while other bodies, being more elastic and capable of vibration, give back a sound, and repeat it several times successively. These last are said to have a tone; the others are not allowed to have any. The tone of the elastic string, or bell, is notwithstanding nothing more than a similar sound of what the former bodies produced, but with the difference of being many times repeated while their notes are but single. So that, if we would give the former bodies a tone, it will be necessary to make them repeat their sound, by repeating our blows swiftly upon them. This will effectually give them a one; and even an unmusical instrument has often had a fine effect by its tone in our concerts. Let us then suppose, that by swift and equally continued strokes, we give any non-elastic body its tone: it is very obvious, that no alterations will be made in this tone by the quickness of the strokes, though repeated ever so fast. These will only render the tone more equal and continuous. but will make no alteration in it. On the contrary, if we make an alteration in the force of each blow, a different tone will then undoubtedly be excited. The difference will be small, it must be confessed; for the tones of these inflexible bodies are capable of but small variation, however, there will certainly be a difference. A table will return a different sound when struck with a club, from what it will do if struck with a whip. Thus non-elastic bodies return a difference of tone, not in proportion to the swiftness with which their sound is repeated, but in proportion to the greatness of the blow which produced it; for in two equal non-elastic bodies, that body produced the deepest tone which was struck by the greatest blow. Mr. Euler is of opinion, that no sound making fewer vibrations than thirty, or more than 7520 in a second, is distinguishable by the human ear; according to which principle, the extremes of our sense of hearing with respect to acute and grave sounds, is an interval of eight octaves. Tentam. Novum Theor. Mus. cap. i. sect. 13. tones. a cause. but if it be very massy, that will render it still graver; but if massy, wide, and long or high, that will make the tone deepest of all. Thus, then, will elastic bodies give the deepest sound, in proportion to the force with which they strike the air: but if we attempt to increase their force by giving them a stronger blow, this will be in vain; they will still return the same tone; for such is their formation, that they are sonorous only because they are elastic, and the force of this elasticity is not increased by our strength, as the greatness of a pendulum's vibration will not be increased by falling from a greater height. We now then come to the critical question, What is it that produces the difference of tone in two elastic sounding bells or strings? Or, what makes the one deep and the other shrill This question has always been hitherto answered by saying, that the depth or height of the note proceeded from the slowness or swiftness of the vibrations. The slowest vibrations, it has been said, are qualified for producing the deepest tones, while the swiftest produce the highest In this case, an effect has been given for It is in fact the force with which the sounding string strikes the air, that makes the true distinction in the tones of sounds; and which, with greater or less impressions, resembling the greater or less force of the blows upon a nonelastic body, produces correspondent affections of sound. The greatest forces produce the deepest sounds: the high notes are the effect of small efforts. Much also depends upon the constitution, figure, and quantity of the sonorous body, the manner of percussion, &c.; to say nothing of the intervening obstacles, distance, and disposition of the auricular organ. Thus a bell, wide at the mouth gives a grave sound; From the above considerations, the most notable distinctions of sound have been resolved into loud and low, grave and acute, long and short. With respect to the vibration necessary to produce the various concatenations of sound, particularly in the ascending series, it has been found from the nature of elastic strings, that the longest strings have the widest vibrations, and consequently go backward and forward slowest ; while, on the contrary, the shortest strings vibrate the quickest, or come and go in the shortest intervals. Hence it has been asserted, that the tone of the string depends upon the length or shortness of the vibrations. This, however, is not the case. The same string, when struck upon, must always, like the same pendulum, return precisely similar vibrations; but it is well known, that the same string does not always return precisely the same tone: so that in this case, the vibrations follow one rule, and the tone another. The vibrations must be invariably the same in the same string, which does not return the same tone invariably, as is well known to musicians in general. In the violin, for instance, we easily alter the tone of the string an octave, or eight notes higher, by a softer method of drawing the bow; and, some are known thus to bring out the most charming airs imaginable; or, those peculiar tones called flute notes. The only reason, it has been alleged, that can be assigned for the same string thus returning different tones, must certainly be the different force of its strokes upon