Waves and ripples in water, air, and æther

SOUND AND MUSIC.

OUR discussion of waves and ripples in the air would be very incomplete if we left it without any further reference to the difference between those motions in the air which constitute noise or sound, and those to which we owe the pleasure-producing effects of musical tones. I propose, therefore, to devote our time to-day to a brief exposition of the properties and modes of production of those air-vibrations which give rise to the class of sensations we call music. Sufficient has already been said to make it clear to you that one essential difference between sound or noise and music, as far as regards the events taking place outside of our own organism, is that, in the first case, we have a more or less irregular motion in the air, and, in the second, a rhythmical movement, constituting a train of air waves. The greater pleasure we experience from the latter is, no doubt, partly due to their rhythmic character. We derive satisfaction from all regularly repeated muscular movements, such as those involved in dancing, skating, and rowing, and the agreeable sensation we enjoy in their performance is partly due to their periodic or cyclical character.

In the same way, our ears are satisfied by the uniformly repeated and sustained vibrations proceeding from an organ-pipe or tuning-fork in action, but we are irritated and annoyed by the sensations set up when irregular vibrations of the air due to the bray of a donkey or the screech of a parrot fall upon them. Before, however, we can advance further in an analysis of the nature of musical sounds, two things must be clearly explained. The first of these is the meaning of the term natural period of vibration, and the second is the nature of the effect called resonance. You see before you three small brass balls suspended by strings. One string is 1 foot long, the second 4 feet, and the third 9 feet. These suspended balls are called simple pendulums. Taking in my hands the balls attached to the 1-foot and the 4-foot strings, I withdraw them a little way from their positions of rest and let them go. They vibrate like pendulums, but, as you see, the 1-foot pendulum makes two swings in the time that the 4-foot makes one swing. Repeating the experiment with the 1-foot and the 9-foot pendulum, we find that the short one now makes three swings in the time the long one makes one swing. The inference immediately follows that these pendulums, whose respective lengths are 1, 4, and 9 feet, make their swings from side to side in times which are respectively in the ratio of 1, 2, and 3.

Again, if we withdraw any of the pendulums from its position of rest and let it swing, we shall find that in any stated period of time, say 1 minute, it executes a certain definite number of oscillations which is peculiar to itself. You might imagine that, by withdrawing it more or less from its position of rest, and making it swing over a larger or smaller distance, you could make these swings per minute more or less as you please. But you would find, on trying the experiment, that this is not the case, and that, provided the arc of vibration is not too great, the time of one complete swing to and fro is the same whether the swing be large or small.

In scientific language this is called the isochronism of the pendulum, and is said to have been discovered by Galileo in the Cathedral at Pisa, when watching the swings of a chandelier die away, whilst counting their number by the beats of his pulse. This periodic time of vibration, which is independent of the amplitude of vibration, provided the latter is small, is called the natural time of vibration of the pendulum, or its free periodic time.

In the case of the simple pendulum the free periodic time is proportional to the square root of the length of the pendulum. Accordingly, a short pendulum makes more swings per minute than a long one, and this rate of swinging is quite independent of the weight of the bob. We can, of course, take hold of the bob with our hand and force it to vibrate in any period we please, and thus produce a forced vibration; but a free vibration, or one which is unforced, has a natural time-period of its own.

In order that any body may vibrate when displaced and then set free, two conditions must exist. In the first place, there must be a controlling force tending to make the substance return to its original position when displaced. In the second place, the thing moved must have mass or inertia, and when displaced and allowed to return it must in consequence overshoot the mark, and acquire a displacement in an opposite direction. In the case of the pendulum the elastic control or restoring force is the weight of the bob, which makes it always try to occupy the lowest position. We can, however, make a pendulum of another kind. Here, for instance, is a heavy ball suspended by a spiral spring (see Fig. 53). If I pull the ball down a little, and then let it go, it jumps up and down, and executes vertical vibrations. The elastic control here is the spring which resists extension. In this instance, also, there is a natural free time of vibration, independent of the extent of the motion, but dependent upon the weight of the ball and the stiffness of the spring.

