Sympathetic Vibratory Physics

"The nature of anything is seldom to be discovered in the thing itself;" it is in its relations to other things that we penetrate into and discern its true nature. According to this general truth, in order to understand the musical scales we require to go back to the system of vibrations. And as the laws of oscillatory and vibratory motions are derived from the law of a falling body; so, in order to understand the system of musical vibrations, we require to go back to the primary laws of mechanics, geometry, and arithmetic.
     In the laws of quantities and motions there are three primary ratios from which the musical system of vibrations is developed.
     Pendulums, from the slowness and continuance of their motions, are well adapted to give an ocular demonstration of the relative motions of each of these three primary ratios when compared and combined with the unity and with each other. The numbers 2 and 4 express the first condition in the first ratio; as, in falling bodies, when the times are 2 the distances are 4. In the case of two pendulums, when the length of the one is one fourth part of the other the motions are 1:2; and when two is counted for the upper one, the oscillations of these two pendulums will meet at one. The numbers 3 and 9 express the first condition of the second ratio; as, in falling bodies, when the times are 3 the distances are 9. In the case of two pendulums, when the length of the one is the ninth part of the other, the motions are 1:3; and when three is counted for the upper one, the oscillations of these two pendulums will meet at one. The numbers 5 and 25 express the first condition in the third ratio; as, in falling bodies, when the times are 5 the distances are 25. In the case of two pendulums, when the length of the one is twenty-fifth part of the other, the motions are 1:5; and when five is counted for the upper one, the oscillations of these two pendulums will meet at one.
     In the system of motions in pendulums, the three primary ratios indicated in the foregoing paragraph, namely, 2:4, 3:9, and 5:25, are compared and combined with three different units. In their comparison, 1 is the unit of quantities, that is lengths, and 1 is the unit of motions. The numbers 1/4, 1/9, and 1/25, when taken together with 1 as unity, express the first comparison and combination of quantities; and the numbers 2, 3, and 5, taken together with 1 as unity, express the first comparison and combination of motions.

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Ramsay - Pendulum Illustrations of Ratios