the air. In one case, it has double the tone of the other; because upon the soft touches of the bow, only half its elasticity is put into vibration. Thus, say the authors of this theory, we shall be able clearly to account for many things relating to sounds that have hitherto been inexplicable. For instance, if it be asked, when two strings are stretched together of equal lengths, tensions, and thickness, how does it happen, that one of them being struck, and made to vibrate throughout, the other shall vibrate throughout also? The answer is obvious: the force that the string struck receives, is communicated to the air, and the air communicates the same to the similar string; which therefore receives all the force of the former; and the force being equal, the vibrations must be so too. Again, if one string be but half the length of the other, and be struck, how will the vibrations be? The answer is, the longest string will receive all the force of the string half as long as itself, and therefore it will vibrate in proportion, that is, through half its length. In the same manner, if the longest string were three times as long as the other, it would only vibrate in a third of its length; or if four times, in a fourth of its length. In short, whatever force the smaller string impresses upon the air, the air will impress a similar force upon the longer string, and partially excite its vibrations. Hence also we may account for those gradations of sound in the Folian lyre; an instrument (says Sir John Hawkins) lately obtruded upon the public as a new invention, though described above a century ago by Kircher. (ACOUSTICS, Pl. I. fig. 1.) This instrument is easily made, being nothing more than a long narrow box of thin deal, about thirty inches long, five inches broad, and one inch and three fourths deep, with a circle in the middle of the upper side or belly about one inch and a half in diameter, pierced with small holes. On this side are seven, ten, or (according to Kircher) fifteen or more strings of very fine gut, stretched over bridges at each end, like the bridge of a fiddle, and screwed up or relaxed with screw-pins. The strings are all tuned to one note, and the instrument is placed in a current of air, where the wind can brush over its strings with freedom. A window, with the sash just raised, to give the air admission, will answer this purpose exactly. When the air blows upon these strings with different degrees of force, there will be excited different degrees of sound; sometimes the blast brings out all the tones in full concert: sometimes it sinks them to the softest murmurs; it feels for every tone, and by its gradations of strength solicits those gradations of sound, which art has taken different methods to produce. See EOLIAN HARP. The same observations may be applied to the loudness and lowness, or, as musicians speak, the strength and softness of sound. In vibrating elastic strings, the loudness of the tone is in proportion to the deepness of the note; that is, in two strings, all things in other circumstances alike, the deepest tone will be loudest. In musical instruments, upon a different principle, as in the violin, it is otherwise; the tones are made in such instruments, by a number of small vibrations crowded into one stroke. The resined bow, for instance, being drawn along a string, its roughnesses catch the string at very small intervals, and excite its vibrations. In that instrument, therefore, to excite loud tones, the bow must be drawn quick, and this will produce the greatest number of vibrations. But it must he observed, that the more quickly the bow passes over the string, the less apt will the roughness of its surface be to touch the string at every instant; to remedy this, therefore, the bow must be pressed the harder as it is drawn quicker, and thus its full sound will be brought out from the instrument. If the swiftness of the vibrations, in an instrument thus excited, exceed the force of the deeper sound in another, then the swift vibrations will be heard at a greater distance, and as much farther off as the swiftness in them exceeds the force in the other. By this theory, it is alleged, may all the phenomena of musical sounds be easily explained. It was an ancient opinion that music was first discovered by the beating of different hammers upon the smith's anvil, Without pursuing the fable, let us suppose an anvil, or several similar anvils, to be struck upon by several hammers of different weights or forces. The hammer, which is of double weight, upon striking the anvil, will produce a sound, double that of another, of half its weight: this double sound musicians have agreed to call an octave. The ear can judge of the difference or resemblance of these sounds, with great ease, the numbers being as one and two, and therefore very readily com If red. Suppose a hammer, three times less than the first, strikes the anvil, the sound produced by this will be three times less than the first: so that the ear, in judging the similitude of these sounds, will find somewhat more dithculty; because it is not so easy to tell how often one is contained in three, as it is to tell how often it is contained in two. Again, suppose the hammer four times less, or five time less, the dificulty of judging will be still greater. the hammer be six or seven times less, the difficulty still