SUPPL

SUPPL. VOL. II. PART II.

that the notes of pipes were related to their lengths in the same manner as those of wires, and he expressly makes this remark. Newton, having discovered that sound moved at the rate of about 960 feet per second, observed that, according to the experiments of Mr Sauveur, the length of an open pipe is half the length of an aereal pulse. This he could easily ascertain by dividing the space described by sound in a second by the number of pulses.

Daniel Bernoulli, the celebrated promoter of the Newtonian mechanics, discovered, or at least was the first who attentively marked, some other circumstances of resemblance between the undulations of the air in pipes and the vibrations of wires. As a wire can be made, not only to vibrate in its full length, sounding its fundamental note, but can also be made to subdivide itself, and vibrate like a portion of the whole, with points of rest between the vibrating portions, when it gives one of its harmonic notes; so a pipe cannot only have such undulations of air going on within it as are competent to the production of its fundamental note, but also those which produce one of its harmonic notes. Every one knows that when we force a flute, by blowing too strongly, it quits its proper note, and gives the octave above. Forcing still more, produces the 12th. Then we can produce the double octave or 15th, and the 17th major, &c. In short, by attending to several circumstances in the manner of blowing, all the notes may be produced from one very long pipe that we produce from the trumpet marine, and in precisely the same order, and with the same omissions and imperfections. This alone is almost equivalent to a proof that the mechanism of the undulations of air in a pipe are analogous to that of the vibrations of an elastic cord. Having with so great success investigated the mechanism of the partial vibrations of wires, and also another kind of vibrations which we shall mention afterwards, incomparably more curious and more important in the philosophy of musical sounds, Mr Bernoulli undertook the investigation of those more mysterious motions of air which are produced in pipes; and in a very ingenious dissertation, published in the Memoirs of the Academy of Paris for 1762, &c. he gives a theory of them, which tallies in a wonderful manner with the chief phenomena which we observe in the wind instruments of the flute and trumpet kind. We are not, however, so well satisfied with the truth of his assumptions respecting the state of the air, and the precise form of the undulations which he assigns to it; but we see that, notwithstanding a probability of his being mistaken in these circumstances (it is with great deference that we presume to suppose him mistaken), the chief propositions are still true; and that the changes from note to note must be produced in the order, tho' perhaps not in the precise manner, assigned by him.

It is by no means easy to conceive, with clearness, the way in which musical undulations are excited in the various kinds of trumpets. Many who have reputation as mechanicians, suppose that it is by means of vibrations of the lips, in the same manner as in the hautboy, clarionette, and reed pipes of the organ, where the air, say they, is put in motion by the trembling reed. But this explanation is wrong in all its parts; even in the reed-pipes of an organ, the air is not put in motion by the reeds. They are indeed the occasions of its musical undulation, but they do not immediately impel it into

into those waves. This method (and indeed all methods but the vibrations of wires, bells, &c.) of producing sound is little understood, though it is highly worthy of notice, being the origin of animal voice, and because a knowledge of it would enable the artists to entertain us with sounds hitherto unknown, and thus add considerably to this gift of our Bountiful Father, who has shewn, in the structure of the larynx of the human species, that he intended that we should enjoy the pleasures of music as a laborum dulce lenimen. He has there placed a micrometer apparatus, by which, after the other muscles have done their part in bringing the glottis nearly to the tension which the intended note requires, we can easily, and instantly, adjust it with the utmost nicety.

We trust, therefore, that our readers will indulge us while we give a very cursory view of the manner in which the tremulous motion of the glottis, or of a reed in an organ pipe, produces the sonorous undulations with a constant or uniform frequency, so as to yield a musical note.

If we blow through a small pipe or quill, we produce only a whizzing or hissing noise. If, in blowing, we shut the entry with our tongue, we hear something like a solid blow or tap, and it is accompanied with some faint perception of a musical pitch, just as when we tap with the finger on one of the holes of a flute when all the rest are shut. We are then sensible of a difference of pitch according to the length of the pipe; a longer pipe or quill giving a graver sound. Here, then, is like the beginning of a sonorous undulation. Let us consider the state of the air in the pipe: It was filled by a column of air, which was moving forward, and would have been succeeded by other air in the same state. This air was therefore nearly in its state of natural density. When the entry is suddenly stopped by the tongue, the included air, already in motion, continues its motion. This it cannot do without growing rarer, and then it is no longer a balance for the pressure of the atmosphere. It is therefore retarded in its motion, totally stopped (being in a rarefied state), and is then pressed back again. It comes back with an accelerated motion, and recovers its natural density, while the state of rarefaction goes forward through the open air like any other aerial pulse. Its motions are somewhat, but not altogether, like that of a spiral wire, which has been in like manner moving uniformly along the pipe, and has been stopped by something catching hold of its hindermost extremity. This spring, when thus caught behind, stretches itself a little, then contracts beyond its natural state, and then expands again, quivering several times. It can be demonstrated that the column of air will make but one quiver. Suppose this accomplished in the hundredth part of a second, and that at that instant the tongue is removed for the hundredth part of a second, and again applied to the entry of the pipe. It is plain that this will produce such another pulse, which will join to the former one, and force it out into the air, and the two pulses together will be like two pulses produced by the vibration of a cord. If, instead of the tongue, we suppose the flat plate of an organ-reed to be thus alternately applied to the hole and removed, at the exact moments that the renewals of air are wanted, it is plain that we shall have sonorous undulations of uniform frequency, and therefore a musical note. This is the way in which reeds pro-

duce their effect, not by impelling the air into alternate states of motion to and fro, and alternate strata of rarefied and condensed air, but by giving them time to acquire this state by the combination of the air's elasticity with its progressive motion.

The adjustment of the succeeding puff of air to the pulse which precedes it, so that they may make one smooth and regular pulse, is more exact than we have yet remarked; for the stoppage of the hole not only occasions a rarefaction before it, but by checking the air which was just going to enter, makes a condensation behind the door (so to speak); so that, when the passage is again opened, the two parcels of air are fitted for supporting each other, and forming one pulse.

Suppose, in the next place, that the reed, instead of completely shutting the hole each time, only half shuts it. The same thing must still happen, although not in so remarkable a degree. When the passage is contracted, the supply is diminished, and the air now in the pipe must rarefy, by advancing with its former velocity. It must therefore retard; by retarding, regain its former density; and the air, not yet got into the pipe, must condense, &c. And if the passage be again opened or enlarged in the proper time, we shall have a complete pulse of condensed and rarefied air; and this must be accompanied by the beginning of a musical note, which may be continued like the former.

This will be a softer or more mellow note than the other; for the condensed and rarefied air will not be suddenly changed in their densities. The difference will be like the difference of the notes produced by drawing a quill along the teeth of a comb, and that produced by the equally rapid vibrations of a wire. For let it be remarked here, that musical notes are by no means confined, as theorists commonly suppose, to the regular cycloidal agitations of air, such as are produced by the vibrations of an elastic cord; but that any crack, snap, or noise whatever, when repeated with sufficient frequency, becomes ipso facto a musical sound, of which we can tell the pitch or note. What can be less musical than the solitary cracks or snaps made by a stiff door when very slowly opened? Do this briskly, and the crack changes to a chirp, of which we can tell the note. The sounds will be harsh or smooth, according as the snaps of which they are composed are abrupt or gradual.

