When the lengths of four pendulums are 1, 1/4, 1/9, and 1/25, their relative motions are 1, 2, 3, and 5; and when 5 is counted for the highest, the oscillations of these four pendulums will meet at one.
In the second comparison and combination of the three primary ratios, 1/9 is the unit of quantities, and 3 is the unit of motions; and the same primes and the same process as already used in the first comparison and combination will give the same relative quantities; and the motions will be 3, 6, 9, and 15, compared with the original unit.
In the third comparison and combination of the three primary ratios, 1/27 is the unit of quantities, and 9 is the unit of motions; and the same primes and the same process, again as before, will give the same relative quantities; and in this increasingly rapid range of oscillations the motions will be 9, 18, 27, and 45, compared with the original unit.1
These three combinations of the three primary ratios, when taken together with 1 as unity, produce ten different quantities and motions - 1, 2, 3, 5, 6, 9, 15, 18, 27, and 45; and by producing the octaves of these primes and products, dividing by 4 for the quantities, and multiplying by 2 for the motions2 up to 64, we have 15 additional quantities and motions - 4, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40, 48, 54, 60, and 64. The numbers which express the motions of these twenty-five quantities have among themselves nineteen different ratios, or rates of meeting; and when these ratios are represented by the oscillations of twenty-five pendulums, at the number of 64 for the highest one, they will all have finished their periods, and meet at one for a new series. This is an illustration, in the low silence of pendulum-oscillations, of what constitutes the System of musical vibration in the much higher region of vibrating strings and other elastic bodies, and determines the number of undeveloped sounds which form the harmonious halo of one sound, more or less faintly heard, or altogether eluding our dull mortal ears; and which determines the number of sounds which, when developed, constitute the System of musical sounds.3
Ramsay "To say that I was surprised at what Mr. Keely has discovered would be saying very little indeed ... It would appear that there are three different spheres in which the laws of motion operate. 1 - The first is the one in which Nature plays her grand fugue on the silent harp of Pendulums. In one period of Nature's grand fugue, as illustrated by pendulums, there are 19 ratios in 25 circles of oscillations ranging over 6 octaves; but all in silence. [Scientific Basis and Build of Music, page 86]
together on radial lines from the center they appear grouped in various chords and combinations, dropping out and coming in in such succession as to constitute what Ramsay, whose genius was given to set this thus before us, calls "Nature's Grand Fugue." Beginning at F in the center at the top, and moving either to the right or to the left, after a run of 7 notes we have 4 consecutive Octaves, and then comes the Minor fifth, A-E, followed by the Major fifth, G-D; and this by another Major fifth, F-C; the combinations keep changing till at the quarter of the circle we come to F, A, C, E, G, a combination of the subdominant and tonic Major; and after another varied series of combinations we have at the half of the circle the elements of 2 minor chords, D, F, A and A, C, E, and one Major chord, C, E, G; at the third quarter we have a repetition of the first quarter group; and the various chords and combinations dropping out and coming in, fugue-like; finally we return to where we began, and end with the three-times-three chord, in which the whole 25 notes are struck together, and make that wondrous and restful close of this strange Fugue. No one can hear the thrice-threefold chord of this close and ever forget it; it is "the lost chord" found; and leads the saintly heart away to the Three in One who is the Lord of Hosts; Maker of Heaven and Earth, and all the host of them. [Scientific Basis and Build of Music, page 103]
Fig. 1 - The pendulums in this illustration are suspended from points determined by the division of the Octave into Commas; the comma-measured chords of the Major key being S, 9, 8, 9, 5; T, 9, 8, 5, 9; D, 8, 9, 5, 9. The pendulums suspended from these points are tuned, as to length, to swing the mathematical ratios of the Diatonic scale. The longest pendulum is F, the chords being properly arranged with the subdominant, tonic, and dominant, the lowest, center, and upper chords respectively. Although in "Nature's Grand Fugue" there are 25 pendulums engaged, as will be seen by reference to it, yet for the area of a single key 13 pendulums, as here set forth, are all that are required. It will not fail to be observed that thus arranged, according to the law of the genesis of the scale, they form a beautiful curve, probably the curve of a falling projectile. It is an exceedingly interesting sight to watch the unfailing coincidences of the pendulums perfectly tuned, when started in pairs such as F4, A5, and C6; or started all together and seen in their manifold manner of working. The eye is then treated to a sight, in this solemn silent harp, of the order in which the vibrations of sounding instruments play their sweet coincidences on the drum of the delighted ear; and these two "art senses," the eye and the ear, keep good company. Fig. 2 is an illustration of the correct definition of a Pendulum Oscillation, as defined in this work. In watching the swinging pendulums, it will be observed that the coincidences [Scientific Basis and Build of Music, page 104]
It is very interesting to observe how the number seven, which is excluded from the genesis of the system of vibration, comes into view after the genesis is completed, not only in the seven seconds of the melodic scale, but also in the seven of each of the intervals. As there are seven days in the week, though the seventh was only after the genesis of creation was finished, so there are six intervals, but seven of each, as we have seen; and in each 7-fold group three magnitudes determined by the three genetic magnitudes of the seconds. There is much symbolic meaning in all this. Any of the intervals may be used in melody; in harmony also, either in simple or compound chords, they all have the honor of fulfilling a part; and even those, such as seconds and sevenths, which are less honorable in themselves, have great honor in compound chords, such as dominant sevenths and compound tonics, which fulfill exceedingly interesting functions in the society of chords. [Scientific Basis and Build of Music, page 110]