The basic idea of mouthpiece position on a sax is that pushing the mouthpiece farther down on the cork will make your intonation higher while pulling it out will make your intonation lower.
To quote Jimmy, “You are basically playing a long tube when you play a saxophone. Saxophone is all about the speed of the air like any wind instrument. The length of the tube determines its harmonics. Since EVERYONE has a different throat ,mouth , teeth, and tongue, it is a truly individual journey to find out what works for you. I meet a lot of people who are looking for a standard but in fact there is only one way that will work for you.”
So it’s not as simple as pushing the mouthpiece on farther or pulling it out more.
The temperature and the environment affect intonation too- if it’s hot outside the air will move faster and the sax will be more sharp, if it’s cold- it will tend to be flat.
And a saxophone in tune on one note will not necessarily be in tune on another note, if not adjustments are made by the player. With practice, you’ll develop control and these things will happen subconsciously.