or MARGNY. This is a stringed instrument, invented in the 16th century by an Italian artist Marino or Marigni, and called a trumpet, because it takes only the notes of the trumpet, with all its omissions and imperfections, and can therefore execute only such melodies as are fitted for that instrument. It is a very curious instrument, though of small musical powers, because its mode of performance is totally unlike that of other stringed instruments; and it deserves our very particular attention, because it lays open the mechanism of musical sounds more than anything we are acquainted with; and we shall therefore make use of it in order to communicate to our readers a philosophical theory of music, which we have already treated in detail as a liberal or scientific art.

The trumpet marine is commonly made in the form of a long triangular pyramid, ABCD, fig. A, on which a single string EFG is strained over a bridge F by means of the finger pin L. At the narrow end are several frets 1, 2, 3, 4, 5, &c., between E and K, which divide the length EF into aliquot parts. Thus E1 is \( \frac{1}{3} \) of EF, E2 is \( \frac{1}{4} \), and so on. The bow is drawn lightly across the cord at H, and the string is flopped by pressing it with the finger immediately above the frets, but not so hard as to make it touch the fret. When the open string is sounded, it gives the fundamental note. If it is flopped, in the way now described, at \( \frac{1}{4} \) of its length from E2, it yields the 12th of the fundamental; if flopped at \( \frac{1}{8} \)th, it gives the double octave; if at \( \frac{1}{16} \)th, it gives the 17th major, &c. In short, it always gives the note corresponding to the length of the part between the fret and the nut E. The sounds resemble those of a pipe, and are indeed the same with those known by the name harmonica, and now executed by every performer on instruments of the viol or violin species. But in order to increase the noise, the bridge F is constructed in a very particular manner. It does not rest on the sound-board of the instrument through its whole breadth, but only at the corner a, where it is firmly fixed. The other extremity is detached about \( \frac{1}{10} \) of an inch from the sound-board; and thus the bridge, being made to tremble by the strong vibration of the thick cord, rattles on the sound-board, or on a bit of ivory glued to it. The usual way in which this motion is procured, is to have another string passing under the middle of the bridge in such a manner that, by straining it tight, we raise the corner b from the sound-board to the proper height. This contrivance increases prodigiously the noise of the instrument, and gives it somewhat of the smart sound of the trumpet, tho' very harsh and coarse. But it merits the attention of every person who wishes to know any thing of the philosophy of musical sounds, and we shall therefore say as much on the subject as will conduce to this effect.

Galileo, as we have observed in the article TEMPERAMENT, Suppl., was the first who discovered the real connection between mathematics and music, by demonstrating that the times of the vibrations of elastic cords of the same matter and size, and stretched by equal weights, are proportional to the lengths of the strings. He inferred from this that the musical pitch of the sound produced by a stretched cord depended solely on the frequency of the vibrations. Moreover, not being able to discover any other circumstance in which those sounds physically resembled each other, and reflecting that all sounds are immediately produced by agitations of air acting on the ear, he concluded that each vibration of the cord produced a sonorous pulse in the air, and therefore that the pitch of any sound whatever depended on the frequency of the aerial pulses. In this way alone the sound of a string, of a bell, of an organ pipe, and the bellow of a bull, may have the same pitch. He could not, however, demonstrate this in any case but the one above mentioned. But he was encouraged to hope that mathematicians would be able to demonstrate it in all cases, by his having observed that the same proportions obtained in organ pipes as in strings stretched by equal weights. But it required a great progress in mechanical philosophy, from the state in which Galileo found it, before men could speculate and reason concerning the pulses of air, and discover any analogy between them and the vibrations of a string. This analogy, however, was discovered, and its demonstration completed, as we shall see by and by. In the meantime, Galileo's demonstration of the vibrations of elastic cords became the foundation of all musical philosophy. It must be thoroughly understood before we can explain the performance of the trumpet marine.