This distinction of sounds is most satisfactorily confirmed by experiment. If the tongue of the organ reed is quite flat, and if, in its vibrations, it apply itself to the whole margin of the hole at once, so as completely to shut it (as is the case in the old-fashioned regal stop of the organ), the note is clear, smart, and harsh or hard; but if the lips of the reed are curved, or the tongue properly bent backward, so that it applies itself to the edges of the hole gradatim, and never completely shuts the passage, the note may have any degree of mellow sweetness. This remark is worth the attention of the instrument-makers or organ builders, and enables them to vary the voice of the organ at pleasure. We only mention it here as introductory to the explanation of the sounds of the trumpet.

We trust that the reader now perceives how the air, proceeding along a pipe, may be put in the state of alternate strata of condensed and rarefied air, the particles, in the mean time, proceeding along the pipe with a very moderate velocity; while the state of undulation is propagated at the rate of eleven or twelve hundred feet

Musical Trumpet. feet in a second; just as we may sometimes see a stream of water gliding gently down a canal, while a wave runs along its surface with much greater rapidity.

It will greatly assist the imagination, if we compare these æreal undulations with the undulations of water in an open canal. While the water is flowing smoothly along, suppose a sluice to be thrust up from the bottom quite to the surface, or beyond it. This will immediately cause a depression on the lower side of the sluice, by the water's going along the canal, and a heaping up of the water on the other side. By properly timing the motion of this sluice up and down, we can produce a series of connected waves. If the sluice be not pushed up to the surface but only one-half way, there will be the same succession of waves, but much smoother, &c. &c.

It is in this state, though not by such means, that the air is contained in a sounding trumpet. It is not brought into this state by any tremor of the lips. The trumpeter sometimes feels such a tremor; but whenever he feels it, he can no longer found his note. His lips are painfully tickled, and he must change his manner of winding.

When blowing with great delicacy and care, the deepest notes of a French horn, or trombone, we sometimes can feel the undulations of the air in the pipe distinctly fluttering and beating against the lips; and it is difficult to hinder the lips from being affected by it: but we feel plainly that it is not the lips which are fluttering, but the air before them. We feel a curious instance of this when we attempt to whistle in concert. If our accompanier intonates with a certain degree of incorrectness, we feel something at our own lips which makes it impossible to utter the intended note. This happens very frequently to the person who is whistling the upper note of a greater third. In like manner, the undulations in a pipe react on the reed, and check its vibrations. For if the dimensions of a pipe are such that the undulations formed by the reed cannot be kept up in the pipe, or do not suit the length of the pipe, the reed will either not play at all, or will vibrate only in starts. This is finely illustrated by a beautiful and instructive experiment. Take a small reed of the vox humana stop of an organ, and set it in a glass foot, adapted to the windbox of the organ. Instead of the common pipe above it, fix on it the sliding tube of a small telescope. When all the joints are thrust down, touch the key, and look attentively to the play of the reed. While it is sounding, draw out the joints, making the pipe continually longer. We shall observe the reed thrown into strange fits of quivering, and sometimes quite motionless, and then thrown into wide sonorous vibrations, according as the maintainable pulse is commensurate or not with the vibrations of the reed. This plainly shows that the air is not impelled into its undulations by the reed, but that the reed accommodates itself to the undulations in the pipe.

We acknowledge that we cannot explain with distinctness in what manner the air in a trumpet is first put into musical undulations. We see that it is only in very long and slender tubes that this can be done. In short tubes, of considerable diameter, like the cowherd's horn, we obtain only one or two very indistinct notes, of which it is difficult to name the pitch; and this requires great force of blast; whereas, to bring

out the deep notes of the French horn, a very gentle and well regulated blast is necessary. The form of the lips, combined with the force of the blast, form all the notes. But this is in a way that cannot be taught by any description. The performer learns it by habit, and feels that the instrument leaps into its note without him, when he gradually varies his blast, and continues sounding the same note; although he, in the mean time, makes some small change in his manner of blowing. This is owing to what Mr Bernoulli observed. The tube is suited only to such pulses, and can only maintain such pulses as correspond to aliquot parts of its length; and when the embouchure is very nearly, but not accurately, suited to a particular note, that note forms itself in the tube, and, reacting on the lips, brings them into the form which can maintain it with ease. We have a proof of this when we attempt to found the note corresponding to one seventh of the length. Not having a distinct notion of this note, which makes no part of our scale of melody, we cannot easily prepare for it in the way that habit teaches us to prepare for the others: whereas, from what we shall see presently, the notes one-sixth and one-eighth are both familiar to the mind, and easily produced. When, therefore, we attempt to produce the note one-seventh, we slide, against our will, into the one-sixth or one-eighth.

Nor can we completely illustrate the formation of musical pulses by waves in water. A canal is equally susceptible of every height and length of progressive waves; whereas we see that a certain length of tube will maintain only certain determined pulses of air.

We must therefore content ourselves for the present with having learned, by means of the reed pipes, how the air may exist progressively in a tube, in an alternate state of condensation and rarefaction; and we shall now proceed to consider how this state of the air is related to the length of the tube. And here we can do no more than give an outline of Mr Bernoulli's beautiful theory of flutes and trumpets, but without a mathematical examination of the particular motions. We can, however, shew, with sufficient evidence, how the different notes are produced from the same tube. It requires, however, a very steady attention from the reader to enable him to perceive how the different portions of this air act on each other. We trust that this will now be given.

The conditions which must be implemented, in order to maintain a musical pulse, are two: 1. That the vibrations of the different plates of air be performed in equal times, otherwise they would all mix and confound each other. 2. That they move all together, all beginning and all ending at the same instant. It does not appear that any other state of vibration can exist and be maintained.

The column of air in a tube may be considered as a material spring (having weight and inertia). This spring is compressed and coiled up by the pressure of the atmosphere. But in this coiled state it can vibrate in its different parts, as a long spiral wire may do, though pressed a little together at the ends. It is evident that the air within a pipe, shut at both ends, may be placed in such a situation, in a variety of ways, that it will vibrate in every part, in the same manner as a chord of the same length and weight, strained by a force equal to the pressure of the atmosphere. Thus, in the shut pipe AB (fig. 1.), suppose a harmonic curve ACB, or

a wire of the same weight with the air, throwing itself into the form of this curve. The force which impels the point C to the axis is to that which impels the point $c$ as $CE$ to $ce$. Now, suppose the air in this pipe divided into parallel strata or plates, crossing the tube like diaphragms. In order that these may vibrate in the same manner (not across the tube, but in the direction of its axis), all that is necessary for the moment is, that the excess of the pressure of the stratum $dd$ above that of the stratum $ff$ may be to the excess of the pressure of $DD$ above that of $FF$ as $ce$ to $CE$. In this case, the stratum $ce$ will be accelerated in the direction $ef$, and the stratum $EE$ is accelerated in the same direction, and in the due proportion. Now this may be done in an infinite variety of ways for a single moment. It depends, not on the absolute density, but on the variation of density; because the pressure by which a particle of air is urged in any direction arises from the difference of the distances of the adjoining particles on each side of it. But in order to continue this vibration, or in order that it may obtain at once in the whole pipe, this variation of density must continue, and be according to some connected law. This circumstance greatly limits the ways in which the vibration may be kept up. Mr Bernoulli finds that the isochronism and synchronism can be maintained in the following manner, and in no other that he could think of:

Let $AB$ (fig. 2.) be a cylindrical pipe, shut at $A$, and open at $B$. Then, in whatever manner the sound is produced in the pipe, the undulations of the contained air must be performed as follows: Let $aa$ be a plate of air. This plate will approach to, and recede from, the shut end $A$, vibrating between the situations $bb$ and $cc$, the whole vibration being $bc$, and the plate will vibrate like a pendulum in a cycloid. The greater we suppose the excursions $ab$, $ac$, the louder will the sound be; but the duration of them all must be the same, to agree with the fact that the tone remains the same. The motion will be accelerated in approaching to $aa$ from either side, and retarded in the recede from it. Let us next consider a plate $ss$, more remote from $A$. It must make similar vibrations from the situation $ss$ to the situation $yy$. But these vibrations must be greater in proportion as the plate is farther from $A$. It cannot be conceived otherwise: For suppose the plate $ss$ to make the same excursions with $aa$, and that the rest do the same. Then they will all retain the same distances from each other; and thus there will be no force whatever acting on any particles to make them vibrate. But if every particle make excursions proportional to its distance from $A$, the variation of density will, in any instant, be the same through the whole pipe, and each particle in the vibrating plate $ss$ will be accelerated or retarded in proportion to its distance from $A$; while the accelerations and retardations over all will, in any instant, be proportional to the distance of each particle from its place of rest. All this will appear to the mathematician, who attentively considers any momentary situation of the particles. In this manner all the particles will support each other in their vibrations.