A good illustration of the above principles may be found in the construction of a clock or a watch. A clock contains a pendulum which vibrates in a certain fixed time. The arrangements we call the “works” of a clock are only a contrivance for counting the swings, and recording them by the “hands” of the clock. Owing, however, to the friction of the “works,” the pendulum would soon come to rest, and hence we have a mainspring or “weights” which apply a little push to the pendulum at each swing, and keep it going. In a watch there is no pendulum, but there is a “balance-wheel and hair-spring,” or a wheel which has a spiral spring attached to it, so that it can swing backwards and forwards through a small angle. The so-called “escapement” is a means by which the swings are counted, and a little impulse given to the wheel to keep it swinging. The watch “keeps time” if this hair-spring is of the right degree of stiffness, and the balance-wheel of the right weight and size. Thus a clock can be made to go faster or slower by slightly altering the length of its pendulum, and the watch by slightly changing the stiffness of its hair-spring.

It may be noted in passing that our legs, in walking, swing like pendulums, and every particular length of leg has its own natural time of vibration, so that there is a certain speed at which each person can walk which causes him or her the least amount of fatigue, because it corresponds with the natural free or unforced period of vibration of the leg considered as a pendulum.

We now pass on to notice another very important matter. If we have any pendulum, or mass suspended by a spring, having therefore a certain natural period of vibration, we can set it in motion by administering to it small repeated blows or pushes. If the interval between these impulses corresponds with the natural time-period of oscillation, it will be found that quickly a very large swing is accumulated or produced. If, on the other hand, the interval between the blows does not correspond with the natural time of vibration, then their effect in producing vibration is comparatively small. This may be illustrated with great ease by means of the ball suspended by a spring. Suppose that by means of an indiarubber puff-ball I make a little puff of air against the suspended ball. The small impulse produces hardly any visible effect. Let this puff be repeated at intervals of time equal to that of the natural free period of vibration of the suspended ball. Then we find that, in the course of a very few puffs, we have caused a very considerable vibration or swing to take place in the heavy ball. If, however, the puffs of air come irregularly, they produce very little effect in setting the ball in motion. In the same manner a pendulum, consisting of a heavy block of wood, may be set swinging over a considerable range by a very few properly timed taps of the finger. We may notice another instance of the effect of accumulated impulses when walking over a plank laid across a ditch. If we tread in time with the natural vibration-period of the flexible plank, we shall find that very soon we produce oscillations of a dangerously large extent. Whereas, if we are careful to make the time of our steps or movement disagree with that of the plank, this will not be the case.

It is for this reason that soldiers crossing a suspension bridge are often made to break step, lest the steady tramp of armed men should happen to set up a perilous state of vibration in the bridge. It is not untruthful to say that a boy with a pea-shooter could in time break down Charing Cross Railway Bridge over the Thames. If we suppose a pea shot against one of the sections of this iron bridge, there is no doubt that it would produce an infinitesimal displacement of the bridge. Also there is no question that the bridge, being an elastic and heavy structure, has a natural free time of vibration. Hence, if pea after pea were shot at the same place at intervals of time exactly agreeing with the free time-period of vibration of the bridge, the effects would be cumulative, and would in time increase to an amount which would endanger the structure. Impracticable and undesirable as it might be to carry out the experiment, it is nevertheless certainly true, that a boy with a pea-shooter, given sufficient patience and sufficient peas, could in time break down an iron girder bridge by the accumulation of properly timed but infinitely small blows.

The author had an instance of this before him not long ago. He was at a place where very large masts were being erected. One of these masts, about 50 feet long, was resting on two great blocks of wood placed under each end. This mast was a fine beam of timber, square in section, and each side about 2 feet wide. The mast, therefore, lay like a bridge on its terminal supports. Standing or jumping on the middle of this great beam produced hardly any visible deflection. The writer, however, placed his hand on the centre of the log and pressed it gently. Repeating this pressure at intervals, discovery was soon made of the natural time-period of vibration, and by repeating the pressures at the right moment it was found that large oscillations could be accumulated. If he had ventured to proceed far with this operation, it is certain that, with properly timed impulses, it would have been possible, by merely applying the pressure of one hand, to break in half this great wooden mast.

We have constant occasion in mechanical work to notice that whereas one pull or push of great vigour will not create some desired displacement of an object, a number of very small hits, or properly timed pushes or pulls, will achieve the requisite result. We might summarize the foregoing facts by saying that it is a maxim in dealing with bodies capable of any kind of free vibration that impulses, however small, will create oscillations of any required magnitude, if only applied at intervals equal to the natural free period of vibration of the body in question.

