Honestly, it's pretty complicated in the amount of "stuff" there is but pretty easy when you focus on what you want to do. Just copy/pasting my last post on this:
I wouldn't fret too much about them. Basically, all shards have two (well, three) components:
1) A Quality type; Cool, Cute, Keen, Hard, Buff, Barbed, Wicked, etc.
2) A Shard type; Attack, Mind, Defense, Resist, Aim, Evade, Rise, Exceed, etc.
3) A value; [1], [2], [3], etc.
When you dualize a shard with a piece of equipment, you'll get two immediate bonuses:
1) A
permanent stat boost dependent on the Quality type; the Quality you get on the item is determined by the Quality of the Shard and the Quality of the item being dualized.
2) A
temporary bonus effect dependent on the Shard type and value.
After fighting some battles, the equipment will become tempered (the name turns yellow) and you'll get an extra effect of +Damage% Dealt (Weapons) or -Damage% Taken (Armor). This, like the bonus effect, is temporary.
Now, if you dualize together two tempered pieces of equipment, you'll extract the bonus effect from them and create a Gem with both their effects. The bonus effects from the equipment involved will be lost (as will the tempering bonus, though the Gem doesn't get this bonus) but the stat boosts remain. Now, you return to step 1 and repeat the process.
I'd just try it out to get a hang of it. Take an item you want to dualize and check what your shard qualities give you; the game shows you them when you select them so you won't be doing it blind. There's also a list in the Library > User's Guide > Qualities menu. It's a lot easier than it sounds since it's basically just:
1) Dualize shard onto equipment.
2) Fight some battles to temper equipment.
3) Dualize two tempered pieces of equipment.
4) Repeat.
To give an example, let's take
this list of Qualities and apply some things. The first thing to note on how the process works is all shards are split into "tiers" based off of how many letters are in their quality. So the lowest tier is for Cute, Buff, Hard, Keen, etc. with 4-letters and goes up to the 8-letter ones of Fabulous, Dazzling, Perilous, etc. The Easygoing, Tolerable, Diffident, Impartial, etc. ones are special cases so don't worry about the letter count in them.
[strike[These bonuses are from memory so sorry if some of them don't match up in-game.[/s] Oh neat, they're in the Google Doc I just linked. I should read what I actually link more often :|
So, first we'll Dualize a Shard with the Buff Quality onto a Weapon with the starting Quality of Cute:
Cute + Buff => Olden (+3 P.Atk, +3 C.Atk, +4 Accuracy)
So our item will end up with the Olden quality and gain those stats
permanently. Let's fight some battles to temper it and gain +5% Damage Dealt and we'll Dualize this piece of equipment with another tempered one to remove the effect (and tempered bonus) from the Shard. Again, these stats will remain and the item will still be of Olden Quality at this point.
So, next, we'll dualize this Olden Weapon with a Shard with the Rigid Quality:
Olden + Rigid => Wicked (+8 P.Atk, +3 C.Atk, +2 Accuracy)
And, again, we'll do some fights until it is tempered and gains +7% Damage Dealt. And, again, this will be dualized with another tempered equipment slot to remove the effect of the Shard but those stat bonuses will remain and it will still be of Wicked Quality.
Now, we'll dualize this with a Barbed Quality:
Wicked + Barbed => Epochal (+9 P.Atk, +3 C.Atk, +2 Accuracy)
After tempering it'll get +8% Damage Dealt yadda yadda blah blah. Now, to demonstrate what happens when you dualize a lower tier shard, we'll use an Olden Shard here:
Epochal + Olden => Olden (+3 P.Atk, +3 C.Atk, +4 Accuracy)
So here it'll just take the effect of the lowest "tier" Shard Quality in the mix.
To sum up what we did though, we started with a Cute weapon and then went to Olden, then to Wicked, then to Epochal, and then back to Olden. This means you have a +4 Weapon (since you did 4 total dualizes) and you permanently added the following stats:
+23 P.Atk, +12 C.Atk, +12 Accuracy. Since your current Quality is Olden, you'll end up with a +5% Damage Dealt Tempering bonus once it gets tempered. The effect on the item is dependent on the kind of Shard you used; I'll just say we used a Rise Shard so we have +1 Minimum CC.
For the record, I chose Olden on purpose here to get us back to the starting point. This way you can then go into Wicked again by dualizing a Rigid Quality with the weapon and then go into Epochal by used a Barbed Quality next. If you had a lot of Barbed Shards though, you could skip that Olden step entirely and go Epochal -> Barbed -> Epochal -> Barbed -> Epochal -> Barbed, etc. to really get a lot of +P.Atk out of doing this over and over. This is basically the process I'd do to get a lot of P.Atk and some C.Atk/Accuracy onto a weapon.