increases, insomuch, that the ear cannot readily determine the precise gradation. Now, of all comparisions those which the mind makes most easily, and with least labour, are the most pleasing. And as the ear is but an instrument of the mind, it is therefore most pleased with the combination of two sounds, the difference of which it can most readily distinguish. It is more pleased with the concord of two sounds, which are to each other as one and two, than of two sounds which are as one and three, or one and four, or one and five, or one and six or seven. Upon this pleasure which the mind takes in comparison, all harmony depends. The variety of sounds is infinite; but because the ear cannot compare two sounds so as readily to distinguish their discriminations when they exceed the proportion of one and seven, musicians have been content to confine all harmony within that compass, and have allowed but seven notes in musical composition. Let us now then suppose a stringed instrument fitted up in the order mentioned above. For instance: let the first string be twice as long as the second; let the third string be three times shorter than the first; let the fourth be four times, the fifth string five times, and the sixth six times as short as the first. Such an instrument would probably give us a representation of the lyre, as it came from the hands of the inventor. This instrument will give us all the seven notes following each other, in the order in which any two of them will accord together most pleasingly; but yet it will be a very inconvenient and disagreeable instrument; for in a compass of seven strings only, the first must be seven times as long as the last; and also seven times as loud; so that when the tones are to be played in a different order, loud and soft sounds would be intermixed with most disgusting alternations. In order to improve the first instrument, therefore, succeeding musicians very judiciously threw in all the other strings between the two first, or, in other words, between the two octaves, giving to each, the same proportion it would have had in the first natural instrument. This made the instrument more portable, and A he sounds more even and pleasing. They proportion to their elasticity. Tins is absurd, herefore disposed the sounds between the octave If we allow the difference of tone to proceed in their natural order, and gave each its own from the force, and not the frequency, of the viproportional dimensions. Of these sounds, brations, this difficulty will admit of an easy sowhere the proportion between any two of them lution. These sounds, though they seem to is most obvious, the concord between them will exist together in the string, actually follow each he most pleasing. Thus octaves, which are as two other in succession: while the vibration has to one, have a most harmonious effect: the greatest force, the fundamental tone is brought fourth and fifth also, sound sweetly together, and forward; the force of the vibration decaying, they will be found, upon calculation, to bear the the octave is produced, but almost only instantasame proportion to each other that octaves do. neously; to this succeeds, with diminished force, The true cause why concord is pleasing, must the twelfth; and lastly, the seventeenth is heard arise from our power, in such a case, of mea- to vibrate with great distinctness, while the suring more easily the differences of the tones. three other tones are always silent. These In proportion as the note can be measured with sounds, thus excited, are all of them the harits fundamental tone by large and obvious dis- monic tones, whose differences from the fundatinctions, then the concord is most pleasing; mental tone are, as was said, strong and dison the contrary, when the ear measures the dis- tinct. On the other hand, the discordant tones criminations of two tones by very small parts, cannot be heard. Their differences being but or cannot measure them at all, it loses the beauty small, they are overpowered, and in a manner of their resemblance: the whole is discord and drowned in the tones of superior difference. Yet pain. not always; for Daniel Bernoulli has been able, from the same stroke, to make the same string bring out its harmonic and its discordant tones also. So that from hence we may infer, that every note whatsoever is only a succession of tones; and that those are most distinctly heard, whose differences are most easily perceivable. There is another property in the vibration of musical strings which is held to confirm the foregoing theory. If we strike the string of a harpsichord, or any other elastic sounding chord it returns a continuing sound. This till of late, was considered as one simple uniform tone; but all musicians now confess, that it constantly returns four tones. The notes are, besides the fundamental tone, an octave above, a twelfth above, and a seventeenth. One of the bass notes of the harpsichord has been dissected in this manner by Rameau, and the actual existence of these tones proved beyond a possibility of being controverted. The experiment is easily tried for if we smartly strike one of the lower keys of a harpsichord, and then take the finger briskly away, a tolerable ear will be able to distinguish, that, after the fundamental tone has ceased, three other shriller tones will be distinctly heard; first, the octave above, then