For a sax player, just temperament, tuning by ear, is the rule. For a piano or fretted guitar, it is all about equal temperament. Most tuners also tune to equal temperament. there is a lot on the internet about all of this and why. Here is a good explanation.

I wondered if I was playing in tune by ear (just temperament) and I looked at a tuner (equal temperament) at that moment, what would it say? I didn't find this chart on the internet, so I posted it here for reference.

So, if I play a major scale and my assistant notes on my tuner that I am 4 cents sharp on the Maj 2nd and 13.5 cents flat on the Maj 3rd, I am spot on.

Interval __________ Cent Difference

Fundamental _______ 0.00

Minor Second ______ 11.8

Major Second ______ 3.9

Minor Third ________ 15.6

Major Third ________ -13.7

Fourth ____________ -1.9

Augmented Fourth ___ -9.7

Diminished Fifth ______ 9.8

Fifth ______________ 1.9

Minor Sixth ________ 13.7

Major Sixth ________ -15.6

Minor Seventh ______ 17.6

Major Seventh _____ -11.7

Octave ____________ 0.0

This is one reason experienced saxophonists recommend putting away the tuners and learning to listen.

It also makes me wonder how many intonation problems are due to tuner operator error rather than the saxophone.

[Edit] There are a number of methods used to calculate the "correct" ratios for just temperament. The ratio for the minor second was originally calculated as 25/24 but appears now as 16/15. Both are correct, but the latter is more common. I also added the augmented fourth.

The cent system is exponential, so the difference between 0 and 1 cent is much smaller than the difference between 99 and 100 cents. That said, most people cannot distinguish two notes played sequentially less than about 5-6 cents, and simultaneously with about 2-3 cents.