Okay, this is the last time I'm answering the same, banal question. I mean no personal offense, but you're turning this into something much more complicated than it really is. I've answered this half a dozen times, so here were are again for the last time.
First of all, no the link itself does not make all truths suddenly unveiled...no, one has to take that knowledge of basic harmonics and actually apply it. So let's go with some examples. I pluck an open E on a bass guitar and record it for playback. The low E consists of the fundamental 41.2hz, and varying projections of the overtone series ~82hz, ~124hz, ~165hz etc. Let's say my microphone which I used to record the note was a piece of garbage and only registers frequencies down to 70hz, omitting the 41hz fundamental. Now, using next three harmonics, don't try and say you cannot find the fundamental from those and are "assuming". It is obvious that the common difference between them is ~41hz, yet the lowest note detected is 82. First of all you can simply figure out which harmonic is which by dividing from the difference 165/41 =~4. Hence that is the 4th harmonic in the overtone series. Now in practice, yes, some harmonics from instruments tend not to result from perfect integers, but they are very close. So if I repeated the pattern with 41hz, 81hz, 121hz, 161hz, etc (accounting for some skew [flatness in this case] of upper partials), the world is not suddenly thrown into chaos and the detectable patter shattered! No any sensible person with knowledge of these harmonics can still tell which is which. Maybe in the case of instruments that produce blatant non-integer overtones (like resonant drums and gongs) this might be true, but for the sake of most instruments it is not.
So now on to instruments you propose which suppress the fundamental by nature and produce only odd harmonics (such as a flute or clarinet you say?) I'm honestly not following you at this point. I checked up on some sources regarding flute acoustics and they are capable off producing all integer harmonics. http://www.phys.unsw.edu.au/jw/fluteacoustics.html#harmonics Assuming an instrument does sound such that only odd integers are produced, (low E example again) you get 41.2hz (first/fundamental), 124hz (third), 205hz (fifth), and the 41.2hz is filtered out via the bad recording. Lets divide the two upper harmonics to see how they correspond to one another relatively. 205/124. Our result, 5/3. Immediately it becomes apparent that our integers are 5th and 3rd. What is missing? The fundamental. You can take either 205/5 or 124/3, your pick. Either way you get ~41hz. Not so hard...and certainly not an assumption either. And how should the device know not to fill in 82hz? Because there was no 4th harmonic between the 3rd and 5th, so why should there be one between 1 and 3. It's all a very simple analysis of patterns and ratios. It's nothing more than knowing the math to apply it to very common scenarios.
Now, with the example of even only harmonics, I am not clear on what you mean. Do you mean to say an instrument that produces a fundamental and then evens from thereon in or one that only sounds evens and no fundamental...in which case, for the sake of analysis, wouldn't the second become the new fundamental, the fourth the new second? etc. For the purposes of mathematical analysis it would certainly work. If however, it sounded a fundamental at 41hz, then harmonics at 82hz, 165hz, etc. That would be trickier and probably require relative sampling of things such as the relative amplitude between those two detected harmonics (as higher harmonics diminish in amplitude) and possibly between the rest of the musical content, then some inconsistencies may result.
"Or maybe the bottom line, does that device employ a DSP"
Yes, I believe so. They are a type of harmonic content PROCESSOR. :P
I have not been arguing that these things will reproduce accurate musical content (as I can think of several reasons why they would not), only that there are definitive ways of analysis to find a fundamental pitch. I'm done having this side discussion though. You have repeatedly asked me the same question several times over. I can answer time and time again, but as long as you refuse my proposed explanations I cannot help you. I thank you for your patience and willingness to respond, now let's please just drop it.