According to WikiPedia it's not the frequencies in the harmonic series that change - it's the relative amplitude of them. The frequencies are largely the same apart from very subtle variations. Apparently.
Hee hee - do I sound like I know what I'm talking about - I just read it
"The relative amplitudes of the various harmonics primarily determine the timbre of different instruments and sounds, though formants also have a role. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the even-numbered harmonics are suppressed, which produces a purer tone. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. Of course, the differences in resonance between the wood of the clarinet and the brass of the saxophone also affect their tones. The inharmonic ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments."
There you go - may as well get it from source...
Still - the fact remains that the major third of any tonic (in a diatonic scale) is more prominent in the harmonic series, thus the chord sounds 'stronger'.
I think taking recourse to alternative temperaments is really stretching the point a little, 'well' and 'even' temperament still have major thirds that are relatively closer to a prominent frequency in the given tonic's harmonic series than the relevant minor third.
Lets 'just avoid' 'just intonation'.
Besides, I think we're breaking the 'rules' if we start to drift into unusual or experimental temperaments too much