If we open the upper end of the pipe, it at once emits a note which has a wave-length equal to double the length of the pipe. Hence the note emitted by an open-ended organ-pipe is an octave higher than that given out by a closed organ-pipe of the same length.

The action of an open organ-pipe is not quite so easy to comprehend as that of a closed pipe. The difficulty is to see how stationary air waves can be set up in a pipe which is open at both ends. The easiest way to comprehend the matter is as follows: When the blast of air against the lip of the pipe begins to partially exhaust the air in it, the rarefaction so begun does not commence everywhere in the pipe at once. It starts from the mouthpiece end, and is propagated along the pipe at a rate equal to the velocity of sound. The air at the open ends of the pipe moves in to supply this reduced pressure, and, in so doing, overshoots the mark, and the result is a region of compression is formed in the central portions of the pipe (see Fig. 58). The next instant this compressed air expands again, and moves out at the two open ends of the pipe. We have thus established in the pipe an oscillatory state which, at the central region of the pipe, consists in an alternate compression and expansion or rarefaction of the air, whilst at the open end and mouthpiece end there is an alternate rushing in and rushing out of the air. Hence in the centre of the pipe we have little or no movement of the air, but rapid alternations of pressure, or, which is the same thing, density; and at the two ends little or no change in density, but rapid movement of the air in and out of the pipe.

An analogy between the vibration of the air in a closed and open organ-pipe might be found in considering the vibration of an elastic rod—first, when clamped at one end, and secondly, when clamped at the two ends. The deflection of the rod at any point may be considered to represent change of air-pressure, and the fixed point or points the open end of the pipe at which there can be no change of density, because there it is in close communication with the open air outside the pipe. It is at once evident that the length of the open organ-pipe, when sounding its fundamental tone, is one-half of the length of the air wave it produces. Accordingly, from the formula, wave-velocity = frequency × wave-length, we see that, since the velocity of sound at ordinary temperature is about 1120 feet per second, an approximate rule for obtaining the frequency of the vibrations given out by an open organ-pipe is as follows:⁠—

Frequency = 1120 divided by twice the length of pipe.

We say approximate, because, as a matter of fact, for a reason rather too complicated to explain here, the wave-length of the air-vibrations is equal to rather more than double the length of the pipe. In fact, what we may call the effective length of the pipe is equal to its real end-to-end length increased by a fraction of its diameter, which is very nearly four-fifths.

We can confirm by experiments the statements made as to the condition of the air in a sounding organ-pipe. Here is a pipe with three little holes bored in it at the top, middle, and bottom (see Fig. 59). Each of these is covered with a thin indiarubber membrane, and this, again, by a little box which has a gas-pipe leading to it and a gas-jet connected with it. If we lead gas into the box and light the jet, we have a little flame, as you see. If, then, the indiarubber membrane is pressed in and out, it will cause the gas-flame to flicker. Such an arrangement is called a manometric flame, because it serves to detect or measure changes of pressure in the pipe. The flicker of the flame when the organ-pipe is sounded is, however, so rapid that we cannot follow it unless we look at the image in a cubical revolving mirror of the kind already used. When so regarded, if the flame is steady, we see a broad band of light.

If we sound the organ-pipe gently and look at the bands of light corresponding to the three flames, we see that the flames at the top and bottom of the pipe are nearly steady, but that the one at the middle of the pipe is flickering rapidly, the band of light being changed to a saw-tooth-like form (see Fig. 60).

This shows us that rapid changes of pressure are taking place at the centre of the pipe.

Again, if we prepare a little tambourine (by stretching parchment-paper over a wooden ring), and lower it by a string into the sounding organ-pipe, we shall find that grains of sand scattered over this tambourine jump about rapidly when the membrane is held near the top or the bottom of the pipe, but are quiescent when it is at the middle.

This shows us that there is violent movement of the air at the ends, but not in the centre, thus confirming the deductions of theory.

It should be noted that if the pipe is over-blown or sounded too strongly, harmonics will make their appearance, and the simple state of affairs will no longer exist.

The celebrated mathematician, Daniel Bernoulli, discovered that an organ-pipe can be made to yield a succession of musical notes by properly varying the pressure of the current of air blown into it. If the pipe is an open one, then, if we call the frequency of the primary note 1, obtained when the pipe is gently blown, if we blow more strongly, the pipe yields notes which are the harmonics of the fundamental one, that is to say, have frequencies represented by 2, 3, 4, 5, etc., as the blast of air increases in force.

