With all due respect to Rascher there is no need for tone imagination or any of that BS.
Just play a lower regular range G and then that's what to aim for an octave up.
What's needed, is to feel out the Altissimo G using oral manipulations (voicing) and then once the G starts happening to sound it will probably be a bit shaky and the chances of instantly hitting it again are probably not that high, so the player just has to keep at it for say 10 minutes a day and then eventually they will register in their mind/physical coordination how to play the note more predictably and can then prepare their oral physical coordination (voicing) ahead of sounding the note and can then reproduce the note at any time.
Jazz Is All brings up a very good point about different reeds and different mouthpieces and different saxes resulting in different Altissimo note success outcomes, and that comes about when a player has only a limited or not very good oral and airstream control (voicing ability) and can't adjust very well to a new setup.
The technical terms would be that the player can't match their own oral impedance (voicing) to the note/overtone impedances associated with a particular reed/mouthpiece/sax combo very well, whereas another player with experience might be able to.
When I was trying to do Altissimo for the first time, I would be able to get a Front F based Altissimo F# note after trying for a couple of weeks but I had about 6 different Saxes and various mouthpieces and reeds, and I would move to a new reed/mouthpiece/sax setup and the Altissimo F# would come out like a mouse fart or some strange multiphonic or something else, and I tried adjusting the Front F key vent height and all of that and it helped in some cases a bit, but eventually all of these problems between different setups went away because I just practiced every day for 10 minutes and ended up with more oral/airstream control (voicing control) and more predicting power about how to hit the various Altissimo notes on various setups, so it takes practice and technique adaptation.
Altissimo G on Tenor is well known as an awkward unstable note and if someone is overshooting it and getting a D, then their airstream/oral voicing is not right for that particular reed/mouthpiece/sax combo and so they need to adjust some part of their airstream/oral voicing, but it's a feel thing and it can't really be calculated, so more practice is needed.
Softer reeds are generally harder to voice than harder reeds and it's because of impedances.
The player has an oral impedance (which they can vary to tune into a particular Altissimo note) and the reed/mouthpiece/sax has varying impedances for every regular note and every overtone and the high overtones have lower gear impedance than the regular range notes do (which is why Altissimo is harder to play than regular range) and every sax note needs a certain level of impedance to exist before the note can sound (that's why the Altissimo range overtones are hard to play, because they don't have enough impedance for a note to sound easily, as regular range does), so the player has to increase their own oral impedance for the higher overtones to sound and that's called voicing.
Voicing an overtone means that the player is trying to tune their own oral impedance (voicing) to the overtones impedance (gear impedance) so that the note will sound.
Softer reeds make the reed/mouthpiece/sax combo gear impedance level lower to some degree, which means that the player has to make up for the missing impedance by using more fine control of their own oral impedance and increasing their own oral impedance (voicing).
High mouthpiece baffles tend to increase the gear impedance, so if a player switches from a high baffle piece to a lower baffle piece or vice versa, then they will need to adjust their own oral impedance (voicing) for Altissimo.
It's pretty easy for a low baffle player to overshoot the lower Altissimo overtones when first using a high baffle, because the gear note and overtone impedances have changed to some degree due to the high baffle mouthpiece.
A new vs old reed changes the gear impedance to some degree and so do different reed types, so if the player can get Altissimo happening on a wide variety of different gear setups, then they have pretty good voicing (oral impedance control) abilities, otherwise they might tend to hang onto certain gear setups that might work for them in some way and might tend to be very finicky about reeds and mouthpieces etc etc, because if the reed or mouthpiece was changed then they might have trouble adjusting to the new gear impedance because their voicing ability (oral impedance control) is on the weaker side.
Impedance levels can also effect the intonation of the note and it's tone to some degree, and so softer reeds tend to be harder to intonate when playing higher overtones (Altissimo).
The players own oral impedance (voicing) can influence how an overtones intonation and tone will turn out to some degree, because impedance levels can have an effect on intonation and tone.
The mouse farts and strange multiphonics and unstable notes and animal sounds are all a result of the players own oral impedance (voicing) not being the right match for a particular reed/mouthpiece/sax overtones impedance (if the sax is working ok).
If an overtone has the right impedance associated with it (done by the player getting the right oral impedance match for the particular gear overtone impedance) then the overtone will generally sound and be stable.