We can illustrate these principles by a few experiments which have special reference to musical instruments. If we fasten one end of a rope to a fixed support, we find we can produce a wave or pulse in the rope by jerking the free end up and down with the hand. The speed with which a pulse or wave travels along a rope depends upon its weight per unit of length, or, say, on the number of pounds it weighs per yard, and on the tension or pull on the rope. The tighter the rope, the quicker it travels; and for the same tension the heavier the rope, the slower it travels.

It is not difficult to show that the speed with which the pulse travels is measured by the square root of the quotient of the tension of the rope by its weight per unit of length, or, as it may be called, the density of the rope.

We have already explained that, in a medium such as air, a wave of compression is propagated at a speed which is measured by the square root of the quotient of the air-pressure, or elasticity, by its density. In exactly the same way the hump that is formed on a rope by giving one end of it a jerk, runs along at a speed which is measured by the square root of the quotient of the stretching force, or tension, by the density. The propagation of a pulse or wave along a string is most easily shown for lecture purposes by filling a long indiarubber tube with sand, and then hanging it up by one end. The tube so loaded has a large weight per unit of length, and accordingly, if we give one end a jerk a hump is created which travels along rather slowly, and of which the movement can easily be watched. We may sometimes see a canal-boat driver give a jerk of this kind to the end of his horse-rope, to make it clear some obstacle such as a post or bush.

If we do this with a rope fixed at one end, we shall notice that when the hump reaches the end it is reflected and returns upon itself. If we represent by the letter l the length of the rope, and by t the time required to travel the double distance there and back from the free end, then the quotient of 2l by t is obviously the velocity of the wave. But we have stated that this velocity is equal to the square root of the tension of the rope (call it e) by the weight per unit of length, say m. Hence clearly⁠—

Supposing, then, that the jerks of the free end are given at intervals of time equal to t, or to the time required for the pulse to run along and back again, we shall find the rope thrown into so-called stationary waves. If, however, the jerks come twice as quickly, then the rope can accommodate itself to them by dividing itself into two sections, each of which is in separate vibration; and similarly it can divide itself into three, four, five, or six, or more sections in stationary vibration. The rope, therefore, has not only one, but many natural free periods of vibration, and it can adapt itself to many different frequencies of jerking, provided these are integer multiples of its fundamental frequency.

The above statements may be very easily verified by the use of a large tuning-fork and a string. Let a light cord or silk string be attached to one prong of a large tuning-fork which is maintained in motion electrically as presently to be explained. The other end of the cord passes over a pulley, and has a little weight attached to it. Let the tuning-fork be set in vibration, and various weights attached to the opposite end of the cord.

It is possible to find a weight which applies such a tension to the cord that its time of free vibration, as a whole, agrees with that of the fork. The cord is then thrown into stationary vibration. This is best seen by throwing the shadow of the cord upon a white screen, when it will appear as a grey spindle-shaped shadow. The central point A of the spindle is called a ventral point, or anti-node, and the stationary points N are called the nodes (see Fig. 54). Next let the tension of the string be reduced by removing some of the weight attached to the end. When the proper adjustment is made, the cord will vibrate in two segments, and have a node at the centre. Each segment vibrates in time with the tuning-fork, but the time of vibration of the whole cord is double that of the fork. Similarly, by adjusting the tension, we may make the cord vibrate in three, four, or more sections, constituting what are called the harmonics of the string.

The string, therefore, in any particular state as regards tension and length, has a fundamental period in which it vibrates as a whole, but it can also divide itself into sections, each of which makes two, three, four, or more times as many vibrations per second.

In the case of a violin or piano string, we have an example of the same action. In playing the violin, the effective length of the string is altered by placing the finger upon it at a certain point, and then setting the string in vibration by passing along it a bow of horsehair covered with rosin. The string is set in vibration as a whole, and also in sections, and it therefore yields the so-called fundamental tone, accompanied by the harmonics or overtones. Every violinist knows how much the tone is affected by the point at which the bow is placed across the string, and the reason is that the point where the bow touches the string must always be a ventral point, or anti-node, and it therefore determines the harmonics which shall occur.