Thus, if the pipe is one about 2 feet in length, it will yield a note near to the middle C on a piano. If more strongly blown, it gives a note, C¹, an octave higher, having double the frequency. If more strongly blown still, it yields a note which is the fifth, G¹, above the last, and has three times the frequency of the primary tone; and so on.

If the pipe is closed at the top, then over-blowing the pipe makes it yield the odd harmonics, or the tones which are related in frequency to the primary tone in the ratio of 1, 3, 5, etc. Hence, if a stopped pipe gives a note, C, its first overtone is the fifth above the octave, or G¹.

It is usual, in adjusting the air-pressure of an organ-bellows, to allow such a pressure as that some of the overtones, or harmonics, shall exist. The presence of these harmonics in a note gives brilliancy to it, whereas an absolutely pure or simple musical tone, though not disagreeable to the ear, is not fully satisfying. Any one with a good ear can detect these harmonics or overtones in a single note sounded on a piano or organ due to the subdivision of the vibrating string or air-column into sections separated by nodes.

It will be seen that the acoustic action of the organ-pipe depends essentially upon some operation tending at the commencement to make an expansion of the air in the pipe at one end, and subsequently to cause an increase of air-pressure in it.

This can be effected not only by blowing into the pipe, but in another way, by introducing a hot body into a pipe open at both ends. We can show here as an illustration of this an interesting experiment due to Lord Rayleigh. A long cast-iron water-pipe about 4 inches in diameter and 8 feet long is suspended from the ceiling. About 1 foot up the tube from the lower end a piece of iron-wire gauze is fixed (see Fig. 61). By means of a gas-burner introduced into the tube, we heat the gauze red hot, and on withdrawing the lamp the tube suddenly emits a deep organ-like note for a few moments. The heated metal creates an up-draught in the tube at the lower end, and, as in the case of the open organ-pipe, causes also an in-suction of air at the upper end. The column of air is thus set vibrating with a point of alternate condensation and rarefaction in the centre, and in-draughts and out-rushes of air at the ends. Indeed, this rush of air into and out of the pipe at the lower end during the time it is sounding its note is so violent that if the hands are placed just below the bottom end of the tube they will feel chilled, as if placed near an electric fan, by the blast of air. Closing the bottom end of the pipe with a sheet of metal at once stops the air-movement, and with it the musical note.

In another form the experiment has long been known under the name of a singing flame. A small jet of burning hydrogen gas is introduced into a glass tube about 3 feet in length. The jet must consist of a long narrow brass tube, and the proper position for the jet must be found by trial (see Fig. 62). When this is done, however, the tube emits a clear musical note, due to the tube acting as an open organ-pipe. If the flame is examined in a revolving mirror when the tube is singing, it will be found to be in vibration in sympathy with the movement of air in the tube. The tube often refuses to start singing, but may be made to do it by giving it a little tap. The actions taking place in the tube are something as follows: When the flame is introduced, it heats and rarefies the air around it. This causes an in-rush of air both at the top and bottom of the tube. A state of steady oscillation is then established, in which the air at the centre undergoes periodical expansions and compressions, and the pressure of the air round the flame changes in the same manner. The flame is therefore alternately expanded and contracted. When it expands, it heats the air more. When it is compressed, it heats it less. This variation of the flame causes air to be sucked in or expelled from both open ends of the tube, and establishes the state of steady vibration in accordance with the length of the tube. The flame and the air-column act and react on each other, and establish a state of stationary aerial oscillation in accordance with the natural time-period of the column of air. The tube can be made to give out not only its fundamental note, but a series of harmonics, or overtones, with frequencies 2, 3, 4, 5, etc., times the fundamental note, by varying the position of the flame, which must always be just under the place where a node, or place of alternate condensation and rarefaction, occurs.

We may, in the next place, with advantage briefly examine the principles of construction of one musical instrument, and allude to some recent improvements. One of the most interesting of all the musical appliances devised by human ingenuity is the violin, comprising as it does in its construction an art, a science, and a tradition. In principle the violin is nothing but a wooden box, along the top of which are stretched four strings, which are strained over a piece of wood called a bridge. These strings have their effective length altered in playing by placing the finger of the performer at some place on them, and they are set in vibration by drawing over them a well-rosined bow made of horsehair. The vibrating string communicates its vibrations to the surface of the box or body by means of the bridge, and this again to the air in the interior. The body thus serves two purposes. It acts as a resonating-chamber, and also it affords a large surface of contact with the surrounding air, whereby a greater mass of air is set simultaneously in wave-motion. The four strings are normally tuned in fifths, so that the fundamental note of each is an interval of a fifth above the next.

