Help me understand the interplay between spellcasting and melee combat
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For the next six seconds, you cannot do any of the above.
You may, however, always attack the same round, providing the casting time is low and/or you have plenty of attacks.
So with your example, you would Fire Whirlwind, click attack on target opponent, and then spend two rounds dealing fire damage, likely with all, or at very least most, of your APR
This also means that when you cast Whirlwind Attack, assuming the casting time is 1, this causes a short delay in your attacking. When you start attacking again, this should be an entirely new attack round. However, that casting time does still cut into your DPS, in the sense that you weren't attacking for those 0.6 seconds.
If you have a casting time of 0, however, it might truncate an attack for a different reason, because that instantaneous casting time prevented you from beginning a new attack round. If you have made 2 attacks out of your normal 3 APR, then Whirlwind Attack might not start a new attack round, and thus you might lose an attack because of the transition.
Long story short, it should effectively give you at worst 9 APR, and that likely only applies to very slow weapons like two-handed swords, and even then, only if you've attacked one or two seconds before the spell.
I wouldn't sweat it, honestly. This kind of truncation happens with any spell--your custom spells will behave just as any others. If you want to be absolutely sure, though, that your fighter with 5 APR gets to land 10 hits with fire damage, you might increase the duration of that spell to 13 seconds.
Potions and items effectively count as spells, with the notable difference that they aren't affected by Improved Alacrity. If you've begun casting a spell within the last 6 seconds, you can't drink a potion, even with IA active. IA will, however, let you start casting a new spell right after drinking a potion. So a mage could attack and cast a spell within 1.5 seconds, but could not drink a potion less than six seconds after starting to cast that spell.
Also, speed factor alone should not be sufficient to let your mage land a hit within 1.5 seconds. Not unless your mage has high APR. APR makes the first attacks in a round occur earlier, and makes a much bigger impact than speed factor.
Certain actions rely on your "aura". This terminology comes from spellcasting. Normally the aura is clear and clean, but when a character begins casting a spell their aura gets clogged up with magical energy. While the magical energy is clogging a character's aura they cannot cast another spell until the energy drains out of the aura and it becomes clean again. It takes 1 round for the aura to completely clear regardless of the power of the spell. IE games are programmed so that certain other actions will clog the aura as well as spells. Thus, performing any of these actions will clog your aura and you will not be able to perform another aura clogging action until the aura clears. These are the actions that will clog the aura:
Casting a 'spell' of any sort (this includes all priest spells, wizard spells, and innate spells and abilities).
Using an item of any sort (potions, equipped items, wands, any quick slot items).
You will notice that attacking is not listed there. You can cast a spell and use the remainder of the round to perform any attacks you have left in your round. This needs further explanation. To see how this stuff works for yourself in game I would recommend turning off all auto pause except "round end". A round timer begins when you start any action. This can be attacking, casting a spell, or using an item. When the round timer starts it will take its course until all 6 seconds of the round are up and then, if there is another action in the queue, another will start. Attacks will always come at specific points during the round. When the attacks come depends on their speed factor and how many are going to be in the round (aka APR). If you preform a non attacking action during the time an attack would normally happen then that attack is wasted. Lets look at a few example rounds. Keep in mind that rounds are divided into tenths (for my purpose I will call these 10 tenths of a round "segments" and number them sequentially, starting at 0 and ending at 9).
Character has 3/2 APR
Round 1:
Segment 0: casting of magic missile starts, aura clogged for remainder of round, Attack roll scheduled for this time replaced by casting.
Segment 1: Magic Missile casting is completed.
Segment 2-4: nothing happens.
Segment 5: Scheduled attack roll happens.
Segment 6-8: nothing happens.
Segment 9: Nothing happens, Round concludes at the instant segment 9 ends and the aura is cleansed.
Round 2:
Segment 0: Scheduled attack roll happens.
Segment 1-8: Nohing happens.
Segment 9: Nothing happens, Round concludes at the instant segment 9 ends and the aura is cleansed.
Round 3
Segment 0: Scheduled attack roll happens.
Segment 1: Potion is drunk, aura clogged for remainder of round.
Segment 2-4: Nothing happens.
Segment 5: Scheduled attack roll happens.
segment 6-8: Nothing happens.
Segment 9: Nothing happens, Round concludes at the instant segment 9 ends and the aura is cleansed.
Lets use Greater Whirlwind now. Greater whirlwind has an instant casting time so it does not interfere with any remaining scheduled attacks in the round.
Round 4:
Segment 0: GWW is cast, Arua is clogged for the remainder of the round, Scheduled attack roll happens.
segments 1-8: Scheduled attack roll happens.
Segment 9: Scheduled attack roll happens, Round concludes at the instant segment 9 ends and the aura is cleansed.
Now lets throw improved alacrity into the mix. Improved alacrity automatically cleanses the aura after a spell of any sort clogs it. It does NOT cleanse the aura from item use!
Round 5:
Segment 0: Casting of IA starts, aura clogged for remainder of the round (at least until IA casting concludes), Scheduled attack roll replaced by casting of IA.
segments 1-8: casting of IA continues.
Segment 9: Casting of IA completes at the beginning of the segment, aura immediately cleansed, Magic Missile Casting begins, aura immediately cleansed, Magic Missile Casting is completed at conclusion of this segment.
Now lets combine GWW with IA with Attacks.
Round 6 (IA still active):
Segment 0: GWW cast, Aura cleansed, Scheduled attack roll happens.
Segments 1-9: Scheduled attack roll happens.
Just for fun lets toss more crap in there.
Round 7 (IA still active):
Segment 0: GWW cast, aura cleansed, Assassination cast, Aura cleansed, Critical Strike cast, Aura cleansed, Scheduled attack roll happens.
Segment 1-9: Attack roll happens.
(High level FMTs are really good aren't they?)
So after having fun with my thought experiment rounds I kinda lost my place. I hope this clears it up for you. Basically the trick with Combat abilities like GWW or Quivering Palm is to give them an instant casting time and then the attacks can happen as they should. Some modders will add a very short duration (like 1 second) aura cleansing effect to certain abilities so that they un-clog the aura the spell dirties up as soon as that spell is cast. I believe SCS does this for their instant shapeshifting component (or whatever it is called).. So there is really a lot you can do here.
Any questions, class?
By equipping the robe, the single class arcane caster will cast earlier in a round but he won't cast more spells, except under Improved Alacrity.
By equipping the robe, the multiclass or dualclass will spend less time casting and more time attacking, usually resulting in 1-4 more APR.