The demonstration of Galileo is remarkable for that beautiful simplicity and perspicuity which distinguish all the writings of that great mechanician, and it is the elementary proposition in all mechanical treatises of music. Few of them indeed contain anything more; but it is extremely imperfect, and is just only on the supposition that all the matter of the string is collected at its middle point, and that the rest of it has elasticity without inertia. This did not suit the accurate knowledge of the last century, after Huyghens and Newton had given the world a taste of what might be done by profiting the Gallican mechanics. When a musical cord has its middle point drawn aside, and it is strained into the shape of two straight lines, if it be let go, it will be observed not to vibrate in this form. It may easily be seen in the extremity of its excursions, where it rests, before it return by its elasticity. The reason is this (see fig. B.) When the middle point C of the cord is drawn aside, and the cord has the form of two straight lines AC, CB, this point C, being pulled in the directions CA, CB, at once, is really accelerated in the direction CD, which bisects the angle ACB; and if it were then detached from the rest of the material cord, it would move in that direction. But any other point f between C and B has no accelerating force whatever acting on it. It is equally pulled in the directions fC and fB. The particle C therefore is obliged to drag along with it the inert matter of the rest of the cord; and when it has come to any intermediate situation e, the cord cannot have the form of two straight lines A e, e B, with the particle f situated in f. This particle will be left somewhat behind, as in e, and the cord will have a curved form A e f B; and in this form it will vibrate, going to the other side, and assuming not the rectilineal form ADB, but the curved form A f B. Trumpet A.B. That every particle of the curve A.C.B is now accelerated toward the axis A.B is evident, because every part is curved, and the whole is strained toward A and B, which tends to straighten every part of it. But in order that the whole may arrive at the axis in one moment, and constitute a straight line A.B, it is evidently necessary that the accelerating force on every particle be as the distance of the particle from that point of the axis at which it arrives. It is well known to the mathematician that the accelerating force by which any particle is urged towards a rectilinear position, with respect to the adjoining particles, is proportional to the curvature.

Our readers who are not familiar with such discussions, may see the truth of this fundamental proposition by considering the whole of A.C.B as only a particle or minute portion of a curve, magnified by a microscope. The force which strains the curve may be represented by c.A or AE. Now it is well known (and is the foundation of Galileo's demonstration) that the straining force is to the force with which c accelerates in the direction c.E as A.c to c.D, or as AE to c.D, or as AE to twice c.E. Now c.E is the measure of the curvature of A.C.B, being its deflection from a right line. Therefore when the straining force is the same all over the curve, the accelerating force, by which any portion of it tends to become straight, is proportional to the curvature of that portion. And if r be the radius of a circle passing through A, c, and B, and coinciding with this element of a curve, it is plain that c.D : A = c.A : r, or that the radius of curvature is to the element c.A as the extending force to the accelerating force; and c.D = \(\frac{cA^2}{r}\); and is inversely as r, or directly as the curvature.

Hence we see the nature of that curve which a musical chord must have, in order that all its parts may arrive at the axis at once. The curvature at c must be to the curvature at f as E.c to g.f. But this may not be enough. It is farther necessary that when c has got half way to E, the curvature in the different points of the new curve into which the cord has now arranged itself, be also, in every point, proportional to the distance from the axis. Now this will be the case if the extreme curve has been such. For, taking the cord in any other successive shape, the distance which each point has gone in the same moment must be proportional to the force which impelled it; therefore the remaining distances of all the points from the axis will have the same proportions as before. And the geometrical and evident consequence of this is, that the curvatures will also be in the same proportion.

Therefore a cord that is once arranged in this form will always preserve it, and will vibrate like a cycloidal pendulum, performing its oscillations in equal times, whether they be wide or narrow. Therefore since this perfect isochronism of vibrations is all that is wanted for preserving the same musical pitch or tone, this cord will always have the same note.

