How to Make Home-Made HarmonographsHave you ever heard of the harmonograph? If not, or if at the most you have very hazy ideas as to what it is, let me explain. It is an instrument for recording on paper, or on some other suitable surface, the figures described by two or more pendulums acting in concert.
The simplest form of harmonograph is shown in Fig. 168. Two pendulums are so suspended on points that their respective directions of movement are at right angles to one another--that is, pendulum A can swing only north and south, as it were, and pendulum B only east and west. On the top of B is a platform to carry a card, and on the upper end of A a lever is pivoted so as to be able to swing only vertically upwards and downwards. At its end this lever carries a pen, which when at rest lies on the centre of the card platform.
FIG. 168. - Simple Rectilinear Harmonograph.
The bob, or weight, of a pendulum can be clamped at any point on its rod, so that the rate or "period" of swing may be adjusted or altered. The nearer the weight is brought to the point of suspension, the oftener will the pendulum swing to and fro in a given time--usually taken as one minute. From this it is obvious that the rates of swing of the two pendulums can be adjusted relatively to one another. If they are exactly equal, they are said to be in unison, and under these conditions the instrument would trace figures varying in outline between the extremes of a straight line on the one hand and a circle on the other. A straight line would result if both pendulums were released at the same time, a circle, if one were released when the other had half finished a swing, and the intermediate ellipses would be produced by various alterations of "phase," or time of the commencement of the swing of one pendulum relatively to the commencement of the swing of the other.
But the interest of the harmonograph centres round the fact that the periods of the pendulums can be tuned to one another. Thus, if A be set to swing twice while B swings three times, an entirely new series of figures results; and the variety is further increased by altering the respective amplitudes of swing and phase of the pendulums.
[Footnote 1: It should be pointed out here that the presence of friction reduces the "amplitude," or distance through which a pendulum moves, at every swing; so that a true circle cannot be produced by free swinging pendulums, but only a spiral with coils very close together.]
We have now gone far enough to be able to point out why the harmonograph is so called. In the case just mentioned the period rates of A and B are as 2: 3. Now, if the note C on the piano be struck the strings give a certain note, because they vibrate a certain number of times per second. Strike the G next above the C, and you get a note resulting from strings vibrating half as many times again per second as did the C strings--that is, the relative rates of vibration of notes C and G are the same as those of pendulums A and B-- namely, as 2 is to 3. Hence the "harmony" of the pendulums when so adjusted is known as a "major fifth," the musical chord produced by striking C and G simultaneously.
In like manner if A swings four times to B's five times, you get a "major third;" if five times to B's six times, a "minor third;" and if once to B's three times, a "perfect twelfth;" if thrice to B's five times, a "major sixth;" if once to B's twice, an "octave;" and so on.
FIG. 169. - Goold's Twin Elliptic Pendulum Hamonograph
So far we have considered the figures obtained by two pendulums swinging in straight lines only. They are beautiful and of infinite variety, and one advantage attaching to this form of harmonograph is, that the same figure can be reproduced exactly an indefinite number of times by releasing the pendulums from the same points.
But a fresh field is opened if for the one-direction suspension of pendulum B we substitute a gimbal, or universal joint, permitting movement in all directions, so that the pendulum is able to describe a more or less circular path. The figures obtained by this simple modification are the results of compounded rectilinear and circular movements.
The reader will probably now see even fresh possibilities if both pendulums are given universal movement. This can be effected with the independent pendulums; but a more convenient method of obtaining equivalent results is presented in the Twin Elliptic Pendulum invented by Mr. Joseph Goold, and shown in Fig. 169. It consists of --(1) a long pendulum, free to swing in all directions, suspended from the ceiling or some other suitable point. The card on which the figure is to be traced, and the weights, are placed on a platform at the bottom
FIG. 170. - Benham's miniature Twin Elliptic Pendulum Harmonograph.
of this pendulum. (2) A second and shorter free pendulum, known as the "deflector," hung from the bottom of the first.
This form of harmonograph gives figures of infinite variety and of extreme beauty and complexity. Its chief drawback is its length and weight, which render it more or less of a fixture.
Fortunately, Mr. C. E. Benham of Colchester has devised a Miniature Twin Elliptic Pendulum which possesses the advantages of the Goold, but can be transported easily and set up anywhere. This apparatus is sketched in Fig. 170. The main or platform pendulum resembles in this case that of the Rectilinear Harmonograph, the card platform being above the point of suspension.
