TRUMPET MARINE, or MARIGNY — Suppl. 3rd (1801)

TRUMPET MARINE, or MARIGNY. 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 any thing 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 $\frac{1}{12}$ of EF, E2 is $\frac{1}{12}$ , and so on. The bow is drawn lightly across the cord at H, and the string is stopped 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 be stopped, in the way now described, at $\frac{1}{4}$ of its length from E, it yields the 12th of the fundamental; if stopped at $\frac{1}{2}$ th, it gives the double octave; if at $\frac{2}{3}$ 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 harmonics, 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 $\frac{1}{12}$ 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 mean time, 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 distinguishes all the writings of that great mechanician, and it is the elementary proposition in all mechanical treatises of music. Few of them indeed contain any thing 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 prosecuting the Galilean 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 rectilinear form ADB, but the curved form A e B.

A B. That every particle of the curve $A c c' B$ is now accelerated toward the axis AB 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 AB, 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 is accelerated 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 : c 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 $c D = \frac{c A^2}{r}$ ; 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 Taylorian 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 mistake, although it is allied to the trochoid in the same manner that the figure of lines is allied to the cycloid. Its physical property intitles 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 KHFI parallel to QCR, and make HI, HK, each equal to the arch EG. Let this be done for every point of the quadrantal 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 g c, cutting the inner circle in f, and f b and f l perpendicular to DC, CT, and join E l. 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 K N n' perpendicular to the base, and m' n' parallel to it, and join k n. Lastly, draw X l 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 F r, and L l to r f. Also, because ELX is a right angle, $EX = \frac{EL^2}{EC}$ .

We have $Fr : Ff = CL : CF = CL : CD$ .

$Ff : Gg = CD : CE$

Therefore $Fr : Gg$ , or $KO : O k = CL : CE$ .

The triangles ECL and kOK are therefore similar, as are also kOK and K n m, and consequently ECL and K n m; and because EC is parallel to K n, EL is parallel to K m. For the same reason k m is parallel to E l, and the triangles E l x and m K k are similar, and

$Lx : Kk = LE : Km$ ,

and $Lx : Kk = EC : Kn$ . But farther,

$Lx : Ll = CE : CL$

$Ll : Ff = KN : CD$ , being $= FL : FC$

$Ff : Gg = CD : CE$ , being $= Ff : kO$

$Gg : Kk = CE : CL$ , being $= KO : Kk$ .

Therefore $Lx : Kk = KN \times CE : EL^2 = KN : EX$ .

Therefore $KN : EX = LE : Km$ , and $Km = \frac{EX \cdot LE}{KN}$

and $KN : EX = CE : Kn$ , and $Kn = \frac{EX \cdot CE}{KN}$

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 CE², and then we obtain Km or Ks (which hardly differ) = $\frac{CE^2}{KN}$ , and therefore the curvature is proportional to KN. The small deviation from this ratio would seem to shew 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^st, that the base ACB of the curve is equal to the semicircular arch QER.

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

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

4^th. 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 harpichord 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 1. Let AB the length of the cord be = L, and let CD the $\frac{1}{2}$ breadth of the vibration be B.

We had a little ago $Dm = \frac{CE^2}{CD}$ , but $c : 1 = AB : CE$ , and $CE = \frac{AB}{c}$ , and $cE^2 = \frac{ABc}{c^2}$ . Therefore $Dm = \frac{AB^2}{c^2 \times CD} = \frac{L^2}{9.87 CD}$ 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 straining 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^o 103. cor. 6.) the time in which f would impel the particle Dd along $\frac{1}{2} DC$ , is to the time of one vibration as $1 : c$ . And $\frac{1}{2} DC$ is to z as the square of the time of describing $\frac{1}{2} DC$ , is to the square of the time of describing z; that is, $1 : c^2 = \frac{1}{2} DC : 2z$ , and $c^2 \cdot DC = 2z$ .

Now, by the property of the harmonic curve,

AB : Dm = 2z : AB

\text{But } Dm : Dd = E : f

\text{And } Dd : AB = w : W

\text{Therefore } 2z \cdot E \cdot w = AB \cdot f \cdot W

\text{And } f : w = 2z \times E : AB \times W

\text{But } w : f = 2S : 2z

\text{Therefore } 2S \times E = AB \times W

\text{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^2 : 1^2

\text{Therefore } AB : g = 2E \cdot T^2 : W

\text{And } AB \times W = T^2 \times 2E \times g

\text{Therefore } T^2 = \frac{AB \times W}{2g \cdot E}, \text{ and } T = \sqrt{\frac{AB \times W}{2g \cdot E}}.

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

n = \frac{1}{T} = \sqrt{\frac{2g \cdot E}{AB \cdot W}} = \sqrt{\frac{2g \cdot E}{L \cdot W}}.

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

\text{Therefore } n = \sqrt{\frac{32 \cdot E}{L \cdot W}} \text{ or } \sqrt{\frac{386 \cdot E}{L \cdot W}}. \text{ 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 35.55 inches long, and it weighed 31 grains.

\text{Now } \sqrt{\frac{384 \times 49000}{35.55 \times 31}} = 130.7 = n. \text{ This wire, therefore, ought to make 130.7 vibrations in a second.}

Dr Smith proceeded to ascertain the number of aereal pulses made by this found, 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 e, e b, and then tuned downward the perfect 6th e d. 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 526.5 such vibrations as we have now been considering. The double octave below should make $\frac{1}{2}$ 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 greater 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.

demy. 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 passes on the Continent for the discoverer 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 May 1714. But this demonstration of Mr Sauveur is a mere paralogism, 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 sophisticated demonstration, accommodated to the experiment, and so devoid of any pretensions to argument that this severe critic could not but see its falsity.

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 Marsenius, by Perrault, and others; but Sauveur tells distinctly how to make the observation, and affirms it to be true in all deep notes. Rameau asserts 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 unison 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 unison, 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 stopped at $\frac{1}{4}$ , and the bow drawn lightly across it at H (fig. A), the full vibration at the finger is stopped; 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 ABC 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 fixure 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 stopped by the finger, will destroy each other, and the conspiring 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 stopped at one-third of its length; for there will be the same equilibrium of forces at the two points of division, so that the fixures of these points may be removed, and the string will vibrate in three parts, sounding 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 stopping 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 found every note. It will found 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 found the fundamental, by its whole length;

7. The 21st, which is not in the diatonic scale of our music, $\frac{1}{2}$ its length.
8. The triple octave, or 22d, $\frac{1}{2}$
9. The 23d, or 2d in the scale of the triple octave $\frac{1}{2}$
10. The 24th, or 3d in this scale, $\frac{1}{2}$
11. The 25th, a false 4th of this scale, $\frac{1}{2}$
12. The 26th, a perfect 5th of this scale, $\frac{1}{2}$
13. The 27th, a false 6th of ditto, $\frac{1}{2}$ or $\frac{1}{2}$
14. The 28th, a false 7th minor, $\frac{1}{2}$
15. The 29th, a perfect 7th major, $\frac{1}{2}$
16. The quadraple octave, $\frac{1}{2}$

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 HUANDGUARDY, performs in the same manner. While the wheel rubs 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 Æolian 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 this variety from strings tuned in unison.