This proposition was the discovery of Dr Brooke Taylor, one of the ornaments of our country*, and is published in his celebrated work Methodus Incrementorum. The investigation, however, and the demonstration in that work, are so obscure and so tedious that few had patience to peruse them. It was more elegantly treated afterwards by the Bernoullis and others. The curve got the name of the Tayloren curve; and is considered by many eminent mathematicians as a trochoid, viz., the curve described by a point in the nave or spoke of a wheel while the wheel rolls along a straight line. But this is a misfortune, although it is allied to the trochoid in the same manner that the figure of lines is allied to the cycloid. Its physical property entitles it to the name of the Harmonical Curve. As this curve is not only the foundation of all our knowledge of the vibration of elastic cords, but also furnishes an equation which will lead the mathematician through the whole labyrinth of aerial undulations, and be of use on many other occasions; and as the first mathematicians have, through inattention, or through enmity to Dr Taylor, affected to consider it as the trochoid already well known to themselves—we shall give a short account of its construction and chief properties, simplified from the elegant description given by Dr Smith in his Harmonics.

Let SDTV, QERP (fig. C.), be circles described round the centre C. Draw the diameters QCR, ECP, cutting each other at right angles. From any point G in the exterior circle draw the radius GC, cutting the interior circle in F; draw KHF parallel to QCR, and make HI, HK, each equal to the arch EG. Let this be done for every point of the quadrant arch EGR. The points I, K, are in the harmonic curve; that is, the curve AKDIB passing through the points K and I, determined by this construction, has its curvature in every point K proportional to the distance KN from the base AB.

To demonstrate this, draw FL perpendicular to the axis, and join EL. Take another point g in the outer circle indefinitely near to G. Draw gc, cutting the inner circle in f, and fb and fl perpendicular to DC, CT, and join EL. Then suppose two lines K.m', K.m' perpendicular to the curve in K and k. They must meet in m', the centre of the equicurve circle. Draw KN.n' perpendicular to the base, and m'.n' parallel to it, and join kn. Lastly, draw XL.x perpendicular to EL.

It is plain that k.O, the difference of HK and b.k, is equal to G.g, the difference of GE and g.E, and that KO is equal to Fr, and L.l to r.f. Also, because ELX is a right angle, EX = \(\frac{EL}{EC}\).

We have Fr : Ff = CL : CF, = CL : CD, Ff : Gg = CD : CE.

Therefore Fr : Gg, or KO : Ok = CL : CE.

The triangles ECL and kOK are therefore similar, as are also kOK and K.m, and consequently ECL and K.m; and because EC is parallel to Kn, EL is parallel to Km. For the same reason km is parallel to El, and the triangles Elx and m.Kk are similar, and

\[Lx : Kk = LE : KM,\] and \(Lx : Kk = EC : Kn\). But farther,

\[Lx : Ll = CE : CL,\] \[Ll : Ff = KN : CD, \text{ being } FL : FC,\] \[Ff : Gg = CD : CE, \text{ being } Ff : KO,\] \[Gg : Kk = CE : CL, \text{ being } KO : Kk.\]

Therefore \(Lx : Kk = KN \times CE : EL = KN : EX\).

Therefore \(KN : EX = LE : KM\), and \(KM = \frac{EX}{KN}\),

and \(KN : EX = CE : Kn\), and \(Kn = \frac{EX \cdot E}{KN}\). In the very narrow vibrations of musical cords, CD is exceedingly small in comparison with CE, so that EXEL or EXCE may, without sensible error, be taken for CE, and then we obtain KM or KN (which hardly differ) = \(\frac{CE^2}{KN}\), and therefore the curvature is proportional to KN. The small deviation from this ratio would seem to show that this construction does not give the harmonic curve with accuracy. But it is not so. For it will be found that although the curvature is not as KN, it is still proportional to the space which any particle K must really describe in order to arrive at the axis. These paths are lines whose curvatures diminish as they approach to DC.