Another good illustration of the action of properly intermittent small impulses in creating vibrations may be found in the following experiment with two electrically controlled tuning-forks: A large tuning-fork, F (see Fig. 54), has fixed between its prongs an electro-magnet, E, or piece of iron surrounded with silk-covered wire. When an electric current from a battery, B, traverses the wire it causes the iron to be magnetized, and it then attracts the prongs and pulls them together. The circuit of the battery is completed through a little springy piece of metal attached to one of the prongs which makes contact with a fixed screw. The arrangement is such that when the prongs fly apart the circuit is completed and the current flows, and then the current magnetizes the iron, and this in turn pulls the prongs together, and breaks the circuit. The fork, therefore, maintains itself in vibration when once it has been started. It is called an electrically driven tuning-fork. Here are two such forks, in every way identical. One of the forks is self-driven, but the current through its own electro-magnet is made to pass also through the electro-magnet of the other fork, which is, therefore, not self-driven, but controlled by the first. If, then, the first fork is started, the electro-magnet of the second fork is traversed by intermittent electric currents having the same frequency as the first fork, and the electro-magnet of the second fork administers, therefore, small pulls to the prongs of the second fork, these pulls corresponding to the periodic time of the first fork. If, as at present, the forks are identical, and I start the first one, or the driving fork, in action it will, in a few seconds, cause the second fork to begin to sound. Let me, however, affix a small piece of wax to the second fork. I have now altered its proper period of vibration by slightly weighting the prongs. You now see that the first fork is unable to set the second fork in action. The electro-magnet is operating as before, but its impulses do not come at the right time, and hence the second fork does not begin to move.

If we weight the two forks equally with wax, we can again tune them in sympathy, and then once again they will control each other.

All these cases, in which one set of small impulses at proper intervals of time create a large vibration in the body on which they act, are said to be instances of resonance. A more perfect illustration of acoustic resonance may be brought before you now. Before me, on the table, is a tall glass cylindrical jar, and I have in my hand a tuning-fork, the prongs of which make 256 vibrations per second when struck (see Fig. 55). If the fork is started in action, you at a distance will hear but little sound. The prongs of the fork move through the air, but they do not set it in very great oscillatory movement. Let us calculate, however, the wave-length of the waves given out by the fork. From the fundamental formula, wave-velocity = wave-length × frequency; and knowing that the velocity of sound at the present temperature of the air is about 1126 feet per second, we see at once that the length of the air wave produced by this fork must be nearly 4·4 feet, because 4·4 × 256 = 1126·4. Hence the quarter wave-length is nearly 1·1 foot, or, say, 1 foot 1 inch.

I hold the fork over this tall jar, and pour water into the jar until the space between the water-surface and the top of the jar is a little over 1 foot, and at that moment the sound of the fork becomes much louder. The column of air in the jar is 1·1 foot in length and this resounds to the fork. You will have no difficulty in seeing the reason for this in the light of previous explanations. The air column has a certain natural rate of vibration, which is such that its fundamental note has a wave-length four times the length of the column of air. In the case of the rope fixed at one end and jerked up and down at the other so as to make stationary vibrations, the length of the rope is one quarter of the wave-length of its stationary wave. This is easily seen if we remember that the fixed end must be a node, and the end moved up and down must be an anti-node, or ventral segment, and the distance between a node and an anti-node is one quarter of a wave-length. Accordingly the vibrating column of air in the jar also has a fundamental mode of vibration, such that the length of the column is one quarter of a wave-length. Hence the vibrating prongs of the 256-period tuning-fork, when held over the 1·1 foot long column of air, are able to set the air in great vibratory movement, for the impulses from the prongs come at exactly the right time. Accordingly, the loud sound you hear when the fork is held over the jar proceeds, not so much from the fork as from the column of air in the jar. The prongs of the fork give little blows to the column of air, and these being at intervals equal to the natural time-period of vibration of the air in the jar, the latter is soon set in violent vibration.

We can, in the next place, pass on now to discuss some matters connected with the theory of music. When regular air-vibrations or wave-trains fall upon the ear they produce the sensation of a musical tone, provided that their frequency lies between about 40 per second and about 4000. The lowest note in an organ usually is one having 32 vibrations per second, and the highest note in the orchestra is that of a piccolo flute, giving 4752 vibrations per second. We can appreciate as sound vibrations lying between 16 and 32,000, but the greater portion of these high frequencies have no musical character, and would be described as whistles or squeaks.

