Sound

§ 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:

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. Fig. 166. 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 Fig. 167. 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. Fig. 175. 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 185385—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 REVERSIBILITY87

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.