We see, \(1/7\), that the base ACB of the curve is equal to the semicircular arch QER.

\(2d\). Also that the tangent KZ in any point K is perpendicular to EL.

\(3d\). We learn that the curvature at A and B is nothing, for in these two points KN is nothing.

\(4th\). The radius of curvature at D is precisely \(= \frac{CE^2}{CD}\).

Therefore, as the string approaches the axis, and CD diminishes, the curvature diminishes in the same proportion. The vibrations therefore are performed like those of a pendulum in a cycloid, and are isochronous, whether wide or narrow, and therefore the musical pitch is constant.

This is not strictly true, because in the wide vibrations the extension or extending force is somewhat greater. Hence it is that a string when violently twanged sounds a little sharper at the beginning. Dr Long made a harpsichord whose strings were stretched by weights, by which this imperfection was removed.

It is proper to exhibit the curvature at D in terms of the length AB, and of the greatest excursion cD. Therefore let \(c\) be the circumference of a circle whose diameter is \(t\). Let AB the length of the cord be \(L\), and let CD the breadth of the vibration be \(B\).

We had a little ago \(Dm = \frac{CE^2}{CD}\), but \(c : t = AB : CE\), and \(CE = \frac{AB}{c}\), and \(cE^2 = \frac{AB^2}{c^2}\). Therefore \(Dm = \frac{AB^2}{c^2} \times CD = \frac{L^2}{987CD}\) nearly.

We can now tell the number of vibrations made in a second by a string. This we obtain by comparing its motion, when impelled by the accelerating force which acts on it, with its motion when acted on by its weight only. Therefore let L be the length of a string, and W its weight, and let E be the drawing weight, or extending force. Let \(f\) be the force which accelerates the particle Dd of the cord, and \(w\) the weight of that particle, while W is the weight of the whole cord. Let \(z\) be the space which the particle Dd would describe during the time of one vibration by the uniform action of the force \(f\), and let S be the space which it would describe in the same time by its weight \(w\) alone. Then (Dynamics, Suppl. n° 103, cor. 6.) the time in which \(f\) would impel the particle Dd along \(DC\) is to the time of one vibration as \(1 : c\). And \(DC\) is to \(z\) as the square of the time of describing \(DC\) is to the square of the time of describing \(z\); that is, \(1 : c^2 = \frac{DC}{2z}\), and \(c^2DC = 2z\).

Now, by the property of the harmonic curve,

\[AB : Dm = 2z : AB\]

But \(Dm : Dd = E : f\)

And \(Dd : AB = w : W\)

Therefore \(2z : Ew = AB : W\)

And \(f : w = 2z : E \times W\)

But \(w : f = 2S : 2z\)

Therefore \(2S \times E = AB \times W\)

And \(2E : W = AB : S\).

That is, a musical cord, extended by a force E, performs one vibration DCV in the time that a heavy body describes a space S, which is to the length of the cord as its weight is to twice the extending force.

Now let \(g\) be the space through which a heavy body falls in one second; and let the time of a vibration (estimated in parts of a second) be T. We have

\[AB : S = 2E : W\]

\[S : g = T : 1\]

Therefore \(AB : g = 2ET : W\)

And \(AB \times W = T^2 \times 2E \times g\)

Therefore \(T^2 = \frac{AB \times W}{2gE}\), and \(T = \sqrt{\frac{AB \times W}{2gE}}\).

Let \(n\) be the number of vibrations made in a second.

\[n = \frac{1}{T} = \sqrt{\frac{2gE}{AB \times W}} = \sqrt{\frac{2gE}{LW}}.\]

If the length of the cord be measured in feet, \(2g\) is very nearly 32. If in inches, \(2g\) is 386, more nearly.