It follows from this description that the air in the tube is alternately rarefied and condensed. But these changes are very different in different parts of the tube. They must be greatest of all at $A$; because, while all the plates approach to $A$, they concur in condensing the air immediately adjoining to $A$; while the air in

$aa$ and $ss$ is less condensed by the action of the plates beyond it. The air at $B$ is always of its natural density, being in equilibrio with the surrounding air. At $B$, therefore, there is a small parcel of air, of its natural density, which is alternately going in and out.

This account is confirmed by many facts. If the bottom of the pipe be shut by a fine membrane, stretched across it like a drumhead, with a wire stretched over it, either externally or internally, in the same manner as the catgut is stretched across the bottom of a drum, it will be thrown into strong vibrations, making a very loud noise, by rattling against the cross wire. The same thing happens if the membrane be pasted over a hole close to the bottom, leaving a small space round the edge of the hole without paste, so that the membrane may play out and in, and rattle on the margin of the hole. This also makes a prodigious noise. Now, if the membrane be pasted on a hole far from the bottom, the agitations will be much fainter; and when the hole is near the mouth of the pipe, there will be none. When a pipe has its air agitated in this manner, it is giving the lowest note of which it is susceptible.

Let us next consider a pipe open at both ends. Let $CB$ (fig. 3.) be this pipe. It is plain that, if there be a partition $A$ in the middle, we shall have two pipes $AB$, $AC$, each of which may undulate in the manner now described, if the undulations in each be in opposite directions. It is evidently possible, also, that these undulations may be the same in point of strength in both, and that they may begin in the same instant. In this case, the air on each side of the partition will be in the same state, whether of condensation or rarefaction, and the partition $A$ itself will always be in equilibrio. It will perfectly resemble the point $C$ of the musical cord $BFCGH$ (fig. 6.), which is in equilibrio between the vibrating forces of its two parts. In the pipe, the plates of air on each side are either both approaching it, or both receding from it, and the partition is either equally squeezed from both sides, or equally drawn outwards. Consequently this partition may be removed, and the parcels of air on each side will, in any instant, support each other. There seems no other way of conceiving these vibrations in open pipes which will admit of an explanation by mechanical laws. The vibrations of all the plates must be obtained without any mutual hindrance, in order to produce the tone which we really hear; and therefore such vibrations are impressed by Nature on each plate of air.

But if this explanation be just, it is plain that this pipe $CB$ must give the same note with the pipe $AB$ (fig. 2.) of half the length, shut at one end. But the sound, being doubled, with perfect consonance, must be clear, strong, and mellow. Now this is perfectly agreeable to observation; and this fact is an unequivocal confirmation of the justness of the theory. If we take a slender pipe, about six inches long and one half of an inch wide, shut at one end, and sound it by blowing across its mouth, as we whistle on the pipe of a key, or across a hole that is close to the mouth, and formed with an edge like the sound-hole of a German flute, we shall get a very distinct and clear tone from it. If we now take a pipe of double the length, open at both ends, and blow across its mouth, we obtain the same note, but more clear and strong. And the note produced by blowing across the mouth is not changed by a hole made exactly in the middle, in respect of its musical

pitch, although it is greatly hurt in point of clearness and strength. Also a membrane at this hole is strongly agitated. All this is in perfect conformity to this mechanism.

Thus we have, in a great measure, explained the effect of an open and a shut pipe. The shut pipe is always an octave, graver than an open pipe of the same length; because the open pipe is in union with a shut pipe of half the length.

Let AC (fig. 4.) be a pipe shut at both ends. We may consider it as composed of two pipes AB, BC, stopped at A and C, and open at B. Undulations may be performed in each half, precisely as in the pipe AB of fig. 2.; and they will not, in the smallest degree, obstruct each other, if we only suppose that the plates in each half are vibrating at once in the same direction. The condensation in AB will correspond with the rarefaction in BC, and the middle parcel B will maintain its natural density, vibrating to, and again across the middle; and two plates $a,\; a$, which are equally distant from B, will make equal excursions in the same direction.

We may produce sound in this pipe by making an opening at B. Its note will be found to be the same with that of BC of fig. 2. or of AB of fig. 2.

In the next place, let a pipe, shut at one end, be considered as divided into any odd number of equal parts, and let them be taken in pairs, beginning at the stopped end, so that there may be an odd one left at the open end. It is plain that each of these pairs may be considered as a pipe stopped at both ends, as in fig. 4.

For the partitions will, of themselves, be in equilibrio, and may be removed, and vibrations may be maintained in the whole, consistent with the vibration of the odd part at the open end; and these vibrations will all support each other, and the plates of air which are at the points of division will remain at rest. Conceive the pipe AB of fig. 2. to be added to the pipe AC of fig. 4. the part A of the first being joined to A of the other. Now, suppose the vibrations to be performed in both, in such a manner that the simultaneous undulations on each side of the junction may be in opposite directions. It is plain that the partition will be in equilibrio, and may be removed; and the plate of air will perform the same office, being alternately the middle plate of a condensed and of a rarefied parcel of air. The two pipes CA, AB will together give the same note that AB would have given alone, but louder.

In like manner may another pipe, equal to AC, be joined to the shut end of this compound pipe, as in fig. 5, and the three will still give the same note that AB would have done alone.

And in the same manner may any number of pipes, each equal to AC, be added, and the whole will give still the same note that AB would have given alone. Musical Trumpet.

Hence it legitimately follows, that if the undulations can be once begun in this manner in a pipe, it may give either the sound competent to it, as a single pipe AB (fig. 2.); or it may give the sound competent to a pipe of $\backslash frac\{1\}\{2\}d$, $\backslash frac\{1\}\{4\}d$, $\backslash frac\{1\}\{8\}d$, &c. of its length; the undulations in each part AB, BC, CD, maintaining themselves in the manner already described. This seems the only way in which they can be preserved, both isochronous and synchronous.

It is known that the gravest tones of pipes are as the lengths of the pipes, or the frequency of the undulations are inversely as their lengths. (This will be demonstrated presently). Therefore these accessory tones should be as the odd numbers 3, 5, 7, &c. and the whole tones, including the fundamental, should form the progression of the odd numbers 1, 3, 5, 7, &c.

This is abundantly confirmed by experiment. Take a German flute, and stop all the finger-holes. The flute, by gradually forcing the blast, will give the fundamental, the 12th, the 17th, the 21st, &c. (&c.).