Value of the Harmonograph. - A small portable harmonograph will be found to be a good means of entertaining friends at home or elsewhere. So lay out some reception chairs and invite a few friends over to create some geometric images. The gradual growth of the figure, as the card moves to and fro under the pen, will arouse the interest of the least scientifically inclined person; in fact, the trouble is rather to persuade spectators that they have had enough than to attract their attention. The cards on which designs have been drawn are in great request, so that the pleasure of the entertainment does not end with the mere exhibition. Sitting in your chairs you can admire an album filled with picked designs, showing different harmonies and executed in inks of various colours, is a formidable rival to the choicest results of the amateur photographer's skill.
Practical Instructions for making Harmonographs.
Pendulums. - For the Rectilinear type of harmonograph wooden rods 5/8 to 3/4 inch in diameter will be found very suitable. They cost about 2d. each. Be careful to select straight specimens. The upper pendulum of the Miniature Twin Elliptic type should be of stouter stuff, say a broomstick; that of the Goold apparatus stouter still.
All pendulums on which weights are slid up and down should be graduated in inches and fractions, reckoning from the point of suspension as zero. The graduation makes it easy to re-establish any harmony after the weights have been shifted.
Suspensions. - For a harmonograph to give satisfaction it is necessary that very little friction should be set up at the point of suspension, so that the pendulums may lose amplitude of swing very slowly.
One-way suspensions are easily made. Two types, the point and knife-edge respectively, are shown in Fig. 168 and the top part of Fig. 172. The point suspension is most suitable for small rods and moderate weights; the knife-edge for large rods and heavy weights which would tend to crush a fine point.
FIG. 171. -- Gimbal giving universal movement: point suspension.
Points should rest in cup-shaped depressions in a metal plate; knife-edges in V -shaped grooves in a metal ring.
Screws turned or filed to a sharp end make convenient points, as they can be quickly adjusted so that a line joining the points lies exactly at right angles to the pendulum. The cups to take the points should not be drilled until the points have been thus adjusted. Make a punch mark on the bedplate, and using this as centre for one of the points, describe an arc of a circle with the other. This will give the exact centre for the other cup. It is evident that if points and cup centres do not coincide exactly there must be a certain amount of jamming and consequent friction.
FIG. 172. -- Knife-edge universal-motion gimbal.
In making a knife-edge, such as that shown in Fig. 172, put the finishing touches on with a flat file drawn lengthwise to ensure the edge being rectilinear. For the same reason the V slots in the ring support should be worked out together. If they are formed separately, the chances are against their being in line with one another.
Gimbals, or universal joints, giving motion in all directions, require the employment of a ring which supports one pair of edges or points (Fig. 172), and is itself supported on another pair of edges or points set at right angles to the first. The cups or nicks in the ring should come halfway through, so that all four points of suspension shall be in the same plane. If they are not, the pendulum will not have the same swing-period in all directions. If a gimbal does not work with equal freedom in all ways, there will be a tendency for the pendulum to lose motion in the direction in which most friction occurs.
By wedging up the ring of a gimbal the motion of the pendulum is changed from universal to rectilinear. If you are making a harmonograph of the type shown in Fig. 168, use a gimbal for the platform pendulum, and design it so that the upper suspension gives a motion at right angles to the pen pendulum. The use of two little wedges will then convert the apparatus in a moment from semirectilinear to purely rectilinear.
Weights. - The provision of weights which can be slipped up and down a rod may present some difficulty. Of iron and lead, lead is the more convenient material, as occupying less space, weight for weight, and being more easily cast or shaped. I have found thin sheet roofing lead, running 2 lbs. to the square foot, very suitable for making weights, by rolling a carefully squared strip of the material round the rod on which it will have to move, or round a piece of brass tubing which fits the rod. When the weight has been rolled, drill four holes in it, on opposite sides near the ends, to take nails, shortened so that they just penetrate all the laps but do not enter the central circular space. These will prevent the laps sliding over one another endways. A few turns of wire round the weight over the heads makes everything snug.
Just one caution here. The outside lap of lead should finish at the point on the circumference where the first lap began, for the weight to be approximately symmetrical about the centre.
An alternative method is to melt up scrap lead and cast weights in tins or flowerpots sunk in sand, using an accurately centred stick as the core. This stick should be very slightly larger than the pendulum rod, to allow for the charring away of the outside by the molten metal. (Caution.--The mould must be quite dry.)