When one note has twice the frequency of another it is called the octave of the first. Thus our range of musical tones is comprised within about seven octaves, or within the limits of the notes whose frequencies are 40, 80, 160, 320, 640, 1280, 2560, and 5120.

These musical notes are distinguished, as every one knows, by certain letters or signs on a clef. Thus the note called the middle C of a piano has a frequency of 248, and is denoted by the sign

The octave is divided into certain musical intervals by notes, the frequencies of which have a certain ratio to that of the fundamental note. This ratio is determined by what is called the scale, or gamut. Thus, in the major diatonic natural scale, if we denote the fundamental note by C, called do or ut in singing, and its frequency by n, then the other notes in the natural scale are denoted by the letters, and have frequencies as below.

Hence if the note C has 248 vibrations per second, then the note D will have 9 × 248 ÷ 8 = 279 vibrations per second. On looking at the above scale of the eight notes forming an octave, it will be seen that there are three kinds of ratios of frequencies of the various notes.

(1) The ratio of C to D, or F to G, or A to B, which is that of 8 to 9.

(2) The ratio of D to E, and G to A, which is that of 9 to 10.

(3) The ratio of E to F, or B to C¹, which is that of 15 to 16.

The first two of these intervals or ratios are both called a tone, and the third is called a semitone. The two tones, however, are not exactly the same, but their ratio to one another is that of ⁸⁄₉ to ⁹⁄₁₀ or of 80 to 81. This interval is called a comma, and can be distinguished by a good musical ear.

Several of these intervals or ratios of frequencies have received names. Thus the interval C to E, = 4:5, is called a major third, and the interval E to G, = 5:6, is called a minor third; the interval C to G, = 2:3, is called a fifth, and that of C to C^¹, = 1:2, is called an octave. For the purposes of music it has been found necessary to introduce other notes between the seven notes of the octave. If a note is introduced which has a frequency greater than any one of the seven in the ratio of 25 to 24, that is called a sharpened note; thus the note of which the frequency is ³⁄₂n × ²⁵⁄₂₄ would be called G sharp, and written G♯. In the same way, if the frequency of any note is lowered in the ratio of 24 to 25, it is said to be flattened. Then the note whose frequency is ³⁄₂n × ²⁴⁄₂₅ would be called G flat, and written G♭.

It is obvious that if we were to introduce flats and sharps to all the eight notes we should have twenty-four notes in the octave, and the various intervals would become too numerous and confusing for memory or performance. Hence in keyed instruments the difficulty has been overcome by employing a scale of equal temperament, made as follows: The interval of an octave is divided into twelve parts by introducing eleven notes, the ratio of the frequency of each note to its neighbours on either side being the same, and equal to the ratio 1 to 1·05946.

The scale thus formed is called the chromatic scale, and by this means a number of the flats and sharps become identical; thus, for instance, C♯ and D♭ become the same note. The octave has therefore twelve notes, which are the seven white keys, and the five black ones of the octave of the keyboard of a piano or organ.

Every one not entirely destitute of a musical ear is aware that certain of these musical intervals, such as the fifth, the octave, or the major third, produce an agreeable impression on the ear when the notes forming them are sounded together. On the other hand, some intervals, such as the seventh, are not pleasant. The former we call concords, and the latter discords. The question then arises—What is the reason for this difference in the effect of the air-vibrations on the ear? This leads us to consider the nature of simple and complex air vibrations or waves.

Let us consider, in the first place, the effect of sending out into the air two sets of air waves of slightly different wave lengths. These waves both travel at the same rate, hence we shall not affect the combined effects of the waves upon the air if we consider both sets of waves to stand still. For the sake of simplicity, we will consider that the wave-length of one train is 20 inches, and that of the other is 21. Moreover, let the two wave-trains be so placed relatively to one another that they both start from one point in the same phase of movement; that is, let their zero points, or their humps or hollows, coincide. Then if we draw two wavy lines (see Fig. 56) to represent these two trains, it will be evident that, since the wave-length of one is 1 inch longer than that of the other—that is, a distance equal to twenty wave-lengths—one wave-train will have gained a whole wave-length upon the other, and in a distance equal to ten wave-lengths, one wave train will have gained half a wave-length upon the other. If we therefore imagine the two wave-trains superimposed, we shall find, on looking along the line of propagation, an alternate doubling or destruction of wave-effect at regular intervals. In other words, the effect of superimposing two trains of waves of slightly different wave-lengths is to produce a resultant wave-train in which the wave-amplitude increases up to a certain point, and then dies away again nearly to nothing, as shown in the lowest of the three wave-lines in Fig. 56.

