Sympathetic Vibratory Physics

Ramsay
"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." [Scientific Basis and Build of Music, page 15]

"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." [Scientific Basis and Build of Music, page 16]

"The subdominant, or lowest chord in the key - F, A, C, is the natural product of the first combination of the three primary ratios (2, 3, 5). Their second combination develops the tonic or middle chord C, E, G. The third combination develops the dominant or highest chord G, B, D.

keynote or root of F

	{ Perfect Fifth }
first combination of the three primary ratios	F	Mj 3rd	A	Mn 3rd	C	Subdominant
second combination of the three primary ratios	C	Mj 3rd	E	Mn 3rd	G	Tonic
third combination of the three primary ratios	G	Mj 3rd	B	Mn 3rd	D	Dominant

from [Scientific Basis and Build of Music, page 17]

In the laws of quantities and motions the three primary ratios, 1:2, 1:3, 1:5, with the three different units, F1, C3, and G9, the roots of the chords of the subdominant, tonic, and dominant, produce the three chords of the musical system major, the one not interfering with the other; and by an inverse process are produced, from B720, E240, and A80, its generating notes, the three chords of the musical system minor; the one chord not interfering with the other. In a similar way the chromatic chords can be produced from three different units, without the one interfering with the other; and, like the subdominant, tonic, and dominant chords of the diatonic scale, they are fifths apart. So we may call them the subdominant, tonic, and dominant chromatic chords. Each of the three chromatic chords has also kinship with the major and minor modes, from the way in which the diatonic minor triad is constituted a chromatic chord by its supplement coming in the one side from the minor, and on the other side from the major system. [Scientific Basis and Build of Music, page 53]

"The organic structure of music is formed by the three ratios of 1:2, 1:3, and 1:5, from the laws of quantities and motions; but as it is only the ratio of 1:2 that has a pure, unmixed, invariable character, and as the notes produced by the first, second, and third powers of THREE have different degrees of centrifugal force, and the character of the notes produced by the first power of FIVE depends on the character of the notes from which they are derived, so the final character of the notes and chords is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system; and no matter how these notes may be afterwards placed, like chemical elements, they never lose their original force. [Scientific Basis and Build of Music, page 95]

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three primary ratios