Failing lead, tin canisters filled with metal scrap may be made to serve. It will in this case be necessary to bore the lid and bottom centrally and solder in a tube fitting the rod, and to make an opening through which the weighting material can be inserted.
Adjustment of Weights. - As lead is too soft a metal to give a satisfactory purchase to a screw--a thread cut in it soon wears out--it is better to support a leaden weight from underneath by means of a brass collar and screw. A collar is easily made out of a bit of tubing thickened at the point where the screw will pass by soldering on a suitably shaped piece of metal. Drill through the reinforcement and tubing and tap to suit the screw used, which may well be a camera tail screw, with a large flat head.
FIG. 173. - Brass Collar and Screw
I experienced some trouble from the crushing of wooden rods by a screw, but got over it as follows. The tubing selected for the collar was large enough to allow a piece of slightly smaller tubing to be introduced between it and the rod. This inner piece was slit from one end almost to the other, on opposite sides, and soldered at one end to the outer tube, a line joining the slots being at right angles to the axis of the screw. The pressure of the screw point was thus distributed over a sufficient area of the wood to prevent indentation. (See Fig. 173.)
Pen Levers. - The pen lever, of whatever kind it be, must work on its pivots with very little friction, and be capable of fine adjustment as regards balance. For the Rectilinear Harmonograph the form of lever pivot shown in Fig. 174 is very suitable. The spindle is a wire nail or piece of knitting needle sharpened at both ends; the bearings, two screws filed flat at the ends and notched with a drill.
FIG. 174. - Pivot for pen lever.
The brass standard should be drilled and tapped to fit the screws fairly tight, so that when once adjusted they may not slacken off. If the lever is made of wood, the tail may be provided with a number of metal pegs on which to place the weights; if of wire, the tail should be threaded so that a brass weight and lock screw may be moved along it to any desired position. It is very important that the pressure of the pen on the card should be reduced to a minimum by proper balancing, as the friction generated by a "heavy" pen slows the pendulum very quickly; and that the centre of gravity should be below the point of suspension, to put the pen in stable equilibrium. The lever shown in Fig. 169 is suitable for the Twin Elliptic Pendulum.
FIG. 175. -- End of pen lever.
In this case the lever is not moved about as a whole. Mr. C. E. Benham advocates the use of wood covered with velvet to rest the lever points on.
For keeping the pen, when not in use, off the platform, a small weight attached to the lever by a thread is convenient. When the pen is working, the weight is raised to slacken the thread.
Attaching Pen to Lever. - In the case of wooden levers, it is sufficient to slit the end centrally for a few inches after drilling a hole rather smaller than the pen, at a point which lies over the centre of the card platform, and quite squarely to the lever in all directions, so that the pen point may rest squarely on the card. (Fig. 175.) Another method is to attach to the end of the lever a vertical halftube of tin, against which the pen is pressed by small rubber bands; but even more convenient is a small spring clip shaped as in Fig. 176.
FIG. 176. - Clip to hold glass pen.
The card platform should be perfectly flat. This is essential for the production of good diagrams. If wood is used, it is advisable to glue two thin pieces together under pressure, with the grain of one running at right angles to the other, to prevent warping.
Another important point is to have the card platform square to the rod. If a piece of tubing fitting the rod is turned up true in the lathe and soldered to a disc screwed to the underside of the table, perpendicularity will be assured, and incidentally the table is rendered detachable.
To hold the card in place on the table, slit a spring of an old photographic printing frame down the middle, and screw the two halves, convex side upwards, by one end near two opposite corners of the platform. (See Fig. 170.) If cards of the same size are always used, the table should be marked to assist adjustment.
Making Pens. - The most satisfactory form of pen is undoubtedly a piece of glass tubing drawn out to a point, which is ground down quite smooth. The making of such pens is rather a tedious business, but if care be taken of the pen when made it will last an indefinite time.
Tubing 3/16 or 1/8 inch in external diameter is suitable. Break it up (by nicking with a file) into 9-inch lengths. Take a piece and hold its centre in the flame of a small spirit lamp, and revolve it till it softens. Then draw the glass out in as straight a line as possible, so that the points may be central. If the drawing is done too fast, the points will be much too long to be of any use: half an inch of taper is quite enough.