Again, let AD (fig. 6.) represent the length of a pipe. Construct on AD an harmonic curve AEBFCGDH, in such a manner that HD may be $\backslash frac\{1\}\{2\}AB$, $=\; \backslash frac\{1\}\{2\}BC$, $=\; \backslash frac\{1\}\{2\}CH$. The small ordinates $m,\; m$, &c. will express the total excursion of the plates of air at the points $m,\; m$, &c. and those ordinates which are above the axis will express excursions on one side of the place of rest, and the ordinates below will mark the excursions in the opposite directions, in the same manner as if this harmonic curve were really a vibrating cord. These excursions are nothing in the points A, B, C, H, and are greatest at the points E, F, G, D, where the little mass of air retains its natural density, and travels to, and again, condensing the air at B, or rarefying it, according as the parcels E and F are approaching to or receding from each other. The points A, B, C, H, may be called Nodes, and the parts E, F, G, D, may be called Beats or Loops. This represents very well to the eye the motion of the plates of air. The density and velocity need not be minutely considered at present. It is enough that we see that when the density is increasing at A, by the approach of the parcel E, it is diminishing at B, by the recede of E and F; and increasing at C, by the approach of F and G, and diminishing at H, by the recede of G. In the next vibration it will be diminishing at A and C, and increasing at B and H. And thus the alternate nodes will be in the same state, and the adjoining nodes in opposite states.

The reader must carefully distinguish this motion from

(A) A little reflection will teach us that these tones will not be perfectly in the scale. A certain proportion between the diameter and length of the pipe produces a certain tone. Making the pipe wider or smaller flattens or sharpens this tone a little, and also greatly changes its clearness. Organ-builders, who have tried every proportion, have adopted what they found best. This requires the diameter to be about $\backslash frac\{1\}\{10\}$th or $\backslash frac\{1\}\{12\}$th of the length. Therefore, when we cause the same pipe to sound different notes, we neglect this proportion; and the notes are false, and even very coarse, when we produce one corresponding to a very small portion of the pipe. For a similar reason, Mr Lambert found that, in order to make his pitch pipe sound the octave to any of its notes, it was not sufficient to shorten its capacity one-half by pushing down the piston; he found that the part remaining must be less than the part taken off by a fixed quantity $1\backslash frac\{1\}\{2\}$ inches. Or, the length which gave any note being $x$, the length for its octave must be $\backslash frac\{x\; -\; 1\backslash frac\{1\}\{2\}\}\{2\}$.

from the undulatory motion of a pulse, investigated by Newton, and described in the article ACOUSTICS, Encycloped. That undulation is going on at the same time, and is a result of what we are now considering, and the cause of our hearing this undulation. The undulation we are now considering is the original agitation, or rather it is the SOUNDING BODY, as much as a vibrating string or bell is; for it is not the trumpet that we hear, but the air trembling in the trumpet. The trumpet is performing the office, not of the string, but of the pin and bridge on which the string is strained. This is an important remark in the philosophy of musical sounds.

There is yet another set of notes producible from a pipe besides those which follow in the order of frequency 1, 3, 5, 7, &c.

Suppose a pipe open at both ends, sounding by blowing across the end, and undulating, as already described, with a node in the middle A (fig. 3.) If we still express the fundamental note of the pipe AB of fig. 2. by 1, it is plain that the fundamental of an open pipe of the same length will have the frequency of its undulations expressed by 2; because an open pipe of twice the length of AB (fig. 2.) will be 1, the two pipes AB (fig. 2.), and CB (fig. 3.), being in unison.

But this open pipe may be made to undulate in another manner; for we have seen that AB of fig. 2. joined to CA of fig. 4. may sound altogether when the partition A is removed, still giving the note of AB (fig. 2.) Let such another as AB (fig. 2.) be added to the end C, and let the partition be removed. The whole may still undulate, and still produce the same note; that is, a pipe open at both ends may found a note which is the fundamental of a pipe like AB (fig. 2.), but only one-fourth of its length. The pipe CB of fig. 3. may thus be supposed to be divided into four equal parts, CE, EA, AF, FB, of which the extreme parts EC and FB contain undulations similar to those in AB (fig. 2.); and the two middle parts contain undulations like those in CA (fig. 4.) The partitions at E and F may be removed, because the undulations in EC and EA will support each other, if they are in opposite directions; and those in FB and FA may support each other in the same manner.

It must here be remarked, that in this state of undulation the direction of the agitations at the two extremities is the same; for in the middle piece EF the particles are moving one way, condensing the air at E, while they rarefy it at F. Therefore, while the middle parcel is moving from E towards F, the air at B must be moving towards F, and the air at C must be moving from E. In short, the air at the two extremities must, in every instant, be moving in the opposite direction to that of the air in the middle.

In like manner, if the pipe CB of fig. 3. be divided into six parts, the two extreme parts may undulate like AB of fig. 2. and the four inner parts may undulate like two pipes, such as CA of fig. 4. and the whole will give the sound which makes the fundamental of a pipe of one-sixth of the length, or having the frequency 6.

We may remark here, that the simultaneous motion of the air at the extremities is in opposite directions, whereas in the last case it was in the same direction. This is easily seen; for as the partition which is between the two middle pieces must always be in equilibrio, the air must be coming in or going out at the ex-

tremities together. This circumstance must give some sensible difference of character to the sounds 4 and 6. In the one, the agitations at each end of the tube are in the same direction, and in the other they are in the opposite. Both produce pulses of sound which are conveyed to the ear. Thus we see that the air in a pipe open at both ends may undulate in two ways. It may undulate with a node in the middle, giving the note of AB (fig. 2.), or of its 3d, 5th, 7th, &c. part; and it may undulate with a loop or bight in the middle, sounding like $\backslash frac\{1\}\{2\}$, $\backslash frac\{1\}\{3\}$, $\backslash frac\{1\}\{4\}$, &c. of AB, fig. 2.

In like manner may this pipe produce sounds whose frequency are expressed by 8, 10, &c. and proceed as the even numbers.

This state of agitation may be represented in the same way that we represented the sounds 1, 3, 5, &c. by constructing on AM (fig. 7.) an harmonic curve, with any number of nodes and loops. Divide the parts AF, FD, DE, EM, equally in C, O, P, B. CB will correspond to the pipe, and the ordinates to the curve GFHDLEN will express the excursions of the plates of air.

If the pipe gives its fundamental note, its length must be represented by CO, and the undulations in it will resemble the vibrations of part CO of a cord, whose length AD is equal to 2CO, and which has a node in F.

If the pipe is sounding its octave, it will be represented by CP, and its undulations will resemble the vibrations of a cord CP, whose length AE is $\backslash frac\{1\}\{2\}$ of CP, having nodes at F and D, &c. &c.

We can now see the possibility of such undulations existing in a pipe as will be permanent, and produce all the variety of notes by a mere change in the manner of blowing, and why these notes are in the order of the natural numbers, precisely as we observe to happen in winding the trumpet or French horn. We have, 1st, the fundamental expressed by 1; then the octave 2; then the 12th, 3; the double octave 4; then the third major of that octave 5, or 17th of the fundamental; then the octave of the 12th, or the 5th of this double octave, = 6. We then jump to the triple octave 8, without producing the intermediate sound corresponding to 4th of the pipe. With much attention we can hit it; and it is a fact that a person void of musical ear stumbles on it as easily as on any other. But the musician, finding this sound begin with hum, and his ear being grated with it, perhaps thinks that he is mistaking his embouchure, and he slides into the octave. After the triple octave, we easily hit the sounds corresponding to $\backslash frac\{1\}\{2\}$ and $\backslash frac\{1\}\{3\}$, which are the 2d and 3d of this octave. The next note $\backslash frac\{1\}\{4\}$ is sharper than a just 4th. We easily produce the note 12, which is a just 5th; 13 is a false 6th; 14 is a sound of no use in our music, but easily hit; 15 and 16 give the exact 7th and 8th of this octave.