the twelfth, and lastly the seventeenth; the octave 1 above is in general almost mixed with the fundamental tone, so as not to be easily perceived, except to an ear long habituated to the minute discriminations of sounds. Thus, the smallest tone is heard last, and the deepest and largest one first: the two others in order. In the whole theory of sounds, nothing has given greater room for speculation, conjecture, and disappointment, than this amazing property in elastic strings. The whole string is universally acknowledged to be in vibration in all its parts; yet this single vibration returns no less than four different sounds. They who account for the tones of strings by the number of their vibrations, are here at the greatest loss. Daniel Bernoulli supposes, that a vibrating string divides itself into a number of curves, each of which has a peculiar vibration; and though they all swing together in the common vibration, yet each vibrates within itself. This opinion, which was supported, as most geometrical speculations are, with the parade of demonstration, was only born to die soon after. Others have ascribed this to an elastic difference in the parts of the air, each of which, at different intervals, thus received different impressions from the string, in Against this theory, however, though of plausible appearance, some strong and insuperable objections seem to offer themselves: the fundamental principle of it is incorrect. No body whatever, whether elastic or non-elastic, yields a greater sound by being struck by a larger instrument, unless either the sounding body, or that part of it which emits the sound is enlarged. In this case the largest bodies always return the gravest sounds. In speaking of elastic and nonelastic bodies in a musical sense, we are not to push the distinction so far as when we speak of them philosophically. A body is musically elastic, all of whose parts are thrown into vibrations so as to emit a sound when only a part of their surface is struck. Of this kind are bells, musical strings, and all bodies whatever that are considerably hollow. Musical non-elastic bodies are such bodies as emit a sound only from that particular place which is struck : thus, a table, a plate of iron nailed on wood, a bell sunk in the earth, are all of them non-elastics in a musical sense, though not philosophically so. When a solid body, such as a log of wood, is struck with a switch, only that part of it emits a sound which comes in contact with the switch; the note is acute and loud, but would be no less so though the adjacent parts of the log were removed. If, instead of the switch, a heavier or larger instrument is made use of, a larger portion of its surface then returns a sound, and the note is consequently more grave; but it would not be so, if the large instrument struck it with a sharp edge, or a surface only equal to that of the small one. In sounds of this kind, where there is only a single stroke, without any repetition, the immediate cause of the gravity or acuteness seems to be the quantity of air displaced by the sounding body; a large quantity displaced, produces a grave sound, and a smaller quantity a more acute one, the force wherewith the air is displaced signify ing very little. This is confirmed by some experiments made by Dr. Priestley, concerning the musical tone of electrical discharges. His remarks are: "As the course of my experiments has required a great variety of electrical explosions, I could not help observing a great variety in the musical tone made by the reports. This excited my curiosity to attempt to reduce this variation to some measure. Accordingly, by the help of a couple of spinets, and two persons who had good ears for music, I endeavoured to ascertain the tone of some electrical discharges; and observed, that every discharge made several strings, particularly those that were chords to one another, to vibrate but, one note was always predominant, and sounded after the rest. As every explosion was repeated several times, and three of us separately took the same note, there remained no doubt but that the tone we fixed upon was at last the true one. The result was as follows: A jar containing half a square foot of coated glass sounded F sharp, concert pitch. Another jar of a different form, but equal surface, sounded A jar of three square feet sounded C below F sharp. A battery consisting of sixtyfour jars, each containing half a square foot, sounded F below the C. The same battery, in conjunction with another of thirty-one jars, sounded C sharp. So that a greater quantity of coated glass always gave a deeper note. Differences in the degree of a charge in the same jar made little or no difference in the tone of the explosion: if any, a higher charge gave rather a deeper note." the same. These experiments prove how much the gravity or acuteness of sounds depend on the quantity of air put in agitation by the sounding body. We know that the noise of the electric explosion arises from the return of the air into the vacuum produced by the electric flash. The larger the vacuum, the deeper the note: for the same reason, the discharge of a musket produces a more acute note than that of a cannon; and thunder is deeper than either. Other circumstances also concur to produce different degrees of gravity or acuteness in sounds. The sound of a table struck with a piece