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Soft cap is where diminishing returns kick in.
https://mugenmonkey.com/darksouls/weapons
Yes, this is dark souls, but the principal is the same. Note that each tier is not necessarily consistent. One weapon with A might actually have a better multiplier than another weapon with A, but A should always be better than B. The letter is just a simple easy to indicate the tier bonus.
Soft cap is where you start seeing diminishing returns. Google that term If you haven't heard it before. Basically, you keep seeing significant bonus when adding skill points until you hit the soft cap, usually 40 or 50. After that, the points you invest just don't add enough bonus to make it worthwhile - your point is better used elsewhere.
Basically, each weapon has a base damage number - this is almost always higher for later-game and higher-class weapons. Scaling is what stats are added to the damage in addition to that. S-E shows the rate of efficiency of that stat adding to damage. So, if you took two weapons with identical base damage and made one S strength scaling and one E strength scaling, the S would do more damage. Raising your Strength stat increases the damage output of either weapon, but it raises the S-scaled weapon's damage more.
Now here's the confusing thing: Many Class 0 or similar weapons have better scaling than later weapons. However, the later weapons have higher base damage, so the later weapons still have a higher damage output. The numbers above the scaling (the "22.0 => 30.1" of your example) represent the amount of damage the weapon does on a light attack and a heavy attack, respectively. Some weapons have weak light attacks but insanely damaging heavy attacks. Those numbers are calculated for your character at the moment you are looking at them: it includes the scaling and adds your current stats in. Those numbers should be your guide to deciding between weapons, as they are the numbers the game will go by. (Do note, however, that most enemies have at least some defense against most attack types, so the numbers you see in combat will usually be a bit lower - and sometimes a lot lower if you are using an attack type the enemy is highly resistant to.) Though you should also keep in mind factors like stamina use, damage type, swing speed and reach in addition to damage. Also how cool they look.
If a weapon does not show scaling with a stat, then it does not add that stat to damage. If it has two or more stats, however, then it adds BOTH of them (or ALL for the rare weapons with three-stat scaling). So, assuming your stats are about equal, a Strength C/Dexterity C weapon is likely better than a Strength B weapon. Again, the "attack" numbers should be your guide in making these kinds of decisions. Double scaling is also important because of the soft cap - essentially, the game stops adding your stat bonuses to damage after that stat hits 50. (It doesn't quite stop doing it, but going from Strength 10 to Strength 11 gets you a higher benefit than going from Strength 50 to Strength 51.) Thus, if you are planning to do multiple playthroughs with the same character on New Game +, it's often wise to find a multiple-scaling weapon and leveling up both stats to the cap point.
In short, if you want to use that Trident, both your Strength and Dexterity will be added to the damage. However, a Dexterity 20/Strength 15 player will do more damage with it than a Strength 20/Dexterity 15 player (due to scaling), so I'd recommend leveling up Dexterity first.
You didn't ask about this, but just for the record: Upgrading a weapon at the blacksmith will improve its base damage. This can lead to confusion at the alchemist's, as anything they transmute will be one upgrade level lower than the current one, and so a weapon may seem to do less damage when it in fact will do more once you upgrade it.
Also for the record: Weapon buffs like Pitchfire and the like do damage based on the base attack of the weapon, so a weapon with high base attack is the best way to use such items.
and you know what i dont get, why is it hard to find a good small sword that can get you through the game. also, do throwing items like bombs, and throwing daggers increase damage with DEX?
The S-scaling kureimoa doesn't match the D-scaling seawolf cutlass until 78 strength. (Assuming linearity.)
All heavy attacks do 1.5x the damage of a light attack. The first number is base attack. The second number is actual attack, having factored in stat scaling.
The best way to see the scaling is to bring up the stat sheet with your weapon damage on it, and scroll over a stat point in the tree of skill. It will turn green and change to the post-upgrade number.
The soft cap for all stats except endurance is 50. Typically the scaling drops to 1/8th, though for some spells it's 1/10. Meaning that 78 strength for the kureimoa above is worth 53.5 pre-softcap points.