partial answers

[David Boyce]

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.