partake of the law of Duality. There are two vibrations going on at the same time.
The perfect character of a musical sound is the result of the harmonious workings of the vibrations of the "perfectly elastic" air with the perfectly elastic string. The forces which act on the air and string being proportional to the distances passed through, makes the times of the vibrations equal,1 and the pitch of the sound the same throughout. These varied forces and distances, with the equal times of the vibrations, and with the simultaneous compressions and expansions of the air and string, are all according to the universal laws of Continuity and Duality.
Rectilinear, oscillatory, and vibratory motions are like the chords in music that have a note in common; but while these motions are like each other in one or two particulars, they are unlike perhaps in ten others; and to suppose that that they are like where they are unlike is inevitable error. To suppose that the vibration of a string is the same as the oscillation of a pendulum is like adding equals to unequals, and supposing the wholes to be equals. The definitions which have been given of strings and pendulums have been wrong
1 "If the particles of the air be slightly disturbed from a position of stable equilibrium, every particle makes an effort to return to that position; and it can be shown that the force of restitution varies as the distance from the position of equilibrium ... a musical note is the consequence of the vibrations excited in the disturbed system, and the permanency of its musical pitch is the consequence of these vibrations being all made in the same time ... If the vibrations gradually slackened in their times, as they do in the extent of these excursions from the effect of the resistances they meet, there would be no note of sustained pitch; the sound would then be a sliding descent in pitch, and music would be impossible." - Penny Cyclopedia, Art. "Vibration."
For the very young student it might be expressed thus - 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. - Editor.