Ed, I think what you say is the nub: "that string vibrates, every available multiple of the lowest frequency is not only a "natural" but also, a logical consequence." It may help to think in terms of numbers of nodes, and to consider that no possible node would be missed out, and that this would mean a harmonic series - "harmonic" is after all a mathematics concept - it's a type of numerical series, just as "arithmetic" and "geometric" are types of numerical series. I'm even confusing myself now...... David.
This PTG archive page provided courtesy of Moy Piano Service, LLC