The performer varies the note given by each string by shortening its vibrating length by pressing the finger upon it. The skilled violinist has also great control over the tone, and can determine the harmonics, or overtones, which shall accompany the fundamental by altering the point on the string at which the bow is applied, and lightly touching it at some other point.

The great art in the construction of the violin rests in the manufacture of the wooden body. Its form, materials, and minute details of construction have been the subject of countless experiments in past ages, and until quite recently no essential improvement was made in the instrument as completed by the masters of violin construction three centuries ago. In classical form the violin consists of a wooden box of characteristic shape, composed of a back, belly, and six ribs. These are shaped out of thin wood, the belly being made of pine, and maple used for the rest. A neck or handle is affixed to one end, and a tail-piece, to which the gut-strings are fastened, to the other.

The strings are strained over a thin piece of wood which rests on two feet on the belly. One of these feet rests over a block of wood in the interior of the box called the sound-post, and this forms a rigid centre; the other foot stands on the resonant part of the belly. The belly is strengthened in addition by a bar of wood, which is glued to it just under the place where the active foot of the bridge rests. The ribs or sides of the box are bent inwards at the centre to enable the playing-bow to get at the strings more easily. The selection of the wood and its varnishing is the most important part of the construction. The wood must be elastic, and its elasticity has to be preserved by the use of an appropriate hard varnish, or else it will not take up the vibrations imparted by the strings. The old makers used wood which was only just sufficiently seasoned, and applied their varnish at once.

An essential adjunct is a good bow, which is of more importance than generally supposed. Something may be got out of a poor violin by a good player, but no one can play with a bad bow.

The process of eliciting a musical tone from the violin is as follows: The player, holding the instrument in the left hand, and with its tail end pressed against the left shoulder, places a finger of the left hand lightly on some point on a string, and sweeps the bow gently across the string so as to set it in vibration, yielding its fundamental note, accompanied by the lower harmonics. The purity and strength of the note depend essentially upon the skill with which this touch of the bow is made, creating and sustaining the same kind of vibration on the string throughout its sweep. The string then presses intermittently on the bridge, and this again turns, so to speak, round one foot as round a pivot, and presses intermittently on the elastic wooden belly. The belly takes up these vibrations, and the air in the interior is thrown into sympathetic vibration by resonance. The sound escapes by the ƒ-holes in the belly. The extraordinary thing about the violin is that the shape of the box permits it to take up vibrations lying between all the range of musical tones. The air-cavity does not merely resonate to one note, but to hundreds of different rates of vibration.

The peculiar charm of the violin is the quality of the sound which a skilled player can elicit from it. That wonderful pleading, sympathetic, voice-like tone, which conveys so much emotional meaning to the trained musical ear, is due to the proper admixture of the harmonics, or overtones, with the fundamental notes. The string vibrates not merely as a whole, but in sections. Hence the place at which the bow touches must always be an anti-node, or ventral point, and the smallest change in this position greatly affects the quality of the tone.

Quite recently an entirely new departure has been made in violin construction by Mr. Augustus Stroh, a well-known inventor. He has abolished the wooden body and bridge, and substituted for them an aluminium trumpet-shaped tube as the resonant chamber, ending in a circular corrugated aluminium disc, on the centre of which rests an aluminium lever pivoted at one point. The strings are strained over this lever, and held on a light tube, which does duty as a point of attachment of all parts of the instrument. The strings are the same, and the manipulation of the instrument identical with that of the ordinary violin. The vibrations of the strings are communicated by the pivoted lever over which they pass to the corrugated aluminium disc, and by this to the air lying in the trumpet-tube. This tube points straight away from the player, and directs the air waves to the audience in front. The tone of the new violin is declared by connoisseurs to be remarkably full, mellow, and resonant. The notes have a richness and power which satisfies the ear, and is generally only to be found in the handiwork of the classical constructors of the ordinary form of violin. One great advantage in the Stroh violin is that every one can be made perfectly of the same excellence. The aluminium discs are stamped out by a steel die, and are therefore all identical. The element of chance or personal skill in making has been eliminated by a scientific and mechanical construction. Thus the musician becomes possessed of an instrument in which scientific construction predominates over individual art or tradition in manufacture, yet at the same time the musical effects which skill in playing can produce are not at all diminished.

