‘The whole theory or principle of finding and producing the harmonic notes is in reality very simple, and such as might be communicated to any intelligent child in two or three short lessons. If the author of the Political Register had been born and bred a professional musician, (as among the possible freaks of fortune why should he not?) he would have set the hope of his family before him, and said,

‘My dear little Son,

‘You are to get your bread by playing on the violin. It will therefore be exceedingly useful to you to know all that can be known about the harmonic notes; by which means you may not only get your bread, but be able to secure its being well buttered also. A violin-player is worth a great deal more when he knows all about the harmonic notes; and in fact, since the appearance of Paganini, the chances are, that a player who does not know it will be worth nothing at all.

‘Do you know what an aliquot part is? I am sure you do not. If you have a cake or an apple, and divide it equally among your companions, whether they be two, three, four, or any other number, then the thing is said to be divided into aliquot parts,—“aliquot” being a word in the old Latin language meaning “some certain number or other,” and implying here that the thing is divided into equal parts of “some certain number or other.” But if you were to divide it among the same so that their shares should not be all alike,—or if you were to give each an equal piece, but there should be a piece left after all which was not equal to one of the pieces you had given away, but was greater or less,—then the thing would be divided into parts, but not into aliquot parts. Now then, my dear little son, you know what is meant by dividing a string into aliquot parts.

‘Tell me now, how you would begin to show me the different places in which a string can be divided into aliquot parts. You would first show me the middle point, which divides it into two equal parts. Then you would divide the string, with your eye or with a pair of compasses, into three equal parts, and show me the two points of division between them. Next you would divide it in the same way into four equal parts, and show me the three points of division. And so on, for five, six, seven, eight, and as many more as you liked to continue. These, then, you would say,—both those I have made and those I might make if I liked—are the points that divide the string into aliquot parts. And if you pleased, you might mark them by writing under each point of division the figure which shows how many equal parts the string is divided into,—as for instance a 2 under the point where the string is divided into two, a 3 under each of the points which divide it into three, and so on. And indeed it will be better that you should do this; for then you cannot help observing, that sometimes more figures than one will fall on the same place—as for instance when the string is divided into four, one of the marks 4 will fall on the same place as the division into 2; when it is divided into six, one of the marks 6 will fall on the same place that was previously marked 2, and two more on places that were marked 3; and so on. All of which will be wanted another time.

‘Now if you touch the string gently with the finger at the distance of any aliquot part from the bridge, (mind I said from the bridge, not at any of the divisions into aliquot parts, but at the distance of one of them from the bridge,) and at the same time pull the string or draw the bow across between this point and the bridge, you will see a curious thing. The string will divide itself into all the aliquot parts of which the point touched by the finger makes one,—into two, or into three, or into four, as the case may be,—and every one of them will move by itself, as if it was a little string held fast at the two ends; the sound produced being the same that would be made by pressing the string down to the neck at the point touched, in the common way. If the divisions are few, as two or three, this may be seen distinctly enough by the eye: but where this is not the case, it may be shown to be the fact by laying a little bit of paper on the string while it is sounded; and if this is laid on any of the points of division into aliquot parts, whether on the one nearest the bridge or any of its fellows, it will lie still and not be thrown off, but if it is laid anywhere else, it will be thrown off directly, which shows that the points of division are at rest, and the others are not.

‘If you want to know how or why this curious thing takes place, I will tell you as nearly as I can; but remember I do not pledge myself that this is the reason, but only that I think it very likely to be the reason, and this principally because I know no other way in which it can be brought about. And this way is, that when one portion of the string is moving in one direction, as for instance from me towards you, the next portion of the string is moving at the same time in the contrary direction, or from you to me; and so with the other portions, whatever their number may be. In this manner it seems possible that the points of division should be kept at rest, and in any other manner it seems to be not possible; and therefore, since the fact is before us that the points of division remain at rest, I conclude that it is in this way it takes place. This is what the feelosofers would call a syllogism. And because this sort of balance can only be kept up by the portions of the string moving backwards and forwards (which the same sort of people call vibrating) in equal times or with equal quickness, and this again cannot take place unless the moving portions of the string are of equal length,—it follows that this sort of motion in parts or portions of the string can only take place when those parts or portions are of equal length, which seems to be the reason why the experiment will only answer when the point touched is one that divides the string into aliquot parts.

‘But this is not all; for there is a more curious thing still. And that is, that if you touch the string at any other of the points of division into aliquot parts, (by which I mean any other than the point of division nearest to the bridge,) the string will divide itself in the self-same way,—always with the exception (now mind the exception) of the cases in which the point touched falls in with a point in some simpler mode of division that has gone before. For instance, you remember observing, that when the string was divided into four equal parts, one of the points marked 4 fell on the same place as the division into 2. Touching the string therefore in this place must make the same sound it did before; which is a different sound from that which it makes when touched at the other two points of division into 4. And in like manner in other cases. But when this agreement with some simpler mode of division does not interfere, all the points of division on being touched produce the same sound. For example, if the division be into five equal parts, inasmuch as none of these will coincide with any of the simpler modes of division, there must be four points in the string, any one of which being touched will produce the same harmonic sound.

