Sympathetic Vibratory Physics

Ramsay
Different writers have put forth different views of what constitute a musical vibration, but their various views do not make any difference in the ratios which the notes of this sound-host bear to each other. Whether the vibrations be counted as single or double vibrations, the ratios of their relative motions are the same. Nevertheless, a musical vibration is an interesting thing in itself, and ought to be correctly defined.
A string when vibrating musically is passing and re-passing the central line of its rest or equilibrium with a certain range of excursion. Some writers have defined a vibration to be the passage of the string from one extreme of its excursion to the other, while some have preferred to define it as the passage of the string from the one extreme of its excursion to the other and back again. D. C. Ramsay has been led in his researches to define a vibration as the movement of the string from its central line of rest to the extreme of its excursion on one side, and back to the central line of rest; and from the central line of rest to the extreme of its excursion on the other side, and back again to the "right line," as he calls it, as a second vibration. His reasoning on this will be seen in what follows. (See Fig. 3, Plate IV.) [Scientific Basis and Build of Music, page 21]

These definitions have been given for strings and pendulums have been wrong for both. Indeed, the vibration of a string is not even once from extreme to extreme; for while the string itself goes from one extreme to the other, it moves in half the time of one vibration and half the time of another. In the first half of its course, that is from the extreme to the right line, or line of its rest, it leaves the air on that side to expand itself; and in the second half of its course, that is from the right line to the other extreme, it compresses the air on the other side. Now, a vibration of a string does not consist in the expanding of a body of air on the one side of a string and the compression of a different body of air on the other side, but in the compression and expansion of the same body of air on the one side of the string; so the vibration of the string are on either side of the line of its rest. The vibration is the movement from the right line, or center of action, to the extreme and back again to the right line, and so on the other side.
This definition of a vibration answers to all the requirements of the case; and in exhibiting the ratio 1:2 with two strings, it brings the two strings into the same position at every second vibration of the higher one. And so with every ratio of the musical system. [Scientific Basis and Build of Music, page 23]

The resisting forces which act on the air and string are proportional to the varying distances between the right line and the extreme of the excursion in each vibration; and thus whilst a wide vibration traverses a wider space to reach the extreme than a narrow vibration, yet in the wide vibration the string is pulled back by a proportionally greater force, and therefore with a proportionally greater velocity, than in the narrower vibration; and the wide vibration is thus accomplished in exactly the same time as the narrow one. The times of the vibrations being thus equal throughout, the pitch of the sound remains the same so long as the vibrations continue. [Scientific Basis and Build of Music, page 24]

Sympathetic Vibratory Physics

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