 304 —— law of vibratory motions in water, 377
Wave-length, definition of, 96 —— determination of the length of the sonorous wave, 102
Wave-length, definition of sonorous wave, 104
Wave-motion, illustration, 128-133 —— stationary waves, 133 —— law of, 377
Waves of the sea, causes of the roar of the breaking, 88 _note_
Weber, Messrs., their researches on wave-motion, 133
Wetterhorn, echoes of the, 49
Wheatstone, Sir Charles, his kaleidophone, 170 —— his apparatus for the compounding of rectangular vibrations, 448
Whistles, range of, for fog-signals, 313
Wind, effect on sound, 361
Wires. See STRINGS
Wood, velocity of sound transmitted through, 74 —— musical sounds transmitted through, 115 —— the claque-bois, 175 —— determination of velocity in wood, 211
Woodstock Park, echoes in, 56
Y
Young, Dr. Thomas, his proof of the relation of the point of a string plucked to the overtones, 155 —— on the curves described by vibrating piano-wires, 160-161 —— his theory of resultant tones, 404