Assuming that a point of satisfactory shape has been attained-- and one must expect some failures before this happens--the pen may be placed in the pen lever and ground down on a perfectly clean wet hone laid on the card platform, which should be given a circular movement. Weight the lever so as to put a fair pressure on the point.
The point should be examined from time to time under a strong magnifying-glass, and tested by blowing through it into a glass of water. For very liquid ink the hole should be as small as you can possibly get it; thick inks, such as Indian, require coarser pens.
The sharp edge is taken off and the width of the point reduced by drawing the pen at an angle along the stone, revolving it all the time. The nearer to the hole you can wear the glass away the finer will be the line made by the pen.
Another method is as follows:--Seal the point by holding it a moment in the flame. A tiny bulb forms on the end, and this has to be ground away till the central hole is reached. This is ascertained by the water test, or by holding the pen point upwards, so that light is reflected from the tip, and examining it under the magnifier. Then grind the edge off, as in the first case.
Care of Pens. -- The ink should be well strained, to remove the smallest particles of "suspended matter," and be kept corked. Fill the pen by suction. On no account allow the ink to dry in the pen. Squirt any ink out of it when it is done with, and place it point downwards in a vessel of water, which should have a soft rubber pad at the bottom, and be kept covered to exclude dust. Or the pen may be cleaned out with water and slipped into a holder made by rolling up a piece of corrugated packing-paper. If the point gets stopped up, stand the pen in nitric or sulphuric acid, which will probably dissolve the obstruction; and afterwards wash it out.
Inks. - I have found Stephens's coloured printer inks very satisfactory, and can recommend them.
Paper and Cards. - The paper or cards used to draw the figures on should not have a coated surface, as the coating tends to clog the pen. The cheapest suitable material is hot pressed paper, a few penny-worths of which will suffice for many designs. Plain white cards with a good surface can be bought for from 8s. to 10s. per thousand.
Lantern Slides. - Moisten one side of a clean lantern slide plate with paraffin and hold it over a candle flame till it is a dead black all over. Very fine tracings can be obtained on the smoked surface if a fine steel point is substituted for the glass pen. The design should be protected by a cover-glass attached to it by a binding strip round the edges.
Details of Harmonographs.
The reader may be interested in details of the apparatus shown in Figs. 168 and 170, made by the writer.
The Rectilinear Harmonograph, shown in Fig. 168, has pendulums of 5/8-inch wood, 40 inches long, suspended 30 inches from the lower ends, and set 10 inches apart, centre to centre. The suspensions are of the point type. The weights scale 5 lbs. each. The platform pendulum is provided with a second weight, which can be affixed above the suspension to slow that pendulum for 2:3, 4:5, 7:8, and higher harmonies.
The baseboard is plain, and when the apparatus is in action its ends are supported on boxes or books laid on two tables, or on other convenient supports. The whole apparatus can be taken to pieces very quickly for transport. The total cost of materials used did not exceed 3s. 6d.
The Twin Elliptic Pendulum of Fig. 170 is supported on a tripod base made of three pieces of 1-1/2 x 1-1/2 inch wood, 40 inches long, with ends cut off to an angle of 72 degrees to give a convenient straddle, screwed at the top to an oak head 3/4 inch thick, and braced a foot below the top by horizontal crossbars 2 inches wide and 1/2 inch thick. For transport this stand can be replaced by a flat baseboard similar to that of the Rectilinear Harmonograph described in the last paragraph.
The main pendulum is a straight ash rod, 33 inches long and 1- 1/4 inches in diameter, suspended 13-1/2 inches from its upper end. Two weights of 4-1/2 lbs. each, made of rolled sheet lead, are provided for this pendulum. According to the nature of the harmony, one only, or both together below the suspension, or one above and one below, are used.
The weight of the lower pendulum, or deflector, is supported on a disc, resting on a pin passing through the bottom of a piece of brass tubing, which is provided with an eye at its upper end. This eye is connected by a hook with several strands of silk thread, which are attached to the upper pendulum by part of a cycle tyre valve. The stem part of the valve was cut off from the nut, and driven into a suitably sized hole in the end of the main pendulum.
The screw collar for holding the valve in place had a little brass disc soldered to the outside, and this disc was bored centrally for the threads to pass through. The edges of the hole had been rounded off carefully to prevent fraying of the threads. (Fig. 177.) The over-all length of the pendulum, reckoning from the point of suspension, is 20 inches. The weights of the lower pendulum are several in number, ranging from l lb. to 3 lbs.
