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Notably the ability modifier is INT for Fighters and Rogues, WIS for Monks and CHA for Barbarians, and their own spellcasting ability modifier for all the spellcasting classes, if the wiki is to be believed.
Haven't tested it exactly in the game, but I suppose it sounds right.
For multi-classing I would have to assume it uses whichever is higher, just like when determining whether to use DEX or STR for weapon damage.
Cantrips scaling is based on total character level (1d12 1-4, 2d12 on 5th level etc)
Spell's spell-casting stat is based on the class of the spell. For example, wall of ice is wizard spell, so it will use INT
If there are multiple classes able to use the spell, they are considered separate. For example, fireball can be either wizard or sorcerer spell. If you learned it as wizard, you get INT check, but if as sorcerer, the check is CHA. In the case you have learned the spell from both classes (whyyy), you would have two separate spells for INT and CHA check.
It is only 5e tho, haven't checked it in bg3 yet. Though i've given Gale priest feat and those spells are WIS.
UPD: scroll/items spell is the real question, if the spells are shared between multiple classes. It should be investigated.
Lv9 wizard + flaming hands scroll = INT roll (reasonable)
Lv1 sorcerer + flaming hands scroll = CHA roll (still ok)
Lv1 wizard Lv1 sorcerer 17 int 12 cha - CHA roll
Lv1 sorcerer Lv1 wizard 12 int 17 cha - INT roll
Lv2 wizard Lv1 sorcerer 17 int 12 cha - still CHA rolls.
Seems like the last class taken is the case for the scrolls. It all goes as i said before, if the spell is learned, though.
thank you for sharing your data. as far as I know, this applies to all spellcasting, not just scrolls.
so if you have a level 1 sorcerer, and then take 1 level in wizard, all your wizard and sorcerer spells will derive from INT. same character at level 3, taking a 2nd level in Sorc, will still use INT, because it derives from the "newest first level" caster. If at level 4 this same character takes Cleric, their spellcasting stat, for all spells, will become WIS.
maybe just a conspiracy theory though