How far down the neck do you want accurate intonation for $45? Just wondering so I can calculate the price of a good uke.
Well, it's not really "down" the neck I'm talking about (assuming you really mean "up" the neck - i.e. toward the body). Intonation at the 12th fret and above is even a bit spotty on some very expensive instruments - you're fighting physics and that's why most electric guitars have individually adjustable saddles.
But, intonation at the first fret should be as near perfect as makes no difference. I'm talking not out by more than a couple of cents
at the most. Most of your Hawaiian ukes, custom ukes, etc., will be perfect at the first fret. Factory ukes tend to come from the factory at around five cents out depending on the factory. Ukes setup by MGM, even $45 Dolphins, will be perfect at the first fret. Personally, I think MGM is possibly the best setup guy outside of a custom luthier's shop. I haven't had ukes by all of the other reputable sellers here, but of the ones I have purchased from all have been well setup with no more than a cent or two off on the intonation at the first fret and some being perfect. The difference is MGM can do it and also get a super "crazy low" action without any buzz.
Intonation at the first fret is almost entirely a factor of how high the slots in the nut are. If they are too high the string gets stretched and goes sharp when pressed at the first fret as illustrated in the video. Obviously, if intonation is perfect at the first fret it will also be so at the second and third frets and on up the board until you start running into that string physics thing the further up the fretboard you go.
Most people, especially beginners, play primarily in the first position so having intonation perfect at the first couple of frets makes all the difference in how quickly their ear will develop. If you play instruments that are poorly intonated, you never will develop a good ear. Once you start playing only instruments that are well intonated, most production guitars, ukes, etc. will drive you absolutely nuts. That G on the Dolphin in the video is so bad it's literally almost painful for me to listen to. Several years ago, before I bought nut files and started setting up all of my instruments for perfect intonation, I probably would barely have noticed.
John