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]
Musical sounds are usually caused in the ear by certain vibrations of the surrounding air, which originate from solid bodies in a state of vibration from some force exerted upon them. Vibrations of the air require to attain a certain rate of speed before they become audible to the human ear; and they require to have certain ratios of rate of rapidity in order to constitute that beautiful host of sounds which constitutes the music of mankind. These musical vibrations may arise in the air from a vibrating organ pipe, or a vibrating tuning fork, or a bell, or a sounding glass, or a strand of wire or gut-string, or other rhythmically vibrating body; but to explain and define the nature of a musical vibration from the action upon it of an elastic string is to explain and define it for all. But before defining what a vibration of a string is, let us hear what others have said about it. Charles Child Spencer, Treatise on Music, p. 6, says- "It is customary in calculating the ratios of vibration of musical strings, and which answer to the waves of the atmosphere, to reckon by double vibrations, so that instead of saying there are 32 single vibrations in the lowest sound, C, writers on this branch of music say there are 16 double vibrations in this sound. This method of calculation, therefore, gives 256 vibrations for the fourth Octave C." Playfair, in his Outlines of Natural Philosophy, p. 282, says- "It is usual to reckon the vibrations of a string different from those of a pendulum; the passage from the highest point on one side to the highest point on the other is reckoned a vibration of a pendulum; the passage from the farthest distance on one side to the farthest distance on the other and back again to its first position, is the accounted a vibration of a musical string. It is properly a double vibration." Holden, in his Rational System of Music, says- "Mr. Emerson reckons the complete vibration the time in which a sounding string moves from one side to [Scientific Basis and Build of Music, page 22]
Figure 8.11 - Four Fundamental Phases of a Wave
Figure 8.9 - Four Fundamental Motions of a Pendulum
Ramsay - PLATE IV - Oscillation and Vibration