Do Ability Scores level?

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It's my and my group's 1st time playing D&D, so I'm a bit new to what happens when a character levels up.

I realize that on every even level, the ability modifiers increase by 1. However, do the base modifiers always remain the same? For example, will having the abilities of 16, 16, 14, 13, 11, and 10 stay that way all the way to level 30, with the modifiers ever-increasing by other modifiers?

Thanks in advance.
At 4th, 8th, 14th, 18th, 24th, and 28th level you may add 1 to two different ability scores of your choice. This adjusts anything based off of the actually score (HP from Con) and the actual modifier (Healing surges from Con, Damage, etc.).

At 11th and 21st levels, all six of your ability scores are increased by 1 point.

ATTENTION:  If while reading my post you find yourself thinking "Either this guy is being sarcastic, or he is an idiot," do please assume that I am an idiot. It makes reading your replies more entertaining. If, however, you find yourself hoping that I am not being even remotely serious then you are very likely correct as I find irreverence and being ridiculous to be relaxing.

It's my and my group's 1st time playing D&D, so I'm a bit new to what happens when a character levels up.

Welcome aboard!, I hope you have many years of fun games ahead of you.

I realize that on every even level, the ability modifiers increase by 1. However, do the base modifiers always remain the same? For example, will having the abilities of 16, 16, 14, 13, 11, and 10 stay that way all the way to level 30, with the modifiers ever-increasing by other modifiers?

Ninjaed
One more related question. When calculating damage, which "modifier" do I use? My group is at level 3, so the Paladin's str is 4, but his mod + 1/2 level is 5. When calculating the damage of his longsword, does he add the 4 or the 5 to it? We have been using the score w/o, but it just feels like damage wont scale too well with their levels.
Damage does not have the 1/2 level added in at all, so you have been doing the correct math on that.

Damage scaling by level is represented by the gradual increase in dice used by damaging powers.

As an example: Brute Strike (Fighter Daily attack level 1) does 3[w], Chains of Sorrow (Fighter Encounter attack level 13) does 3[w] and a status effect.

Another factor in the damage scaling is access to more damage augmenting feats (on the occasions in which they stack) and magic weapons.

ATTENTION:  If while reading my post you find yourself thinking "Either this guy is being sarcastic, or he is an idiot," do please assume that I am an idiot. It makes reading your replies more entertaining. If, however, you find yourself hoping that I am not being even remotely serious then you are very likely correct as I find irreverence and being ridiculous to be relaxing.

One more related question. When calculating damage, which "modifier" do I use? My group is at level 3, so the Paladin's str is 4, but his mod + 1/2 level is 5. When calculating the damage of his longsword, does he add the 4 or the 5 to it? We have been using the score w/o, but it just feels like damage wont scale too well with their levels.

Correct, it's just +4 and woun't increase until his strength modifier does. Damage from any particular power does not scale well with level; unlike the attack roll it only receives the ability bonus.

However, higher level powers deal more dice of damage, so the impact of daily and encounter powers will improve as they are gained (and latter replaced). Also higher enhancements to weapons and implements do become accessible with increased level. Also feat bonuses can help improve things nicely.

Contrarywise, increases to abilities will greatly benefit some secondary effects where the ability modifier is the only component. Things like "..., and you can shift a number of squares equal to your wisdom modifier."