If you investigate this topic, I think you'll conclude that the free reed tongue vibrates only at its fundamental bending mode...
I don't know which model is the closest to a real concertina reed... (snip)
If there were significant contributions from more than one mode, the result would be noise, since their frequencies are not, as a rule, close enough to integer multiples of each other. Thus, one cannot expect to encounter these modes in any instrument that produces acceptable musical tones.
Tom, are you claiming that (bell-shaped) bells don't produce "acceptable musical tones"?
(I was going to wait until I had read Tom's PICA paper before commenting further, but since others have not done so, I'm returning to the "fray".)
I don't think that anyone here is claiming that the reed itself vibrates at any but its fundamental bending frequency. But the frequency spectrum
that is responsible for the timbre doesn't come from that
vibration. It comes from the Fourier decomposition into sinusoidal frequencies of the very non
-sinusoidal pressure wave generated when the vibrating reed chops the flowing airstream, and by the differential resonance and absorption of those frequencies in the reed cavity. (Probably also from differential transmission/absorption outside the cavity -- e.g., through the end, -- but I won't try to treat that here).
The integer-multiples criterion may be valid for chords -- i.e., where the strengths of the different frequencies are of similar magnitudes, -- but modes which are much weaker than the fundamental, though still perceptible, should be able to add "flavor" without masking/disrupting the fundamental pitch or even causing an unpleasant sensation, no matter what their ratio to the fundamental. I suspect that descriptions of the timbre of different instruments as more or less "harsh", "brassy", "rich", "mellow", etc. have more to do with the relations among these relatively weaker higher frequencies than any clash or harmony between them and the fundamental. The fundamental questions of timbre are then: What are the frequencies (or frequency ratios) of these "harmonics", and what are their (relative) intensities?