24, 27, 30, 32, 36, 40, 45, 48.
The meaning of the terms third, fourth, fifth, etc., which we have so often applied to the musical intervals, is now apparent; the term has reference to the position of the note in the scale.
§ 7. Composition of Vibrations
In our second lecture I referred to, and in part illustrated, a method devised by M. Lissajous for studying musical vibrations. By means of a beam of light reflected from a mirror attached to a tuning-fork, the fork was made to write the story of its own motion. In our last lecture the same method was employed to illustrate optically the phenomenon of beats. I now propose to apply it to the study of the composition of the vibrations which constitute the principal intervals of the diatonic scale. We must, however, prepare ourselves for the thorough comprehension of this subject by a brief preliminary examination of the vibrations of a common pendulum.
Such a pendulum hangs before you. It consists of a wire carefully fastened to a plate of iron at the roof of the house, and bearing a copper ball weighing 10 lbs. I draw the pendulum aside and let it go; it oscillates to and fro almost in the same plane.
I say “almost,” because it is practically impossible to suspend a pendulum without some little departure from perfect symmetry around its point of attachment. In consequence of this, the weight deviates sooner or later from a straight line, and describes an oval more or less elongated. Some years ago this circumstance presented a serious difficulty to those who wished to repeat M. Foucault’s celebrated experiment, demonstrating the rotation of the earth.
Nevertheless, in the case now before us, the pendulum is so carefully suspended that its deviation from a straight line is not at first perceptible. Let us suppose the amplitude of its oscillation to be represented by the dotted line a b, Fig. 166. The point d, midway between a and b, is the pendulum’s point of rest. When drawn aside from this point to b, and let go, it will return to d, and in virtue of its momentum will pass on to a. There it comes momentarily to rest, and returns through d to b. And thus it will continue to oscillate until its motion is expended.
The pendulum having first reached the limit of its swing at b, let us suppose a push in a direction perpendicular to a b imparted to it; that is to say, in the direction b c. Supposing the time required by the pendulum to swing from b to a to be one second,[78] then the time required to swing from b to d will be half a second.
Fig. 166. Suppose, further, the force applied at b to be such as would carry the bob, if free to move in that direction alone, to c in half a second, and that the distance b c is equal to b d, the question then occurs, where will the bob really find itself at the end of half a second? It