Alan, clearly it is a very complex subject. There are many parameters to be taken into account. Here are a few that come to mind:
– microphone quality/specs
– algorithm used for spectrum analysis
– distance from microphone
– room reverberation
– air flow
– oral cavity resonance
– pitch of the note
and finally, certainly the physical characteristics of the instrument but also its state of preservation and, eventually, its tuning.
There may be other parameters, but they all influence timbre in subtle ways that are very difficult to see in graphical representations of the harmonic spectrum. But they are clearly perceived by our ears.
In my experience (I took a few measurements of different recorders a few years ago), with a spectrum analyzer we can see only very evident differences and, truth is, the spectrum can be vastly influenced by the player.
I know that all high-end recorder makers have one or more pro players to test new instruments: that makes sense because professional players know what to expect from an instrument, how it should sound _to_them_ (considering their playing style) and thus they have an internal reference. But they can just decided whether or not an instrument works: the sound of another player would never be the same as theirs. To a certain extent, I think the situation is not that different with melodicas…
It would be even worse, in my opinion, to analyze the spectrum of a recording: youTube, for example, has its own compression algorithms which obviously alter the sound.
That said, I find the spectrum analyzer extremely useful to see how we can alter the timbre by shaping our oral cavity or by altering the air flow.