of wood, will not be the same with that produced from a plate of iron struck by the same piece of wood, even if the blows should be exactly equal, and the iron perfectly kept from vibrating. Here the sounds are generally said to differ in their degrees of acuteness, according to the specific gravities or densities of the substances which emit them. Thus gold, which is the most dense of all metals, returns a much graver sound than silver; and metalline wires, which are more dense than strings return a proportionably graver sound. But neither does this appear to be a general rule in which we can put confidence. Bell metal is denser than copper, but it by no means appears to yield a graver sound; on the contrary, it seems very probable, that copper will give a graver sound than bell-metal, if both are struck upon in their non-elastic state; and we can by no means think that a bell of pure tin, the least dense of all the metals, will give a more acute sound than one of bell-metal which is greatly more dense.-In some bodies hardness seems to have a considerable effect. Glass, which is considerably harder than any metal, gives a more acute sound; bell-metal is harder than gold, lead, or tin, and therefore sounds much more acutely; though how far this holds with regard to different substances, there are not a sufficient number of experiments for us to judge. In bodies musically elastic, the whole substance vibrates with the slightest stroke, and therefore they always give the same note, whether they are struck with a large or with a small instrument; so that striking a part of the surface of any body musically elastic is equivalent, in it, to striking the whole surface of a non-elastic one. If the whole surface of a table was struck with another table, the note produced would be neither more nor less acute whatever force was employed; because the whole surface would then yield a sound, and no force could increase the surface: the sound would indeed be louder in proportion to the force employed, but the gravity would remain the same. In like manner, when a bell, or musical string, is struck, the whole substance vibrates, and a greater stroke cannot increase the substance.-Hence we see the fallacy of what is said concerning the Pythagorean anvils. An anvil is a body musically elastic, and no difference in the tone can be perceived whether it is struck with a large or with a small hammer; because either of them are sufficient to make the whole substance vibrate, provided nothing but the anvil is struck upon: smiths, however, do not strike their anvils, but red hot iron laid upon their anvils; and thus the vibrations of the anvil are stopped, so that it becomes a non-elastic body, and the differences of tone in the strokes of different hammers proceed only from the surface of the large hammers covering the whole surface of the iron, or at least a greater part of it than the small ones. If the small hammer is sufficient to cover the whole surface of the iron as well as the large one, the note produced will be the same, whether the large or the small hammer is used. The argument for the preceding theory, grounded on the production of what are called flute notes on the violin, is also built on a false foundation; for the bow being lightly drawn on an open string, produces no flute notes, but only the harmonies of the note to which the string is tuned. The flute notes are produced by a particular motion of the bow, quick and near the bridge, and by fingering very gently. By this management, the same sounds are produced, though at certain intervals only, as if the vibrations were transferred to the space between the end of the finger-board and the finger, instead of that between the finger and the bridge. Why this small part of the string should vibrate in such a case, and not that which is under the immediate action of the bow, we are ignorant; nor dare we affirm that the vibrations really are transferred in this manner, only the same sounds are produced as if they were. Though these objections seem sufficiently to overturn the foregoing theory, with regard to acute sounds being the effects of weak strokes and grave ones of stronger impulses, we cannot admit that longer or shorter vibrations are the occasion of gravity or acuteness in sound. A musical sound, however lengthened, either by string or bell, is only a repetition of a single one, whose duration by itself is but for a moment, and is therefore inappretiable like the smack of a whip, or the explosion of an electrical battery. The continuation of the sound is nothing more than a repetition of this instantaneous inappretiable noise after the manner of an echo, and it is this echo that makes the sound agreeable. For this reason, music is much more agreeable, when played in a large hall where the sound is reverberated, than in a small room where there is no such reverberation; and for the same reason, the sound of a string is more agreeable when put on a hollow violin than when fastened to a plain board, &c.