FIG. 177. -- Suspension for lower weight of Twin Elliptic Harmonograph.
Working the Harmonograph. - A preliminary remark is needed here. Harmonies are, as we have seen, a question of ratio of swing periods. The larger the number of swings made by the more quickly moving pendulum relatively to that of the slower pendulum in a given time, the higher or sharper is the harmony said to be. Thus, 1:3 is a higher harmony than 1:2, and 2:3 is lower or flatter than 3:8.
The tuning of a harmonograph with independent pendulums is a simple matter. It is merely necessary to move weights up or down until the respective numbers of swings per minute bear to one another the ratio required. This type of harmonograph, if made of convenient size, has its limitations, as it is difficult to get as high a harmonic as 1:2, or the octave with it, owing to the fact that one pendulum must in this case be very much shorter than the other, and therefore is very sensitive to the effects of friction.
FIG. 176a. - Harmonograms illustrating the ratio 1:3. The two on the left are made by the pendulums of a twin elliptical harmonograph when working concurrently; the three on the right by the pendulums when working antagonistically.
FIG. 177a. -- Harmonograms of 3:4 ratio (antagonistically). (Reproduced with kind permission of Mr. C. E. Benham.)
The action of the Twin Elliptic Pendulum is more complicated than that of the Rectilinear, as the harmony ratio is not between the swings of deflector and upper pendulum, but rather between the swings of the deflector and that of the system as a whole.
Consequently "tuning" is a matter, not of timing, but of experiment. Assuming that the length of the deflector is kept constant--and in practice this is found to be convenient--the ratios can be altered by altering the weights of one or both pendulums and by adjustment of the upper weight.
For the upper harmonies, 1:4 down to 3:8, the two pendulums may be almost equally weighted, the top one somewhat more heavily than the other. The upper weight is brought down the rod as the ratio is lowered.
To continue the harmonies beyond, say, 2:5, it is necessary to load the upper pendulum more heavily, and to lighten the lower one so that the proportionate weights are 5 or 6:1. Starting again with the upper weight high on the rod, several more harmonies may be established, perhaps down to 4:7. Then a third alteration of the weights is needed, the lower being reduced to about one-twentieth of the upper, and the upper weight is once more gradually brought down the rod.
Exact figures are not given, as much depends on the proportions of the apparatus, and the experimenter must find out for himself the exact position of the main weight which gives any desired harmonic.
A few general remarks on the action and working of the Twin Elliptic will, however, be useful.
1. Every ratio has two forms. (a) If the pendulums are working against each other-- antagonistically--there will be loops or points on the outside of the figure equal in number to the sum of the figures in the ratio. (b) If the pendulums are working with each other--concurrently-- the loops form inside the figure, and are equal in number to the difference between the figures of the ratio.
To take the 1:3 ratio as an example. If the tracing has 3+1=4 loops on the outside, it is a specimen of antagonistic rotation. If, on the other hand, there are 3- 1=2 loops on the inside, it is a case of concurrent rotation. (Fig. 176, A.)
2. Figures with a ratio of which the sum of the numbers composing it is an even number (examples, 1:3, 3:5, 3:7) are symmetrical, one half of the figure reproducing the other. If the sum is Uneven, as in 1:2, 2:3, 2:7, the figure is unsymmetrical. (Fig. 177, A.)
3. The ratio 1:3 is the easiest to begin upon, so the experimenter's first efforts may be directed to it. He should watch the growth of the figure closely, and note whether the repeat line is made in front of or behind the previous line of the same loop. In the first case the figure is too flat, and the weight of the upper pendulum must be raised; in the second case the weight must be lowered. Immediately an exact harmonic is found, the position of the weight should be recorded.
Interesting effects are obtained by removing the lower pendulum and allowing the apparatus to describe two elliptical figures successively, one on the top of the other, on the same card. The crossing of the lines gives a "watered silk" appearance to the design, which, if the pen is a very fine one and the lines very close together, is in many cases very beautiful.
Readers who wish for further information on this fascinating subject are recommended to purchase "Harmonic Vibrations," published by Messrs. Newton and Co., 72 Wigmore Street, London, W. This book, to which I am much indebted, contains, besides much practical instruction, a number of charming reproductions of harmonograms.
Before closing this chapter I should like to acknowledge the kind assistance given me by Mr. C. E. Benham, who has made a long and careful study of the harmonograph.