Therefore \(n = \sqrt{\frac{32E}{LW}}\) or \(\sqrt{\frac{386E}{LW}}\). This may easily be compared with observation. Dr Smith hung a weight of 7 pounds, or 49,000 grains, on a brass wire suspended from a finger pin, and shortened it till it was in perfect unison with the double octave below the open string D of a violin. In this state the wire was 35 inches long, and it weighed 31 grains.

Now \(\sqrt{\frac{384 \times 49000}{35 \times 31}} = 1307 = n\). This wire, therefore, ought to make 1307 vibrations in a second. Dr Smith proceeded to ascertain the number of aerial pulses made by this sound, availing himself of the theory of the beats of tempered consonances invented by himself. On his fine chamber organ he tuned upwards the perfect fifths DA, A, C, &c., and then tuned downward the perfect sixth cd. Thus he obtained an octave to D, which was too sharp by a comma, and he found that it beat 65 times in 20 seconds. Therefore the number of vibrations was \(\frac{65}{20} = 81\), or 263,25. These were complete pulses or motions from D to V and back again, and therefore contained 526 such vibrations as we have now been considering. The double octave below should make \(\frac{5}{2}\)th of this, or 131,6, which is not a complete vibration more than the above theory requires; more accurate coincidence is needless.

This theory is therefore very completely established, and it may be considered as one of the finest mechanical problems which has been solved in this century. We mention it with the greatest minuteness, because the merit of Dr Taylor is not sufficiently attended to. Mr Rameau, and the other great theorists in music, make no mention of him; and such as have occasion to speak of the absolute number of vibrations made by any musical note, always quote Mr Sauveur of the French academy. This gentleman has written some very excellent dissertations on the theory of music, and Sir Isaac Newton in his *Principia* often quotes his authority. He has given the actual determination of the number of vibrations of the note C, obtained in a manner similar to that practised by Dr Smith on his chamber organ, and which agrees extremely well with that measure.

But Mr Sauveur has also given a mechanical investigation of the problem, which gives the same number of vibrations that he observed. We presume that Rameau and others took the demonstration for good; and thus Mr Sauveur passed on the Continent for the discovery of this theorem. But it was not published till 1716, though read in 1713; whereas Dr Taylor's demonstration was read to the Royal Society in 1714. But this demonstration of Mr Sauveur is a mere paradoxism, where errors compensate errors; and the assumption on which he proceeds is quite gratuitous, and has nothing to do with the subject. Yet John Bernoulli, from enmity to Taylor and the English mathematicians, takes not the least notice of this sophistical demonstration, accommodated to the experiment, and so devoid of any pretensions to argument that this severe critic could not but see its fallacy.

Sauveur was one of the first who observed distinctly that remarkable fact which Mr Rameau made the foundation of his musical theory, viz., that a full musical note is accompanied by its octave, its twelfth, and its seventeenth major. It had been casually observed before, by Mersenne, by Perauld, and others; but Sauveur tells distinctly how to make the observation, and affirms it to be true in all deep notes. Rameau affirms it to be universally and necessarily true in all notes, and the foundation of all musical pleasure.

It had been discovered before this time, that not only a full note caused its union to resound, but also that a 12th, being founded near any open string, the string resounded to this 12th. It does the same to a 15th, a 17th major, a 22d, &c.

Dr Wallis added a very curious circumstance to this observation. Two of his pupils, Mr Noble and Mr Pigott, in 1673, amusing themselves with these resonances, observed, that if a small bit of paper be laid on the string of a violin which is made to resound to its union, the paper is thrown off; a proof that the string resounded by really vibrating, and that it is thrown into these vibrations by the pulses of the air produced by the other string. In like manner the paper is thrown off when the string resounds to its octave. But the young gentlemen observed, that when the paper was laid on the middle point of the string, it remained without agitation, although the string still resounded. They found the same thing when they made the string resound to its 12th: papers laid on the two points of division lay still, but were thrown off when laid on any other place. In short, they found it a general rule, that papers laid on any points of division corresponding to the note which was resounded, were not agitated.

