CHAPTER IV
.
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.
[Illustration: FIG. 53.]
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 2_l_ 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—
2_l_/_t_ = √(_e_/_m_); or _t_ = 2_l_ · √(_m_/_e_)
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.
[Illustration: FIG. 54.]
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.
[Illustration: FIG. 55.—An experiment on resonance.]
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
[Illustration]
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.
_do_ _re_ _mi_ _fa_ _sol_ _la_ _si_ _do′_ C D E F G A B C^1 n ⁹⁄₈n ⁵⁄₄n ⁴⁄₃n ³⁄₂n ⁵⁄₃n ¹⁵⁄₈n 2n
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 C^1, 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 ³⁄₂_n_ × 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 ³⁄₂_n_ × 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.
[Illustration: FIG. 56.—The formation of beats by two wave-trains.]
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 C^1 forming the middle octave on a piano. The frequencies and differences of the eight tones in this octave are as follows:—
FREQUENCIES OF THE NOTES OF THE MIDDLE OCTAVE OF A PIANO.
Notes. Frequency. Difference.
C 264 33 D 297 33 E 330 22 F 352 44 G 396 44 A 440 55 B 495 33 C^1 528
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.—
C. B. Fundamental 264 495 First harmonic 528 990 Second ” 792 1475 Third ” 1056 1980 Fourth ” 1320 2475 Fifth ” 1584 2970
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.
[Illustration: FIG. 57.—A closed organ-pipe.]
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.
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.
[Illustration: FIG. 58.—An open organ-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.
[Illustration: FIG. 59.]
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.
[Illustration: FIG. 60.]
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, C^1, an octave higher, having double the frequency. If more strongly blown still, it yields a note which is the fifth, G^1, 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 G^1.
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.
[Illustration: FIG. 61.]
[Illustration: FIG. 62.—Singing flame.]
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.
[Illustration: FIG. 63.—Diagram of the human ear.]
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.
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