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.

[Illustration:  Fig. 176a.—­Hamonograms 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.]

[Illustration:  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.)