We must, then, determine how far apart these points of maximum wave-amplitude or points of no wave effect lie. If the wave-length of one train is, as stated, 20 inches, then a length of ten wave-lengths is 200 inches, and this must be, therefore, the distance from a place of maximum combined wave-effect to a place of zero wave-effect. Accordingly, the distance between two places where the two wave trains help one another must be 400 inches, and this must also be the distance between two adjacent places of wave-destruction. If, therefore, we look along the wavy line representing the resultant wave, every 400 inches we shall find a maximum wave-amplitude, and every 400 inches a place where the waves have destroyed each other. We may call this distance a wave-train length, and it is obviously equal to the product of the constituent wave-lengths divided by the difference of the two constituent wave-lengths.

It follows from this that if we suppose the two wave-trains to move forward with equal speed, the number of maximum points or zero points which will pass any place in the unit of time will be equal to the difference between the frequencies of the constituents. Let us now reduce this to an experiment. Here are two organ-pipes exactly tuned to unison, and when both are sounded together we have two identical wave-trains sent out into the air. We can, however, slightly lengthen one of the pipes, and so put them out of tune. When this is done you can no longer hear the smooth sound, but a sort of waxing and waning in the sound, and this alternate increase and diminution in loudness is called a beat. We can easily take count of the number of beats per second, and by the reasoning given above we see that the number of beats per second must be equal to the difference between the frequencies of the two sets of waves. Thus if one organ-pipe is giving 100 vibrations per second to the air, and the other 102, we hear two beats per second.

Now, up to a certain point we can count these beats, but when they come quicker than about 10 per second, we cease to be able to hear them separately. When they come at the rate of about 30 per second they communicate to the combined sound a peculiar rasping and unpleasant effect which we call a discord. If they come much more quickly than 70 per second we cease to be conscious of their presence by any discordant effect in the sound.

The theory was first put forward by the famous physicist, Von Helmholtz, that the reason certain musical intervals are not agreeable to the trained ear is because the difference between the frequencies of the constituent fundamental tones or the harmonics present in them give rise to beats, approximately of 30 to 40 per second.

In order to simplify our explanations we will deal with two cases only, viz. that of the octave interval and that of the seventh. The first is a perfect concord, and the second, at least on stringed instruments, is a discord. It has already been explained that when a string vibrates it does so not only as a whole, but also in sections, giving out a fundamental note with superposed harmonics. Suppose we consider the octave of notes lying between the frequencies 264 and 528, which correspond to the notes C and C¹ forming the middle octave on a piano. The frequencies and differences of the eight tones in this octave are as follows:⁠—

It will thus be seen that the differences between the frequencies of adjacent notes are such as to make beats between them which have a number per second so near to the limits of 30 to 40 that adjacent notes sounded together are discords.

Suppose, however, we sound the seventh, viz. C and B, together. The frequencies are 264 and 495, and the difference is 231. Since, then, the difference between the frequencies lies far beyond the limit of 30 to 40 per second, how comes it that in this case we have a discord? To answer this question we must consider the harmonics present with the fundamentals. Write down each frequency multiplied respectively by the numbers 1, 2, 3, 4, etc.⁠—

On looking at these numbers we see that although the difference between the frequencies of the two fundamentals is too great to produce the disagreeable number of beats, yet the difference between the frequencies of the fundamental of note B (495) and the first harmonic of note C (528) is exactly 33, which is, therefore, the required number. Accordingly, the discordant character of the seventh interval played on a piano is not due to the beats between the primary tones, but to beats arising between the first harmonic of one and the fundamental of the other. It will be a useful exercise to the reader to select any other interval, and write down the primary frequencies and the overtone frequencies, or harmonics, and then determine whether between any pairs disagreeable beats can occur.

The presence of harmonics or overtones is, therefore, a source of discord in some cases, but nevertheless these overtones communicate a certain character to the sound.

