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Given that every class gets a minimum of 5 ability score improvements each at levels 4, 8, 12, 16, and 19 (Rogue and Fighter get more at other levels), do these ability score improvements occur by character level or by class level?

For example, is it possible to create a character who never once gains a single ability score improvement by reaching level 3 in a class and then switching to a different class repeatedly?

I had originally thought the 5 that every class gets would occur by character level, so for example a sorcerer1/bard3 would earn an ability score improvement for reaching character level 4. But there are others who have expressed the opinion that this is wrong.

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Ability score improvements are a class feature, and are only gained when the class level grants it. Different classes gain these advances at different levels (the Fighter is the clearest example of this), and there is no unified table of advancement by character level.

It's possible to never gain an advancement by always multiclassing into a new class before reaching 4th level in a class. One effect of this is that "dipping" a class for a single level or two is less obviously useful; considering the extra limits on what is acquired from a new class's first level when multiclassing, this is likely a significant reason for the design, apart from being able to give classes different advancement rates.

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The effect is that this will penalize dippers LESS than it penalizes players who actually develop two different classes evenly. For example, a Sorcerer1/Bard19 loses no ability score improvements (and if he waits until his last level-up to dip Sorcerer doesn't even get set back a level in ability advancement) whereas a Sorcerer10/Bard10 in addition to already being gimped by not knowing any 9th-level spells is further gimped by missing out on his 5th ability score improvement and delaying the ones he does get unless he advances each class in increments of 4. –  Dyndrilliac Aug 14 at 21:33
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@Dyndrilliac You're looking at 20-level builds from the back end. Instead, it's exactly the 4-level increments that are the limit in forward progression. It also makes common 3.x builds that involve many single dips less attractive. –  SevenSidedDie Aug 14 at 21:37
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@Dyndrilliac 4 levels sorc, 4 levels bard, then another 4 sorc, and another 4 bard, will give you the same stat progression as everybody else, though you will lose out on the level 19 stat boost but gain your diversity in exchange. –  GMNoob Aug 15 at 6:51

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