A lot of people do not understand how the fretboard has been designed.
The set of frequencies it is designed to produce are part of a mathematical and physics system called a Temperament. There are lots of Temperaments. They all have histories and issues.
The industry standard used to design and set-up your ukulele is called Equal Temperament. it has an in built "error" of around 2 cents for the Perfect Fifth interval and other intervals. it is like an average thing. It sets up your uke to be in the middle of a ballpark, then you use technique to fine tune dynamically as you play. Most people do it sub-consciously after month or so of playing without ever being taught about it.
An interval is a ratio between two audio frequencies, the root and the played frequency. played frequency/root frequency = interval expressed as a ratio.
So intervals are expressed as ratios. Some examples are: octave interval = 2:1, Perfect Fifth interval = 3:2 (1.5), Perfect 4th interval = 4:3. There are 12 intervals in the scales we use today, one for each chromatic note in a scale. The "perfect" set of intervals come from a certain temperament which may be one of the oldest temperaments dating from Pythagorus. They are all about sets of ratios, 12 ratios one for each note, some quite tricky with fractions of numbers like 32. The problem with this Temperament is that it is great in one key, but very troublesome if you want to change key. The frequency sets assigned to notes only work well for one key, when you change key you need to tweak the frequencies, like trying to twiddle the tuner knob half through a tune that has changed key for the bridge.
On an instrument like a violin which has no frets, you can tune by ear using physics. There is a thing called a beat, when you play two frequencies at the same time and they are the right frequencies you hear beats, tuning to a perfect fifth is done by twiddling the violin tuner knob until the beats are not there. The beats go away when there is a perfect 3:2 ratio of frequencies, this is a function of how sinusoidal waves interact. It works great on violins and has been used for centuries.
So many uke players do not realise that this will not work on a ukulele that has been designed and set up to the current industry standard. The fifth interval programmed into the Equal Temperament is 1.49, not 1.5. The fifth note in Equal temperament does not have a perfect fifth interval to the root note. The beats thing wont work unless you have the exact ratio of 1.5. An interval or ratio of 1.49 makes an audio frequency about 2 cents different to what you get if you use 1.5. So we keep getting this continual chatter from those who can tune in perfect fifths by ear, but they do not understand ukulele tuning, they always say there is an intonation error.
They try to tune the G string by ear using the beats and wonder why it ends up about 2 cents off from the note on the C string at fret 7, which should be the same note. Once they get the G string "wrong" it messes up the tuning most noticeably past fret 7. And they spend a lot of time twiddling. This "error" will happen in every uke that has been set up to the industry standard. Instead if you want to use tuning by ear for your ukulele with the best results you need to tune direct to the A note on each string or tune to the string in unison interval not to the Perfect Fifth interval. Unison is an interval or ratio of 1:1, IE the same note.
On top of that there are real intonation problems caused by poor quality manufacture and poor playing technique.
The first fret space is right next to the nut, it is where the string is hardest to push down because it is so close to the nut. it is a very common place to get an intonation error. There is no magic cure. A good set-up will help a lot.
As you get further along the neck, you are getting into tighter fret spacings. Another problem is string height is often highest around fret 12, so there is a lot more potential for bending the string unintentionally.
What you need to realise is that you have to work on technique more to get the intonation as you move along the fret board towards the sound hole. One of the benefits of a well made uke and well set-up uke is that you need to rely less on technique and it is easier to play with good intonation as you move along the fretboard. Some custom ukes are expensive because the maker has spent a lot of doing stuff you never see to make them so much easier to play up the fretboard. But often a good set-up will work as well.
Anyway, I suggest that you do not take my word for it or read wishy washy explanations in other posts. Do some research into these headings: Temperament. Harmonic Musical Series. Rule of 18 (a model for fretboard design). Tables showing Equal Temperament frequencies. Tables showing the 12 ratios and names for each Temperament. Tuning a violin by ear.
