One of the difficulties usually ignored in "discussions" about just tuning and equal tempered tuning is that most physical devices only approximate the "pure" integer frequency relationships between the fundamental tone and their harmonics.

Arguments about any "superiority" of just tuning are almost required to include the assumption that the harmonics are in strict integer frequencies. x1, x2, x3, ... etc.

For a reasonably long string at reasonably high tension, it works out pretty well - which is a reason, perhaps, why strings are much favored in music. It holds pretty well for long "pipes," provided that they don't get too "fat." It doesn't work at all for "free-reed" instruments like harmonicas and accordians.

For a "free-reed," the first harmonic is NOT at twice the frequency of the fundamental tone - and the deviation increases as you consider higher harmonics. This may be one reason why - as offered above - squeezeboxers were among the first to truly accept equal tempered tuning (to the extent that you can "tune" one).

Even for a simple "pipe" like a whistle, the "effective length" of the tube depends on the wavelength. The wave node (or peak, depending on how the tube is stopped) actually extends a little past the end, making the "acoustic length" a little different than the physical length. Since a short wave (high harmonic) doesn't "stick out as much," the higher harmonics of the tube actually tend to go a little sharp relative to the fundamental tone. The situation is even a little worse with a "finger hole" since the pneumatic resistance of the "rest of the tube" makes the effective length pretty "squishy," and significantly different for each "harmonic" contained within the tone.

Reference has been made to the "Barbershop" effect of just tuning, but it is quite possible for your "lead" to sing to an equal tempered scale - if that's what he/she has learned, while the "harmony parts" tune (in just intonation) to each individual note that the lead sings. Most people will not be able to detect the "scale" in use - but will still be impressed with the "great resonance" of the well "intoned" harmony. (Actually, your lead would need a pretty tin ear not to automatically "drift into" a just scale, but I've encountered a few people who are "stuck in equal temper.")

The theory is there to help with quessing what mother nature does; but you've got to remember that "she don't do math," and she can get pretty sarcastic with people who rely on simple math try to do too much. More complex math can get you a little closer, but the instrument-to-instrument (and even note-to-note) variations make it difficult to make general statements with any great confidence.

John