Thus, as we ascend, we introduce more notes into every octave, till at last we can nearly complete a very high octave; but in order to do this with success, and tolerable readiness, we must take an instrument of a very low pitch, that we may be able nearly to fill up the steps of the octave in which our melody lies. Few players can make the French horn or trombone sound its real fundamental, and the octave is generally mistaken for it. The proof of this is, that most players can give

give the 5th of the lowest note that they are able to produce; whereas the 5th of the real fundamental cannot be uttered. Therefore that lowest note is not the fundamental, but the octave to the fundamental.

Few performers can found even this second octave on a short instrument, such as the ordinary military trumpet; and what they imagine to be the fundamental found of this instrument is the double octave above it. This appears very strange; and it may be asked, how we know what is really the fundamental note of a trumpet? The answer to this is to be obtained only by demonstrating, on mechanical principles, what is the frequency of undulation corresponding to a given length of pipe. This is a proposition equally fundamental with its corresponding one in the theory of musical cords; but we have referred it till now, because many readers would stop short at such an investigation, who are able to understand completely what we have now delivered concerning the music of the trumpet.

Suppose therefore a pipe shut at both ends, and that the whole weight of the contained air is concentrated in its middle point, the rest retaining its elasticity without inertia; or (which is a more accurate conception), let the middle point be conceived as extending its elasticity to the two extremities of the pipe, being repelled from each by a force inversely as the distance. Let the length of this pipe be $L$. This may also express the weight of the middle plate of air, which will always be proportional to the length of the pipe, because all is supposed to be concentrated there. Let $E$ be the elasticity of the air. This must be measured by the pressure of the atmosphere, or by the weight of the column of mercury in the barometer. Perhaps the rationale of this will be better conceived by some readers by considering $E$ as the height of a homogeneous atmosphere. Then it is plain that $E$ is to $L$ as the weight of this atmospheric column to the weight of the column of the same air which fills the pipe whose length is $L$. Then it is also plain that $E$ is to $L$ as the external pressure; and consequently, as the elasticity which supports that pressure is to the weight or inertia of the matter to be moved. Let this middle plate or diaphragm be withdrawn from its place of rest to the very small distance $a$. The elasticity or repulsion will be augmented on one side and diminished on the other; and the difference between them is the only force which impels the diaphragm toward the middle point, and causes it to vibrate, or produces the undulation. It is plain that the repulsion on one side is $\backslash frac\{1\}\{2\}L\; \backslash times\; E$, or $\backslash frac\{L\}\{L-a\}\; E$ (for $\backslash frac\{1\}\{2\}L\; -\; a\; :\; \backslash frac\{1\}\{2\}L\; =\; E\; :\; \backslash frac\{1\}\{2\}L\; \backslash times\; E$), and the repulsion on the other side is $\backslash frac\{1\}\{2\}L\; \backslash times\; E$, or $\backslash frac\{L\}\{L+2a\}\; E$. The difference of these repulsions is $E\; \backslash times\; L\; \backslash times\; \backslash frac\{4a\}\{L^2\; -\; 4a^2\}$. But as we suppose $a$ exceedingly small in comparison with $L$, this difference, or the accelerating force, may safely be expressed by $E\; \backslash frac\{4a\}\{L\}$, or $4a\; \backslash frac\{E\}\{L\}$.

Hence we deduce, in the first place, that the undulations will be isochronous, whether wide or narrow; because the accelerating force is always proportional to the distance $a$ from the middle point.

Now, let a pendulum, whose quantity of matter is $L$, and length $a$, be supposed to vibrate in a cycloid by the force $\backslash frac\{4a\}\{L\}\; E$, or $\backslash frac\{4E\}\{L\}\; a$. It must perform its vibrations in the same time with the plate of air; because the moving force, the matter to be moved, and the space along which they are to be similarly impelled, are the same in both cases. Let another pendulum, having the same quantity of matter $L$, vibrate by its weight $L$ alone. In order that these two pendulums may vibrate in equal times, their lengths must be as the accelerating forces. Therefore we must have $\backslash frac\{4E\}\{L\}\; a\; :\; L\; =\; a\; :\; \backslash frac\{aL^2\}\{4Ea^2\}\; =\; \backslash frac\{L^2\}\{4E\}$, which is therefore the length of the synchronous pendulum.

Now, a cord without weight and inertia, but loaded with the weight $L$ at its middle point, and strained by a weight $E$, and drawn from the axis to the distance $a$, is precisely similar in its motion to the diaphragm we are now considering, and must make its oscillations in the same time.

This is applicable to any number of plates of air, by substituting in the cord a loaded point for each of the plates; for when the case is thus changed, both in the pipe and the cord, the space to be passed over by the plate of air bears the same proportion to $a$, which is passed over by the whole air concentrated in the middle point, which the space to be passed over by the corresponding loaded point of the cord bears to that passed over by the whole matter of the cord concentrated in the middle point; and the same equality of ratios obtains in the accelerating forces of the plate of air and the corresponding loaded point of the cord. Suppose, then, a pipe divided into 2, 3, 4, &c. equal parts, by 1, 2, 3, diaphragms, each of which contains the air of the intervening portion of the pipe, the whole weight $L$ being equally divided among them. If there be but one diaphragm, its weight must be $L$; if two, the weight of each must be $\backslash frac\{1\}\{2\}L$; if three, the weight of each must be $\backslash frac\{1\}\{3\}L$; and so on for any number.

By considering this attentively, we may infer, without further investigation, what will be the undulations of all the different plates of air in a pipe stopped at both ends. We have only to compare it with a cord similarly divided and loaded. Increase the number of loaded points, and diminish the load on each, continually—it is evident that this terminates in the case of a simple cord, with its matter uniformly diffused; and a simple pipe, with its air also uniformly diffused over its whole length.

Therefore, if we take an elastic cord, and stretch it by such a weight that the extending weight may bear the same proportion to the accelerating force acting on the whole matter concentrated in its middle point, which the elasticity of the air bears to its accelerating force acting on the whole matter concentrated at the mouth of an open pipe, sounding its fundamental note, the cord and the air will vibrate in the same time. Moreover, since the proportion between the vibrations of a cord so constituted, and those of a cord having its matter uniformly diffused, is the same with the proportion between the undulations in a pipe so constituted, and those of a pipe in which the air is uniformly diffused—it is plain that the vibrations of the cord and of the pipe

Musical Trumpet pipe in their natural state will also be performed in equal tones.

We look on this as the easiest way of obtaining a distinct perception of the authority on which we rest our knowledge of the absolute number of undulations of the air in a pipe of given length. It may be obtained directly; and Daniel Bernoulli, Euler, and others, have given very elegant solutions of this problem, without having recourse to the analogy of the vibrations of cords and undulations of a column of air. But it requires more mathematical knowledge than many readers are possessed of who are fully able to follow out this analogical investigation.

Let us therefore compare this theory with experiment. What we call an open pipe of an organ is the same which we, in this theory, have considered as a pipe open at both ends; for the opening at the foot, which the organ builders call the voice of the pipe, is equivalent to a complete opening. The aperture, and the sharp edge which divides the wind, may be continued all round, and the wind admitted by a circular slit, as is represented in fig. 10. We have tried this, and it gives the most brilliant and clear tones we ever heard, far exceeding the tones of the organ. An open organ pipe, therefore, when sounding its fundamental note, undulates with one node in its middle, and its undulations are analogous, in respect of their mechanism, with the vibrations of a wire of the same length, and the same weight, with the column of air in the pipe, and stretched by a weight equal to that of a column of the same air, reaching to the top of a homogeneous atmosphere, or equal to the weight of a column of mercury as high as that in the barometer.