Dr Wallis (the greatest theorist in music of the last century) justly concluded that these points of the resounding string were at rest, and that the intermediate parts were vibrating, and producing the notes corresponding to their lengths.

From this Mr Sauveur, with great propriety, deduced the theory of the performance of the trumpet marine, the vielle, the clavichord, and some other instruments.

When the string of the trumpet marine is gently flopped at $\frac{1}{4}$, and the bow drawn lightly across it at H (fig. A), the full vibration at the finger is flopped; but the string is thrown into vibrations of some kind, which will either be destroyed or may go on. It is of importance to see what circumstance will permit their continuance.

Suppose an elastic cord put into the situation ABCDE (fig. D), such that AB, BC, CD, DE, are all equal, and that BCD is a straight line. Let the point C be made fast, and the two points B and D be let go at once. It is evident that the two parts will immediately vibrate in two harmonical curves AC and CDE, which will change to ABC and CDE, and so on alternately. It is also evident that if a line FCG be drawn touching the curve ABC, it will also touch the curve CDE; and the line which touches the curve ABC in C, will also touch the curve CDE. In every instant the two halves of the cord will be curves which have a common tangent in the point C. The undoubted consequence of this is, that the point C will not be affected by these vibrations, and its fixture may be taken away. The cord will continue to vibrate, and will give the sound of the octave to its fundamental note.

The condition, then, which must be implemented, in order that a string may resound to its octave, or take the sound of its octave, is simply this, that its two parts may vibrate equally in opposite directions. This is evidently possible; and when the bow is drawn across the string of the trumpet marine at H, and irregular vibrations are produced in the whole string, those which happen to be in one direction on both sides of the middle point, where it is gently flopped by the finger, will destroy each other, and the remaining ones will be instantly produced, and then every succeeding action of the bow will increase them.

The same thing must happen if a string is gently flopped at one-third of its length; for there will be the same equilibrium of forces at the two points of division, so that the fixtures of these points may be removed, and the string will vibrate in three parts, founding the 12th of the fundamental.

We may observe, by the way, that if the bow be drawn across the string at one of the points of division, corresponding to the flopping at the other end of the string, it will hardly give any distinct note. It rattles, and is intolerably harsh. The reason is plain: The bow takes some hold of the point C, and drags it along with it. The cord, on each side of C is left behind, and therefore the two curves cannot have a common tangent at C. The vibrations into which it is thus jogged by the bow destroy each other.

We now see why the trumpet marine will not sound every note. It will sound none but such as correspond to a division of the string into a number of equal parts, and its note will be in unison with a string equal to one of those parts. Therefore it will first of all sound the fundamental, by its whole length;

1. Its octave, corresponding to $\frac{1}{2}$ its length 2. The 12th, 3. The 15th, or double octave, 4. The 17th, 5. The 19th, 6. The 22nd,

7. The 7. The 21st, which is not in the diatonic scale of our music, \( \frac{1}{3} \) its length. 8. The triple octave, or 22d, \( \frac{1}{3} \) 9. The 24th, or 2d in the scale of the triple octave 10. The 25th, or 2d in this scale, \( \frac{1}{3} \) 11. The 26th, a false 4th of this scale, \( \frac{1}{3} \) 12. The 26th, a perfect 5th of this scale, \( \frac{1}{3} \) 13. The 27th, a false 6th of ditto, \( \frac{1}{3} = \frac{1}{3} \) 14. The 28th, a false 7th minor, \( \frac{1}{3} \) 15. The 28th, a perfect 7th major, \( \frac{1}{3} \) 16. The quadruple octave.

Thus we see that this instrument will not execute all music, and indeed will not complete any octave, because it will neither give a perfect 4th nor 6th. We shall presently see that these are the very defects of the trumpet.