Whilst our attention has so far been fixed on the external operations in the air which constitute a train of music-making waves, it seems only appropriate to make, in conclusion, a brief reference to the apparatus which we possess in our ears for appreciating these subtle changes in air-pressure with certainty and pleasure. The ear itself is a marvellous appliance for detecting the existence of waves and ripples in the air, and it embodies in itself many of the principles which have been explained to-day.

The organ of hearing is a sort of house with three chambers in it, or, rather, two rooms and an entrance hall, with the front door always open. This entrance passage of the ear is a short tube which communicates at one end with the open air, being there provided with a sound-deflecting screen in the shape of an external ornamental shell, commonly called the ear. In many animals this external appendage is capable of being turned into different positions, to assist in determining the direction in which the sound wave is coming. The entrance tube of the ear is closed at the bottom by a delicate membrane called the tympanum, or drum. Against this drum-head the air waves impinge, and it is pressed in and out by the changes of air-pressure. This drum separates the outer end from a chamber called the middle ear, and the middle ear communicates, by a sort of back staircase, or tube called the Eustachian tube, with the cavity at the back of the mouth (see Fig. 63).

Behind the middle ear, and buried in the bony structure of the skull, is a third, more secret chamber, called the inner ear. This is separated from the middle ear by two little windows, which are also covered with delicate membranes. In the middle ear there is a chain of three small bones linked with one another, which are connected at one end with the tympanum, or drum, and at the other end with the so-called oval window of the inner ear. Helmholtz has shown that this little chain of bones forms a system of levers, by means of which the movements of the tympanum are diminished in extent, but increased in force in the ratio of 2 to 3.

The internal ear is the real seat of audition, and it comprises the parts called the labyrinth, the semicircular canals, and the cochlea. These are cavities lined with delicate membranes and filled with fluid. In the cochlea there is an organ called Corti’s organ, which is a veritable harp of ten thousand strings. This consists of innumerable nerve-fibres, which are an extension of the auditory nerve. The details of the organic structure are far too complicated for description here. Suffice it to say that air waves, beating against the tympanum, propagate vibrations along the chain of bones into the fluids in the inner ear, and finally expend themselves on these nerve-fibres, which are the real organs of sound-sensation.

Helmholtz put forward the ingenious hypothesis that each fibre in the organ of Corti was tuned, so to speak, to a different note, and that a composite sound falling upon the ear was analyzed or disentangled by this organ into its constituents. Although this theory, as Helmholtz originally stated it, has not altogether been upheld by subsequent observation, it is certain that the ear possesses this wonderful power of analysis. It can be shown by mathematical reasoning of an advanced kind that any musical sound, no matter what its quality, can be resolved into the sum of a number of selected pure sounds such as those given by a tuning-fork.

Consider now for one moment the physical state of the air in a concert-room in which a large orchestra is performing. The air is traversed by a chaos of waves of various wave-lengths. The deep notes of the violincello, organ, and trumpets are producing waves 10 to 20 feet in wave-length, which may be best described as billows in the air. The violin-strings and middle notes of the piano, harp, or flute are yielding air waves from 6 or 8 feet to a few inches long, whilst the higher notes of violins and flutes are air ripples some 3 or 4 inches in length.

If we could see the particles of the air in the concert-room, and fasten our attention upon any one of them, we should see it executing a most complicated motion under the combined action of these air-wave-producing instruments. We should be fascinated by the amazing dance of molecules to and fro and from side to side, as the medley of waves of compression or rarefaction embraced them and drove them hither and thither in their resistless grasp.

The tympana of our ears are therefore undergoing motions of a like complicated kind, and this complex movement is transmitted through the chain of bones in the middle ear to the inner ear, or true organ of sensation. But there, by some wondrous mechanism not at all yet fully understood, an analysis takes place of these entangled motions.

The well-trained ear separates between the effect due to each kind of musical instrument, and even detects a want of tuning in any one of them. It resolves each sound into its harmonics, appreciates their relative intensity, is satisfied or dissatisfied with the admixture. In the inner chamber of the ear physical movements are in some wholly inscrutable manner translated into sensations of sound, and the confused aggregation of waves and ripples in the air, beating against the tympanic membrane there, takes effect in producing impulses which travel up the auditory nerve and expend their energy finally in the creation of sensations of melody and tune, which arouse emotions, revive memories, and stir sometimes the deepest feelings of our minds.