Dr. Smith (see Harmonics, 2d edit. p. 193) found that a brass wire, whose length was 35.55 inches, and weight 31 troy grains, and stretched by 7 pounds avoirdupois, or 49000 grains, was in perfect unison with an open organ pipe whose length was 86.4 inches.

Now 86.4 inches of this wire weighs 75.34 grains. When the barometer stands at 30 inches, and the thermometer at 55° (the temperature at the time of the experiment), the height of a homogeneous atmosphere is 332640 inches. This has the same proportion to the length of the pipe which the pressure of the atmosphere has to the weight of the column of air contained in the pipe.

Now $86.4\; :\; 332640\; =\; 75.34\; :\; 29060$. This wire, therefore, should be stretched (if the theory be just) by 29060 grains, in order to be unison with the other wire, and we should have $35.55^2\; :\; 86.4^2\; =\; 49.00\; :\; 29060$. But, in truth, $35.55^2\; :\; 86.4^2\; =\; 49000\; :\; 289430$. The difference is 630. The error scarcely exceeds $\backslash frac\{1\}\{2\}$%, and does not amount to an error of one vibration in a second.

We must therefore account this theory as accurate, seeing that it agrees with experiment with all desirable exactness.

We may also deduce from it a very compendious rule for determining the absolute number of aereal pulses made by an open pipe of any given length. When considering the vibrations of cords, we found that the number of vibrations made in a second is $\backslash sqrt\{\backslash frac\{386E\}\{LW\}\}$, where

E is the extending weight, W the weight of the cord, and L its length. Let H be the height of a homogeneous

atmosphere. We have its weight $=\; \backslash frac\{HW\}\{L\}\; =\; E$.

Therefore substituting $\backslash frac\{HW\}\{L\}$ for E in the above formula, we have the number of aereal pulses made per second

$=\; \backslash sqrt\{\backslash frac\{386H\}\{L\}\}$, or $=\; \backslash sqrt\{\backslash frac\{386H\}\{L\}\}$. Now $\backslash sqrt\{386H\}$, computed in inches, is 11331. Therefore, if we also measure the length of the pipe L in inches, the pulses in a second are $=\; \backslash frac\{11331\}\{L\}$.

Thus, in the case before us, $\backslash frac\{11331\}\{86.4\}\; =\; 131.12$, or this pipe produces 131 pulses in a second. Dr. Smith found by experiment that it produced 130.9, differing only about $\backslash frac\{1\}\{2\}$th of a pulse.

We see that the pitch of a pipe depends on the height of the homogeneous atmosphere. This may vary by a change of temperature. When the air is warmer it expands, and the weight of the induced column is lessened, while it still carries the same pressure. Therefore the pitch must rise. Dr. Smith found his organ a full quarter tone higher in summer than in winter. The effect of this is often felt in concerts of wind instruments with stringed instruments. The heat which sharpens the tone of the first flattens the last. The harpichord soon gets out of tune with the horns and flutes.

Sir Isaac Newton, comparing the velocity of sound with the number of pulses made by a pipe of given length, observed that the length of a pulse was twice the length of the open pipe which produced it. Divide the space passed over in a second by the number of pulses, and we obtain the length of each pulse. Now it was found that a pipe of 21.9 inches produced 262 pulses. The velocity of sound (as computed by the theory on which our investigation of the undulations in pipes proceeds) is 960 feet. Now $\backslash frac\{960\; \backslash times\; 12\}\{262\}\; =\; 44$ inches.

very nearly, the half of which is 22, which hardly differs from 21.9. The difference of this theoretical velocity of sound, and its real velocity 1142 feet per second, remains still to be accounted for. We may just observe here, that when a pipe is measured, and its length called 21.9, we do really allow it too little. The voice-hole is equivalent to a portion, not inconsiderable of its length, as appears very clearly from the experiments of Mr. Lambert on a variable pitch pipe, and on the German flute, recorded in the Berlin Memoirs for 1775. He found it equivalent to $\backslash frac\{1\}\{2\}$th; and this is sufficient for reconciling these measures of a pulse with the real velocity of sound.

The determination which we have given of the undulations of air in an organ pipe is indirect, and is but a sketch of the beautiful theory of Daniel Bernoulli, in which he states with accuracy the precise undulation of each plate of air, both in respect of position, density, velocity, and direction of its motion. It is a pleasure to observe how the different equations coincide with those which express the vibrations of an elastic cord. But this would have taken up much room, and would not have been suited to the information of many curious readers, who can easily follow the train of reasoning which we have employed.

Mr. Bernoulli applies the same theory to the explanation

Musical trumpet. nation of the undulations in flutes, or instruments whose sounds are modified by holes in the sides of the pipe. But this is foreign to our purpose of explaining the music of the trumpet. We shall only observe, that a hole made in that part of a pipe where a note should form itself, in order to render practicable the undulations competent to a particular note, prevents its formation, and in its place we only get such undulations (and their corresponding sounds) as have a loop in that place. The intelligent reader will perceive that this single circumstance will explain almost every phenomenon of flutes with holes; and also the effects of holes in instruments with a reed voice, such as the hautboy or clarionette.

We now see that the sound or musical pitch of a pipe is inversely as its length, in the same manner as in strings. And we learn, by comparing them, that the sound of a trumpet has the same pitch with an open organ pipe of the same length. A French horn, 16 feet long, has the sound C fa ut, which is also the sound of an open flute-pipe of that length.

The TROMBONE, great trumpet, or SACKBURE, is an old instrument described by Merseenus and other authors of the last century. It has a part which slides (air-tight) within the other. By this contrivance the pitch can be altered by the performer as he plays. This is a great improvement when in good hands; because we can thus correct all the false notes of the trumpet, which are very offensive, when they occur in an emphatic or holding note of a piece of music. We can even employ this contrivance for filling up the blanks in the lower octaves.

We must not take leave of this subject without taking notice of another discovery of Mr Bernoulli's, which is exceedingly curious, and of the greatest importance in the philosophy of music.

Artists had long ago observed that the deep notes of musical instruments are sometimes accompanied by their harmonic sounds. This is most clearly perceived in bells, some of which give these harmonics, particularly the 12th, almost as strong as the fundamental. Musicians, by attending more carefully to the thing, seem now to think that this accompaniment is universal. If one of the finest sounding strings of the bales of a harp-sichord be struck, we can hear the 12th very plainly as the sound is dying away, and the 17th major is the last sound that dies away on the ear. This will be rendered much more sensible, if we divide the wire into five parts, and at the points of division tie round it a thread with a fast knot, and cut the ends off very short. This makes the string false indeed by the unequal loading; but, by rendering those parts somewhat less moveable by this additional matter, the portions of the wire between these points are thus jogged, as it were, into secondary vibrations, which have a more sensible proportion to the fundamental vibration. This is still more sensible in the sound of the strings of a violincello when so loaded; but we must be careful not to load them too much, because this would do much retard the fundamental vibration, without retarding the secondary vibrations, that both cannot be maintained together. (N. B. This experiment always produces a beat in the sound.)—Listening to a fine sounding flute-pipe of the organ, we can also very often perceive the same thing. Mr Rameau, and most other theorists in music, now af-

feet that this is the essence of a musical sound, and necessarily exists in all of them, distinguishing them from harsh noises. Rameau has made this the foundation of his system of music, asserting that the pleasure of harmony results from the successful imitation of this harmony of Nature. (see Music, Enceycl.). But a little logic should convince these theorists that they must be mistaken. If a note is musical because it has these accompaniments, and by this composition alone is a musical note, what are these harmonics? Are they musical notes? This is granted. Therefore they have the same composition; and a musical note must consist at once of every possible sound; yet we know that this would be a jarring noise. A little mathematics, too, or mechanics, would have convinced them. A simple vibration is surely a most possible thing, and therefore a simple sound. No, say the theorists; for though the vibration of the cord may be simple, it produces such undulations in the air as excite in us the perception of the harmonics. But this is a mere assertion, and leaves the question undecided. Is not a simple undulation of the air as possible as the simple vibration of a cord?