This singular stringed instrument has been described in this detail, chiefly with the view of preparing us for understanding the real trumpet. The Vielle, Savoyarde, or Hurdygurdy, performs in the same manner. While the wheel runs one part of the string like a bow, the keys gently press the strings, in points of aliquot division, and produce the harmonic notes.

It is to prevent such notes that the part of harpichord wires, lying between the bridge and the pins, are wrapped round with silk. These notes would frequently disturb the music.

Lastly on this head, the Aeolian harp derives its vast variety of fine sounds from this mode of vibration. Seldom do the cords perform their fundamental or simple vibrations. They are generally sounding some of the harmonics of their fundamentals, and give us all the variety from strings tuned in unison.

Trumpet, Musical, is a wind instrument which sounds by pressing the closed lips to the small end, and forcing the wind through a very narrow aperture between the lips. This is one of the most ancient of musical instruments, and has appeared in all nations in a vast variety of forms. The couch of the savage, the horn of the cowherd and of the poitman, the bugle horn, the luteus and tuba of the Romans, the military trumpet, and the trombone, the cor de chasse or French horn—are all instruments winded in the same manner, producing their variety of tones by varying the manner and force of blowing. The serpent is another instrument of the same kind, but producing part of its notes by means of holes in the sides.

Although the trumpet is the simplest of all musical instruments, being nothing but a long tube, narrow at one end and wide at the other, it is the most difficult to be explained. To understand how sonorous and regulated undulations can be excited in a tube without any previous vibration of reeds to form the waves at the entry, or of holes to vary the notes, requires a very nice attention to the mechanism of aerial undulations, and we are by no means certain that we have as yet hit on the true explanation. We are certain, however, that these aerial undulations do not differ from those produced by the vibration of strings; for they make strings resound in the same manner as vibrating cords do. Galileo, however, did not know this argument for his assertion that the musical pitch of a pipe, like that of a cord, depended on the frequency alone of the aerial undulations; but he thought it highly probable, from his observations on the structure of organs, 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 aerial 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, founding 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, though 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 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 lenimentum. 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 flopped (being in a rarefied state), and is then prefled 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 flopped 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 produce 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 flopping 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 creak changes to a chirp, of which we can tell the note. The sound 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. 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 aerial undulations with the undulations of water in an open canal. While the water is flowing smoothly along, suppose a flake 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 flake, by the water's going along the canal, and a heap up of the water on the other side. By properly timing the motion of this flake up and down, we can produce a series of connected waves. If the flake 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 sound 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 flares. 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 propelled 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 sound 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 shall 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, show, 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 flute pipe AB (fig. 1.), suppose a harmonic curve ACB, or Musical Trumpet

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 c. 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 d above that of the stratum f may be to the excess of the pressure of DD above that of FF as c to CE. In this case, the stratum c will be accelerated in the direction ef, and the stratum EE 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 ee, 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 recession from it. Let us next consider a plate aa more remote from A. It must make similar vibrations from the situation bb to the situation cc. But these vibrations must be greater in proportion as the plate is farther from A. It cannot be conceived otherwise: For suppose the plate aa 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 aa 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 aa is left condensed by the action of the plates beyond it. The air at B is always of its natural density, being in equilibrium 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 passed 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 passed 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 equilibrium. It will perfectly resemble the point C of the musical cord BFGGH (fig. 6.), which is in equilibrium 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 flender 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. 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 unison 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 α₁, α₂, 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 equilibrium, 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 equilibrium, 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 fill the same note that AB would have given alone.

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 3/4, 1/4th, 1/8th, &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 synchronous and synchronous.

It is known that the grave 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. (a).

