This is a very interesting thread. I'm not an expert musician, but as a a logician by both nature and vocation, I can't help adding my two cents.
Disclaimer: I use a digital tuner to tune my uke, mostly because it's easy and frankly I can't tell the difference so it results in more playing time. I won't post an in-or-out-of-tune youtube video any time soon, but if I did it would be my play that offends you, not my tuning.
In the original post, OregonJim made a flawed argument when he wrote that tuning the strings open was a mistake (due to tuning at lowest possible tension), and then proceeded to suggest that one should tune the A string open, then tune to that with fretted strings. You can't have it both ways. If tuning open is inaccurate, then tuning the first string open is still inaccurate. By the original argument, as soon as you fret the A string, it's out of tune (even relative to the other strings that you tuned to it, because it wasn't fretted when you did so). And by the same argument, the 2, 3, and 4 strings will be out of tune whenever they are played open. Either the difference is too small to matter, or the proposed solution results in mistuned 1 when fretted and mistuned 2, 3, and 4 when open.
It was pointed out later in the thread that perhaps a better way to tune is to tune all strings fretted. This is at least a consistent argument, but of course if it matters then that tuning suffers the same fate as the OP's solution when strings are played unfretted. Open strings will always be out of tune by this method if the argument is correct.
Jim also points out that equal temperament was not favored in the "old days", but those days are truly very old, for equal temperament was the tuning of choice for lutes and other plucked string instruments from as long ago as 1400. That's all likely moot, however, as even the method Jim describes will result in equal temperament because the frets determine note spacing. And unless you are playing with a 19th-century British organist, that's probably a good thing.
Then there's the soft vs. loud tuning argument, where Jim states that tuning soft will result in out-of-tune loud play. If it were true that higher pluck tension resulted in a different frequency, then tuning loudly would result in soft play being out of tune, so again it's a can't-win situation. I think Jim was actually suggesting "tune at playing volume", which is reasonable advice, but not for the reason Jim stated. The maximum tension is on the string when you pluck it, but it The length of the string relative to the amplitude of the vibration means that the difference in frequency when played loudly vs. softly is almost certainly negligible. And a string played loudly only has higher tension than a softly-plucked string during part of its vibration (i.e., the part of the amplitude that is greater than the amplitude of the softer note); the rest of the time, the tension on the string is the same (and as the string moves through the zero-amplitude point, they both have minimum tension).
There is a reason Jim is right about tuning loudly enough, though, and that's because both our ears and digital tuners that do Fourier transforms to identify predominant frequencies from vibration data can more easily determine the played notes when the amplitude of the vibration is higher. Literally, we can hear louder notes better than softer notes, so we (and digital tuners) can do a better job when the note is louder.
Having said that, a quick test with two inexpensive digital tuners (a Snark and and a D'Addario) showed no variation in the detected frequency based on plucking softly vs. very loudly. A quick test of me using Jim's method showed I need a lot more practice to get even close to the performance of the digital tuners. By the time I got to the fourth string my incremental error had added up like a game of "telephone".
I'm sure there's nothing wrong with the tuning method Jim suggests (assuming sufficient skill); at the very least it's good ear practice, and it's essentially what every orchestra has done by ear (from the oboe's A) for hundreds of years (and note that they are tuning equally tempered). But I haven't heard any evidence that it provides fundamental improvement or measurable differences in uke or guitar tuning from using a digital tuner. I'd be very interested to hear what frequencies an expert in this method actually ends up tuning to. Then we'd be able to debate what's "better". In any case, I'm pretty sure the differences between digital tuning and expert ear relative tuning will be less than the differences between playing fretted and open (which. A blind test to see if an expert could tell the difference with an ear-tuned instrument and a digitally-tuned instrument (and which they preferred) would also be very interesting.
One other thing that occurred to me: If nuts are such a problem, why not lengthen the neck enough to put another fret and a capo above it. that would eliminate the problem of nut vagaries and allow for perfect tuning. Yes, this is a nutty idea (haha) and it's a good thing I'm not a luthier.
Bottom line: tune your uke the best you can, play it, and enjoy. I already wish I'd spent the time it took to write this playing my uke instead (and perhaps you do, too).