It is, however, a very curious thing, that almost all musical sounds really have this accompaniment of the octave, 12th, double octave, and 17th major; for these are the harmonics that we hear.

The jealousy of Leibnitz and of John Bernoulli, and their unfriendly thoughts respecting all the British mathematicians, made John Bernoulli do every thing in his power to lessen the value of Dr Taylor's investigation of the vibration of a musical cord. Taylor gave him a good opportunity. Perhaps a little vain of his investigation of this abstruse matter, he thought too much of it. He affirmed that the harmonic curve was the essential form of a string giving a musical note. This was denied, without knowing at first whether it was true or false. But as the analytic mathematics improved, it was at length found that there are an infinity of forms into which an elastic cord can be thrown, which are consistent both with isochronous vibrations, whether wide or narrow, and also with the condition of the whole cord becoming a straight line at once. Euler, D'Alembert, and De la Grange, have prosecuted this matter with great ingenuity, and it is one of the finest problems of the present day.

Daniel Bernoulli, of a very different cast of mind from his illustrious friends, admired both Newton and Taylor; and so far from wishing to eclipse Dr Taylor by the additions he had made to his theory, tried whether he could not extend Taylor's doctrine as far as the author had said. When he took a review of what he had done while explaining the partial vibrations of musical cords, he thought it very possible that while a cord is vibrating in three portions, with two nodes or points of rest, and sounding the 12th to its fundamental, it might at the same time be also vibrating as a simple cord, and sounding its fundamental note. It was possible, he thought, that the three portions might be vibrating between the four points with a triple frequency, while the two middle nodes were vibrating across the straight line between the two pins; and thus the vibrating cord might be a moveable axis, to which the rapid vibrations of the three parts might always be referred. This was very specious; and when a little more attentively considered, became more probable; for if

the cord $A\backslash beta\; B\backslash gamma\; C\backslash delta\; D$ (fig. 8.) be vibrating as a 12th to its fundamental $AD$, the points $B$ and $C$ are in equilibrio. If therefore these two points be laid hold of by hooks, and be drawn aside to $\backslash beta$ and $\backslash gamma$, while the string is yet vibrating, this should not hinder the vibrations. If the hooks be annihilated in an instant, the whole should vibrate between $A$ and $D$; and this should be in a way very different from the simple vibration. The question now is, will the cord continue to vibrate with the loops $\backslash beta\backslash gamma$, $\backslash beta\backslash gamma$, &c. in the 900th part of a second (for instance), while the whole string vibrates from $A\backslash beta\backslash gamma\; D$ to $A\backslash gamma\backslash delta\; D$ in the 300th part of a second? or will it at once acquire the form of the simple harmonic curve? The case in which it is most likely to take the latter mode of vibration is when the points $\backslash beta$ and $\backslash gamma$ are let go at the instant that each portion of the string is in the middle of its vibration, and therefore forms the line $A\backslash beta\backslash gamma\; D$. But a moment's consideration will shew us that it cannot do this; for at that instant the point $\backslash gamma$, for instance, which had come from $q$, is moving outwards with a most rapid motion, and therefore will continue to go outward, while $\backslash beta$ and $\backslash gamma$ are approaching the axis. The point $\backslash gamma$, on the contrary, is at this moment approaching the axis with a motion equally rapid. They cannot therefore all come to the axis at once, and the vibration must differ greatly from a simple one. On the other hand, let it be supposed that both species of vibrations can be preserved, and that, at the moment of letting go the points $\backslash beta$ and $\backslash gamma$, the cord has the form $A\backslash beta\backslash gamma\backslash delta\; D$. Then, when $\backslash beta$ and $\backslash gamma$ have come to $B$ and $C$, having made $\backslash frac\{1\}\{2\}$ a vibration, the point $m$ will be in the axis, having made a vibration downward, and a half vibration upwards. $q$, in like manner, is in the axis, having made a whole vibration upwards, and half a vibration downwards. $n$ is like $m$. Thus the whole comes to the axis at once; and in such a manner, that if the points $B$ and $C$ were instantly stopped, the three portions would continue their partial vibrations without any new effort. The result of this compound vibration must be a compound pulse of air, which will excite in us the perception of the fundamental sound and of its 12th. The consequence will be the same if the points $\backslash beta$ and $\backslash gamma$ are stopped any where short of the axis; and therefore (said Bernoulli) the string will really vibrate so if not stopped at all.

But this was refused by Euler, who observed that in the points $\backslash beta$ and $\backslash gamma$ of contrary flexure, having no curvature, there can be no accelerating force. This caused Bernoulli to attempt a direct investigation, examining minutely the curvatures and accelerating forces in the different points.

He had the pleasure of finding that the accelerating forces arising from the curvature in every point, were precisely such as would produce the accelerations necessary in those points for performing the motion that was required. And he exhibited the equations expressive of the state of the cord in all these points. And, on the faith of these equations, he restored the Taylorea curve to the rank which its inventor had given it; and he asserted that in every musical vibration the cord was disposed in a harmonical curve either simple or compound. He farther shewed that the equations which Euler and D'Alembert had given for the musical cord (at least in the cases which they had publish-

ed) were included in his equations, and that their equations only exhibited its momentary states, while his own equations shewed the physical connection of them all; which is, that the whole cord forms a harmonic curve between the two fixed pins, while its different portions form subordinate harmonic curves on the first as an axis. Euler and D'Alembert, although they acknowledge this in the particular cases which they had taken as examples, on account of their simplicity, still insist that no subordinate harmonic vibrations can correspond to all the states of an elastic cord which their equations exhibit as isochronous and permanent. Mr Bernoulli's death put an end to the controversy, and the question (considered as a general theory) is perhaps still undecided. It may very probably be true, that as a simple vibration may be permanent which never has the form of the simple harmonic described by Dr Taylor, so a vibration may exist compounded of such vibrations, and therefore not expressible by any equation deduced from the Taylorea curve.

But, in the mean time, Mr Bernoulli has made the most beautiful discovery in mechanics which has appeared in the course of the present century, and has explained the most curious phenomenon of continued sounds, viz. the almost universal accompaniment of the harmonic notes of any fundamental sound. For this susceptibility of compounded variation is not confined to a 12th, but is equally demonstrable of every other harmonic. Nay, it is evident that the same simple vibration of a cord may furnish a moveable axis to more than one harmonic. For as the simple vibration can have a subordinate harmonic vibration superinduced upon it, so may this compounded vibration have another superinduced on it, and so on to any degree of composition. And farther, as Mr Bernoulli has shewn the complete analogy between the accelerations of the different points of an elastic cord and of the corresponding plates of a column of air, it legitimately follows that all the consequences which we can easily deduce, respecting the vibrations of an elastic cord, may be affirmed respecting the undulations of a column of air in a pipe. Therefore this accompaniment of the harmonics must not be confined to the music of strings and bells, but equally obtains in the music of wind instruments. And thus the doctrine becomes universal.