Again, let AD (fig. 6) represent the length of a pipe. Construct on AD an harmonic curve AEBFCGH, in such a manner that HD may be 1/2 AB, = 1/2 BC, = 1/2 CH. The small ordinates m n will express the total excursion of the plates of air at the points m, n, &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 Bights 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 1/4th or 1/8th of the length. Therefore, when we cause the same pipe to found 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 found the octave to any of its notes, it was not sufficient to shorten its capacity one-half by pulling down the piston; he found that the part remaining must be less than the part taken off by a fixed quantity 1/2 inches. Or, the length which gave any note being x, the length for its octave must be \( \frac{x - 1}{2} \). from the undulatory motion of a pulse, investigated by Newton, and described in the article Acoustics, Encycl. 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, founding 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 AD of fig. 2. joined to CA of fig. 4. may found 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 equilibrium, the air must be coming in or going out at the extremities together. This circumstance must give some sensible difference of character to the sounds 4 and 6. Trumpet. 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 1/3, 1/5, 1/7, &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 founding its octave, it will be represented by CP, and its undulations will resemble the vibrations of a cord CP, whose length AE is 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, 1/4, 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 flounders on it as easily as on any other. But the musician, finding this sound begin with him, 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 9/2 and 15/2, which are the 2d and 3d of this octave. The next note 11/2 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 fleps 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 Now, let a pendulum, whose quantity of matter is $L$, and length $a$, be supposed to vibrate in a cycloid by the force $\frac{4}{E} \cdot E$, or $\frac{4}{L} \cdot a$. 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 $\frac{4}{E} \cdot a : L = a : \frac{4}{L} \cdot a$, 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 $E$, 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 $\frac{1}{2} L$; if three, the weight of each must be $\frac{1}{3} L$; and so on for any number.

By considering this attentively, we may infer, without farther 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, founding 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... We look on this as the easiest way of obtaining a distinct perception of the authority on which we rely 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 proficient in 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 grams, 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 grams.

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 : 290060. This wire, therefore, should be stretched (if the theory be just) by 290060 grams, in order to be unison with the other wire, and we should have 35.55 : 86.4 = 49000 : 290060

But, in truth, 35.55 : 86.4 = 49000 : 289480

The difference is 5120

The error scarcely exceeds 1/50, 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 aerial 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 \( \sqrt{\frac{386H}{LW}} \), 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 \( \frac{HW}{L} = E \).

Therefore substituting \( \frac{HW}{L} \) for \( E \) in the above formula, we have the number of aerial pulses made per second \( = \sqrt{\frac{386H}{L^2}} \), or \( \sqrt{\frac{386H}{L^2}} \). Now \( \sqrt{386H} \), computed in inches, is 11331. Therefore, if we also measure the length of the pipe \( L \) in inches, the pulses in a second are \( \frac{11331}{L} \). Thus, in the case before us,

\( \frac{11331}{86.4} = 131.12 \), or this pipe produces 131 pulses in a second. Dr Smith found by experiment that it produced 1309, differing only about \( \frac{1}{4} \)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 harpsichord 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 \( \frac{960 \times 12}{262} = 44 \) 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 \( \frac{1}{4} \)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 expla- 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 node 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 flat, which is also the sound of an open flute-pipe of that length.

The Trombones, great trumpet, or Sackbut, is an old instrument described by Mercennius 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 bass of a harpsichord 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 flat 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 too 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 assert 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, Encyclo.) 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 harmonies? 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 everything 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 founding the 12th to its fundamental, it might at the same time be also vibrating as a simple cord, and founding 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 Musical Trumpet.

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Musical Trumpet. 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 anything 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 *fourth dominant*, or 4th of the scale, and the 6th where there is no resonance. He says that these notes, "fremissant, quoique elles ne renforcent 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 harpsichord 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 noticed at as the fourth dominant.

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 fingers, and all simple nations, are examples. They not only fix on certain intervals as the steps of their tunes, but are delighted 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 cantus 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, polkaed 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 himself (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, founded 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 flopped, but the string continued to vibrate in two parts, founding 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 founded 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 founding 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 found the 12th very strongly in its neighbourhood, the string instantly divided itself, and vibrated in union, 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 Aeolian 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 necesary 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 ribbon 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 sounded. 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 too 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.