‘But if you want to know how and why this still more curious thing takes place, I can only tell you in a roundabout sort of way as before. If you divide the string, for example, into five equal parts, and touch any of the four points of division you choose, you check and finally prevent the continuance of any motion at the point touched, though at the same time it would appear that the touching (which, to make the experiment answer, must be very light) is not enough to hinder the shaking, or, as the learned people call it, the vibration, given at one end, from being communicated past the point of touch. If, instead of touching the string lightly, you were to lay hold of it with a pair of pincers, then the experiment would fail altogether; the reason of which may be concluded to be, because the motion is presented from being at all communicated beyond the point laid hold of. In fact the art,—for there is an art in everything, from scraping the grains off a cob of Indian corn to sounding a musical string, whatever the difference in importance and dignity of the two things may be,—appears to consist in touching the string in such a manner, and with such a degree of pressure, as shall allow the motion given by pulling or bowing to be communicated past the finger, and yet shall check and finally prevent the continuance of all motion, or, as it was called before, vibration, that is not consistent with the point which is touched remaining at rest. Now if you consider carefully, you will see that the only way in which motion can go on and this point remain at rest, is by the string’s dividing itself into the five equal portions, the movements of which shall balance each other as before described. It does not indeed follow, that because the motion could go on no other way, it must necessarily go on in this; but we have the evidence of the fact that it does go on in this; and the knowledge of the reasons why it could not go on in any other is at all events very useful to make us remember what the effect is that is produced, and how.

‘The next thing is to be able to tell what all the sounds thus produced are. Now you remember that when you were a very little boy, I showed you, that if you stop a string by pressing it down hard in the middle you produce its Octave; where the two sounds (of the original string and its half) are such sounds as are produced by a man and a child when they sing the same tune together, but in very different pitches of voice;—that if, instead of shortening the string in this manner by the half, you shorten it by a third part, you produce the sound which musicians have called the Fifth; if you shorten it by a fourth part, you produce the Fourth; if by the fifth part, the Major Third; if by the sixth part, the Minor Third; with a great deal more which it is not necessary to mention now;—and I told you, too, that the intervals from one of these sounds to another were not the same, or such as to allow of beginning on any you please and making the others serve in the places they happen to fall in, which is attempted to be done by what is called Temperament, a thing that you as a violin player should hold in as much scorn, as an invitation to cut off your two legs for the sake of trying how pleasant it is to hop on wooden ones. If then you want to know what sound any of the harmonics really is, you have only to do this;—double the distance from the bridge to the nearest of the points of division into aliquot parts, over and over, till you get to some length that when pressed down in the common way makes a note which you know, as the Octave, the Fifth, &c.; and then the harmonic will be this note, only raised by as many octaves as there have been doublings. For example, if you touch the thickest or G string of the violin so as to bring out the harmonic at one-fifth of its length from the bridge, and want to know what note this is,—doubling this length once makes two-fifths of the whole string, and doubling it again makes four-fifths, and four-fifths pressed down in the common way make the Major Third or B; therefore the harmonic produced is B two octaves higher than the B on the thickest string, or the same sound as the first B on the thinnest or E string. And in like manner in other cases.

‘The examination of all the different possible harmonic notes might evidently be carried a long way; and it would be very useful to do it if you were intended for a trumpeter, for all the notes on the trumpet or French horn are harmonic notes. But for playing on the violin, as much as is given above appears to be sufficient. It will enable you to trace all the principal harmonic sounds, and in fact all that on the violin are of any practical use; for though there is no absolute end of the number of harmonic notes, inasmuch as you may divide the string into a hundred parts if you please, and then into a hundred-and-one,—yet after the division into five or into six, the sounds on the violin become so feeble as to be of no use except as matters of experiment and curiosity. And it will have this further good effect, that it will make you cease to marvel and to wonder at finding the harmonic sounds on the same string grow sometimes deeper and sometimes shriller, as you move your finger from the bridge towards the head,—as if there was some mystery in it that anybody could not learn in half an hour when they set about it properly.

‘Suppose now you could stop some tune (as for instance “God save the King”) on one string of the violin, as for example the fourth, with your first or second finger, and at the same time always touch the stopped string gently with the little finger of the same hand at one quarter of the way to the bridge so as to bring out the harmonic note;—is it not plain that you would play the tune, only in the Double Octave, or two octaves higher than if played by the simple stopping on the fourth string? There is no doubt that this is very hard, especially for a little boy; it is almost as bad as playing on two violins at once. But still the thing can be done. And if, instead of touching with the little finger at the quarter of the way to the bridge, you should touch at the third, the fifth, or the sixth of the way, you would bring out notes that were not Double Octaves to the sound that would be made by simply pressing down the first finger, but other sounds, which you have it in your power to calculate; all of which might by possibility be very useful, but the other was mentioned as being the simplest. If you asked me what is the use of playing anything in Double Octaves in this manner, or in any other of the harmonic notes,—I should answer, First, because these harmonic notes have a very fine and pure sound,—they do not squall like the sounds made by pressing the strings to the finger-board very near the bridge;—Secondly, because it is much easier to make the sounds in tune in this manner, than by trying to make them by stopping near the bridge,—for where the string is so short, the smallest error in the stopping becomes sensible in proportion;—Thirdly, because (as it is not necessary to be always playing in harmonics) they may be mixed up with the common notes of the violin, and save an immensity of trouble in jumping from one end of the instrument to the other to find the high notes. Look, for instance, at an old-fashioned fiddler playing on the second string, and wanting (suppose) A in alto; and see what a leap he will make to find it on the first string, and what a horrible screech he will bring out after all, when he might produce the note in the most perfect tune and tone by only touching the second open string that he is on already, harmonically at a fourth of the way from the head to the bridge, or at the same place that he would stop D on the second string.’