Mr Bernoulli did not think it enough to shew that these compound vibrations are possible. He endeavours to shew that this accompaniment must be frequent. He illustrates this very prettily, by supposing that a toothed wheel is turned round, and rubs with its teeth on an elastic cord. If the successive dropping of the teeth keep exactly pace with such vibrations as the cord can take and maintain by its elasticity, these will certainly be formed on it. If the intervals do not exactly correspond, a little reflection will shew that the agitation which the cord acquires will approximate to those which it can maintain; and, if when they are exactly so in any place of it, and the wheel be in that instant removed, this vibration will remain and diffuse itself through the rest of the cord; so that the very last dying quiver (so to speak) will be a harmonic. Every harmonic agitation tends, by the very nature of the thing, to continue, while those that are incompatible really do destroy each other; and the very last must be the remainder or superplus of such as could continue,

musical over those which destroyed each other. Accordingly, the harmonic notes of wires are always most distinctly heard as the sound is dying away.

There is no occasion now to say any thing about the fallacy of Rameau's Generation Harmonique as a theory of musical pleasure. Our harmonies please us, not because a sound is accompanied by its harmonics, but because harmonics please. His principle is therefore a tautology, and gives no instruction whatever. His theory is a very forced accommodation of this principle to the practice of musicians, and taste of the Public. He is exceedingly puzzled in the case of the fouddominante, or 4th of the scale, and the 6th where there is no resonance. He says that these notes, "fremissent, quoiqu'elles ne resonnent pas." But this misleads us. They do not resound; because a 4th and a 6th cannot be produced at all by dividing the cord. They tremble; because the false 4th and false 6th are very near the true ones, and the true 4th and 6th would both tremble and resound, if they were made false. A string will both tremble and resound, if very nearly true, as any one observes the 12th and 17th on a harpichord tremble and resound very strongly, though they are tempered notes. The whole theory is overturned at once by tuning the 4th false, so as to correspond to an aliquot division of the cord. It will then resound; and if this had happened to be agreeable, it would have been caught at as the fouddominant.

The physical cause of the pleasure of harmonic sounds is yet to seek, as much as our choice of those notes for melody which give us the best harmony (see TEMPERAMENT, Suppl.). We have no hesitation in saying that, with respect to our choice, the two are quite independent. Thousands enjoy the highest pleasure from melody who never heard a harmonious sound. All the untaught singers, and all simple nations, are examples. They not only fix on certain intervals as the steps of their tunes, but are disgusted when other steps are taken. Nor do we hesitate, for the very same reasons, to say that the rules of accompaniment are dependent on the cantos or air, and by no means on the fundamental bass of Rameau. The dependence assumed by him, as the rule of accompaniment, would, if properly adhered to, according to his own notions of the comparative values of the harmonics, lead to the most fantastic airs imaginable, always jumping by large intervals, and altogether incompatible with graceful music. The rules of modulation which he has squeezed out of his principle, are nothing but forced, very forced, accommodations of a very vague principle to the current practice of his contemporaries. They do not suit the primitive melodies of many nations, and they have caused these national musics to degenerate. This is acknowledged by all who are not perverted by the prevailing habits. We have heard, and could write down, some most enchanting lullabies of simple peasant women, possessed of musical sensibility, but far removed, in the cool sequestered vale of life, from all opportunities of stealing from our great composers. Some of these lullabies never fail to charm, even the most erudite musician, when sung by a fine flexible voice: but it would puzzle Mr Rameau to accompany them secundum artem.

We conclude this subject by describing a most beautiful and instructive experiment.

Mr Watt, the celebrated engineer, was amusing him-

self (about the year 1765) with organ building, and invented a monochord of continued sound, by which he could tune an organ with mathematical precision, according to any proposed system of temperament. It consisted of a covered string of a violincello, sounding by the friction of an ivory wheel. The instrument did not answer Mr Watt's purpose, by reason of the dead harshness of its tone, and a flutter in the string by the unequal action of the wheel. But Mr Watt was amused by observing the string frequently taking, of its own accord, points of division, which remained fixed, while the rest was in a state of strong vibration. The instrument came into the possession of the writer of this article. He soon saw that it gave him an opportunity of making all the experiments which Bernoulli could only relate. When the string was kept in a state of simple vibration, by a very uniform and gentle motion of the wheel, if its middle point was then gently touched with a quill, this point immediately stopped, but the string continued to vibrate in two parts, sounding the octave: And this it continued to do, however strong the vibrations were rendered afterwards by increasing the pressure and velocity of the wheel. The same thing happened if the string was gently touched at one third. It instantly divided itself into three parts, with two nodes, and sounded the 12th. In the same manner the double octave, the 17th, and all other harmonics, were produced and maintained.

But the prettiest experiment was to put something soft, such as a lock of cotton, in the way of the wide vibrations of the cord, at one-third and two-thirds of its length, so as to disturb them when they became very wide. When this was done, the string instantly put off the appearance of fig. 8. performing at once the full vibration competent to its whole length, and the three subordinate vibrations, corresponding to one-third of its length, and sounding the fundamental and the 12th with equal strength. In this manner all the different accompaniments were produced at pleasure, and could be continued, even with strong sounds. And it was amusing to observe, when the wheel was strongly pressed to the string, and the motion violent, the nodes would form themselves on various parts of the string, running from one part to another. This was always accompanied with all the jarring sounds which corresponded to them.

When the string was making very gentle, simple vibrations, and the wheel hardly touching it, if a violincello was made to sound the 12th very strongly in its neighbourhood, the string instantly divided itself, and vibrated in unison, frequently retaining its simple vibration and fundamental tone. We recommend this experiment to every person who wishes to make himself well acquainted with the mechanism of musical sounds. He will see, in a most sensible and convincing manner, how a single string of the Molian harp gives us all the changes of harmony, sliding from one sound to another, according as it is affected in its different parts by an irregular breeze of wind. The writer of this article has attempted to regulate these sweet harmonic notes, and to introduce them into the organ. His success has been very encouraging, and the sounds far exceed in pathetic sweetness any that have yet been produced by that noble instrument. But he has not yet brought them fully under command, nor made them strong enough for any thing but the softest chamber music. Other necessary

necessary occupations prevent him from giving the attention to this subject that it deserves. He recommends it therefore to the musical instrument makers as richly deserving their notice. His general method was this: A wooden pipe is made, whose section is a double square. A partition in the middle divides it into two pipes, along side of each other. One of them communicates with the foot and wind chest, and is shut at the upper end. The other is open at the upper, and shut at the lower end. In the partition there is a slit almost the whole length, and the sides of this slit are brought to a very smooth chamfered or feather edge. A fine catgut is strained in this slit, so as almost to touch the sides. It is evident that when the wind enters one pipe by the foot, it passes through the slit into the other, and escapes at the top, which is open. In its passage it forces the catgut into motion, and produces a musical note, having all the sweetness of the Aeolian harp. The strength of sound may be increased by increasing the body of air which is made to undulate. This was done by using, instead of catgut, very narrow silk tape or ribband varnished: but the unavoidable raggedness of the edges made the sounds coarse and wheezing. Flat silver wire was not sufficiently elastic; flat wire, used for watch balance springs, was better, but still very weak founded. Other methods were tried, which promised better. A thin round plate of metal, properly supported by a spring, was set in a round hole, made in another plate not so thin, so as just not to touch the sides. The air forced through this hole made the spring plate tremble, dancing in and out, and produced a very bold and mellow sound.—This, and similar experiments, are richly worth attention, and promise great additions to our instrumental music.