Helmholtz’s chief conclusions as regards the cause of concord and discord in musical tones were as follows:⁠—

(1) Musical sounds which are pure, that is, have no harmonics mixed up with them, are soft and agreeable, but without brilliancy. Of this kind are the tones emitted by tuning-forks gently struck or open organ-pipes not blown violently.

(2) The presence of harmonics up to the sixth communicates force and brilliancy and character to the tone. Of this kind are the notes of the piano and organ-pipes more strongly blown.

(3) If only the uneven harmonics, viz. the first, third, fifth, etc., are present, the sound acquires a certain nasal character.

(4) If the higher harmonics are strong, then the sound acquires great penetrating force, as in the case of brass instruments, trumpet, trombone, clarionet, etc.

(5) The causes of discord are beats having a frequency of 30 to 40 or so, taking place between the two primary tones or the harmonics of either note.

The pleasure derived from the sound of a musical instrument is dependent, to a large extent, on the existence of the desirable harmonics in each tone, or on the exclusion of undesirable ones.

In the next place, let us consider a little the means at our disposal for creating and enforcing the class of air-waves which give rise to the sensations of musical tones. Broadly speaking, there are three chief forms of musical air-wave-making appliance, viz. those which depend on the vibrations of columns of air, on strings, and on plates respectively.

One of the oldest and simplest forms of musical instrument is that represented by the pan-pipes, still used as an orchestral accompaniment in the case of the ever-popular peripatetic theatrical display called Punch and Judy.

If we take a metal or wooden pipe closed at the bottom, and blow gently across the open end, we obtain a musical note. The air in the pipe is set in vibration, and the tone we obtain depends on the length of the column of air, which is the same as the length of the pipe. The manner in which this air-vibration is started is as follows: On blowing across the open end of the pipe closed at the bottom a partial vacuum is made in it. That this is so, can be seen in any scent spray-producer, in which two glass tubes are fixed at right angles to each other. One tube dips into the scent, and through the other a puff of air is sent across the open mouth of the first. The liquid is sucked up the vertical tube by reason of the partial vacuum made above it. If we employ a pipe closed at the bottom and blow across the open end, the first effect of the exhaustion is that the jet of air is partly sucked down into the closed tube, and thus compresses the air in it. This air then rebounds, and again a partial vacuum is made in the tube. So the result is an alternate compression and expansion of the air in the closed tube. The column of air is alternately stretched and squeezed, and a state of stationary vibration is set up in the air in the tube; just as in the case of a rope fixed at one end and jerked up and down at the other end. The natural time-period of vibration of the column of air in the tube controls the behaviour of the jet of air blown across its mouth, the energy of the jet of air being drawn upon to keep the column of air in the tube in a state of oscillation. Thus a flutter is excited in the air in the tube, which is maintained as long as there is a blast of air across its mouth, and this communicates to the air outside a wave-motion. We have, therefore, a musical note produced, the wave-length of which is four times the length of the closed tube across the mouth of which we are blowing. Accordingly, a very simple musical instrument such as the pan-pipes consists of a row of tubes closed at the bottom, the tubes being of different lengths. A current of air from the mouth is blown across the tubes taken in a certain order, and we can obtain a simple melody by that process of selection.

An organ-pipe is only a more perfect means for doing the same thing. Organ-pipes may be either open or closed pipes. Also they have either a reed or a flute at one end for the purpose of establishing air-vibrations when a current of air is blown into the pipe. The form of organ-pipe most easy to understand is the closed flute pipe. This consists of a wooden tube closed at the upper end, and at the lower end having a foot-tube and mouthpiece as shown in section in Fig. 57. When a gentle current of air is blown in at the foot-tube, it impinges on the sharp edge or chamfer of the mouthpiece, and it acts just as when blown across the open end of a simple closed pipe. That is to say, it sets up a state of alternate compression and expansion of the air in the pipe. At the closed end, period-changes in density in the air are established, but no great movement takes place. At the open end or mouth there are no great changes of density, but the air is alternately moving in and out at the mouthpiece. The steady blast of air against the chamfer, therefore, sets up a state of steady oscillation of the air in the pipe, the air being squeezed up and extended alternately so that there is first a state of compression, and then a state of partial rarefaction in the air at the closed end of the pipe. In this case, also, the wave-motion communicated to the surrounding air has a wave-length equal to four times the length of the pipe.