In order to play in tune for the entire length of the scale, you need to set your intonation properly. It’s not difficult to check to see if your guitar is set up for correct intonation.
Using a tuner to fine tune your intonation, start on the low E and check the 12th fret harmonic and the actual twelve fret E note. If they are not the same, you’ll need to either move the saddle forward towards the neck or backward towards the other end to tune your intonation.
You need to move the saddle back towards the tailpiece if the twelfth fret tone is higher than natural octave overtone. The trick here is to move the saddle only about 1/32″ or less at a time.guitar hardware
The screws that adjust the movement of the saddles are usually mounted at the far end of you hard-tail bridge or tremolo.
Shorten the string length by moving the saddle forward towards the neck of the guitar if the 12th fret tone is lower than the octave overtone.
Other factors that can potentially cause your intonation to be off are string height and change of string gauge.
If the strings are set too high from the fret board then you have to literally push them further to make contact with the frets. Additional stretching of the strings will result in the pitch being too sharp.
Another factor is the gauge of the strings. As the bow of the neck changes with the use of different gauge strings, If this is the case then you need to adjust the neck first before adjusting the intonation.
Lastly, defective saddles will move back and forth on the bridge causing your intonation to be off. In that case replacing your old bridge saddles will be the cure.