-In the sound of a bell we cannot avoid observing this echo very distinctly. The sound appears to be made up of distinct pulses, or repetitions of the same note produced by the stroke of the hammer. It can by no means be allowed, that the note would be more acute though these pulses were to succeed one another more rapidly; the sound would indeed become more simple, but would still preserve the same tone. In musical strings the reverberations are vastly more quick than in bells, and therefore their sound is more uniform or simple, and consequently more agreeable than that of bells. In musical glasses, the vibrations must be inconceivably quicker than in any bell, or stringed instrument : and hence they are of all others the most simple and the most agreeable, though neither the most acute nor the loudest. As far as we can judge, quickness of vibration contributes to the uniformity, or simplicity, but not to the acuteness, nor to the loudness, of a musical note. See HAR MONICA. It may here be objected, that each of the different pulses of which we observe the sound of a bell to be composed, is of a very perceptible length, and far from being instantaneous; so that it is not fair to infer that the sound of a bell is only a repetition of a single instantaneous stroke, seeing it is evidently the repetition of a length ened note. To this it may be replied, that the inappretiable sound, which is produced by striking a bell in a non-elastic state, is the very same whichi, being first propagated round the bell, forms one of these short pulses, that is afterwards re-echoed as long as the vibrations of the metal continue, and it is impossible that the quickness of repetition of any sound can either increase or diminish its gravity. The writers on sound have been betrayed into many difficulties and obscurities, by rejecting the 47th proposition, B. ii. of Newton, as inconclusive reasoning. Of this proposition, however, the ingenious Mr. Young of Trinity College, Dublin, published some time since a clear, explanatory defence. He concedes that the demonstration is obscurely stated, and takes the liberty of varying, in some degree, from the method of Newton. "1. The parts of all sounding bodies (he observes,) vibrate according to the law of a cycloidal pendulum: for they may be considered as composed of an indefinite number of elastic fibres; but these fibres vibrate according to that law. 2. Sounding bodies pro pagate their motions on all sides in directum, by successive condensations and rarefactions and successive goings forward and returning back. ward of the particles. 3. The pulses are those parts of the air which vibrate backwards and forwards; and which, by going forward, strike pulsant against obstacles. The latitude of a pulse is the rectilineal space through which the motion of the air is propagated during one vibration of the sounding body. 4. All pulses move equally fast. This is proved by experiment: and it is found they describe 1070 Paris feet, or 1142 London feet in a second, whether the sound be loud or low, grave or acute. 5. To determine the latitude of a pulse: Divide the space which the pulse describes in a given time by the number of vibrations performed in the same time by the sounding body, the quotient is the latitude. "M. Sauveur, by some experiments on organ pipes, found that a body, which gives the gravest harmonic sound, vibrates twelve times and an half in a second, and that the shrillest sounding body vibrates 51,100 times in a second. At a medium, let us take the body which gives what Sauveur calls his fired sound: it performs 100 vibrations in a second, and in the same time the pulses describe 1070 Parisian feet; therefore the space described by the pulses whilst the body vibrates once, that is, the latitude, or interval of the pulse will be 107 feet. 6. To find the proportion which the greatest space, through which the particles of the air vibrate, bears to the radius of a circle, whose perimeter is equal to the latitude of the pulse. During the first half of the progress of the elastic fibre, or sounding body, it is continually getting nearer to the next particle; and during the latter half of its progress, that particle is getting farther from the fibre, and these portions of time are equal: therefore we may conclude, that at the end of the progress of the fibre, the first particle of air will be nearly as far distant from the fibre as when it began to move; and in the same manner we may infer, that all the particles vibrate through spaces nearly equal to that run over by the fibre. Now, M. Sauveur has found by experiment, that the middle point of a chord which produces his fixed sound, and whose diameter is one sixth of a line, runs over in its smallest sensible vibrations one eighteenth of a line, and in its greatest vibrations seventy-two times that space; that is seventy-two x one eighteenth of a line, or four lines, that is, one third of an inch. The latitude of the pulses of this fixed sound is 107 feet; and since the circumference of a circle is to its radius as 7-10 is to 113, the greatest space described by the particles will be to the radius of a circle, whose periphery is equal to the latitude of the pulse, as one-third of an inch is to 1-7029 feet, or 20-4348 inches, that is, as one to 61.3044. If the length of the string be increased or diminished in any proportion, cæteris paribus, the greatest space described by its middle point will vary in the same proportion For the inflecting force is, to the tending force, as the distance of the string from the middle point of vibration to half the length of the string; and therefore the inflecting and tending forces being given, the string will vibrate through spaces pro |