The saxophone, as all instruments do, includes notes in its range that are inherently out of tune. However, there is no one-size-fits-all approach to fixing them or even listing them. There are a wide range of tuning discrepancies between instrument manufacturers and even variability between models within the same brand. Additionally, soprano, alto, tenor, and baritone saxophones will often exhibit opposite intonation tendencies on certain notes.
It is important to note that the pursuit of exemplary intonation is more complicated than raising flat notes and lowering sharp notes; saxophonists should remain flexible by developing a repertoire of fingering choices for each note in order to have options available in any musical situation. As an example, conventional methods would tell students that G4 is a flat note best corrected by adding the third-finger fork F# to the regular fingering. However, at an extremely quiet dynamic, the G will likely not need to be raised because the natural tendency of the instrument is to play sharper at piano or pianissimo dynamics. Also, if this G is part of an Eb major chord, then it would be desirable for the note to be a little flat; furthermore, if the Eb in this scenario was flat, it would be necessary for the G to be lowered even more.
Thus, the best choice is to approach tuning from the perspective of being adaptable rather than assuming that one way of doing things is always correct. This mindset has the added benefit of encouraging students to listen to ensemble intonation and adjust if it sounds wrong. If the brass section naturally drifts slightly sharp on a forte passage, it is much easier for a saxophonist to temporarily raise notes with a fingering adjustment that it is to re-tune everyone. Even if the saxophone note in this scenario is inherently sharp, it may be necessary to raise it further to compensate for the loud dynamic, which generally causes the saxophone to drift flat.
Some schools of pedagogy favor adjusting via voicing manipulation or embouchure pressure changes rather than fingerings. Many professionals use such methods of adjustment, and they remain a valuable pedagogical approach. However, it is much safer to approach the pursuit of good intonation through fingering adjustment for three reasons.
First, it is a more exact science. If a student has to lower a note by a certain amount, it is much more predictable and reproducible to use a fingering to make the adjustment instead of a manipulation of embouchure or voicing. Similarly, telling a whole section to “lip it down” will produce wildly varying effects due to different interpretations of the same statement. In short, the fingering will have a similar effect in every instance, whereas adjusting the embouchure or voicing is dependent on fatigue, reed resistance, and dynamics, just to name a few factors.
Changing fingerings also preserves timbre better. In certain registers and on certain notes, it is crucial for a constant amount of embouchure pressure be maintained to achieve a good timbre and response. For example, on low notes lipping down creates poor response and can introduce a buzzy sound. On the other hand, lipping up is unable to elicit as big of an adjustment and can make a very thin sound.
Finally, changing fingerings fits better with band pedagogy. A trumpet player adjusts written D4 with the tuning slides, not by lipping it down. Similarly, one would never tell a tubist to lip down a fingering using first and third valves if the instrument has a fourth. Thus, telling the saxophone section to lip it down when a better fingering exists is not the best practice.
To these ends, the fingering chart (pdf below) should be viewed as a simple point of departure for students to use when approaching intonation. Many saxophonists use corrective fingerings that are widely known and accepted, and many have also adapted their fingerings relative to their specific make and model of saxophone. These fingerings should also be viewed in the proper context; if a student always plays flat, there are fundamental tone production problems that are causing this. It is not as simple as fixing every pitch with a corrective fingering.
The notes below written D4 are usually quite sharp on most saxophones and can be problematic to adjust by altered fingerings because of how many fingers it takes to play these notes. If the saxophonist tries to lower the pitch by relaxing the embouchure, response in this register will be greatly diminished and the timbre will become unstable.