Thanks for your insights, but my question was not about subclassing, changing spells and such.
Oh sorry, then definitely just allow your Wizard to cast two spells with every action at 5th level, and when you get to 11th level and some martials get 3 attacks per action, allow the wizard to cast 3 spells in one action. It definitely isn't OP at all. But still if you are worried, then just restrict the level of the spells cast to be no higher that the Wizard's proficiency bonus divided by 2 (rounded down). So at 5th level the wizard can cast 2x 1st level spells or 1x2nd or 3rd level spell per Action, paying the same number of spell slots as per normal. At level 9 they can cast 2x 1st or 2nd level spells or 1x 3rd or 4th level spell, and at level 17 they can cast 3x 3rd, 2nd, or 1st level spells or 1x 4th-9th level spell.
Your restriction suggestion is actually a MUST if more than 1 spell casting is allowed. Something that Focoe wrote to use CON save to see if you can even do it, or having a feat that is similar to second wind so you can only do it 3 times per short/long rest and maybe even to limit the spells or spell levels it affects.
But the other spells where you control summons would also cause issues, because with this things could go out of hand, so these restrictions should also be taken into consideration on those spells. Maybe it requires a complete rework of how magic is used. I think it is better not to touch it! :)
Thank you everyone for your valuable feedbacks!
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"Everything that has a beginning will come to an end!"
I would point out here too, that at 5th level, Wizards get Fireball. A spell that is by far better than hitting one target twice with a sword. D&D is about resource management, and for the Wizards it's about managing spell slots.
You can cast a Cantrip and a Levelled Spell in one turn. At level 5, your cantrips increase in power, generally from say 1d8 to 2d8. A sword does not level up in power at level 5. They just get to swing that 1d8 damaging sword... twice... for 2d8 total damage.
What you're really talking about is "I'd like to roll dice multiple times with my magic attack" but I think you're missing how powerful magic users get above martials, starting at level 5.
I think you also missed my intention with the ability to do more in one round. As I wrote this requires lowering damage and spell DC, and might also require a complete rethinking how magic usage is done in a fight. Maybe a better system would be to use action points instead of spell slots, so a Fireball would cost you 5 AP, while a single target spell would cast 2AP, then you could add more to a round of fight than just 1 spell and then move on to the next character.
Also, in legacy edition - that I am playing with the wizard -, you can only cast a cantrip with a leveled spell, when one of them has bonus action casting time. But I am not talking about pure damage output, but simply more variety.
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"Everything that has a beginning will come to an end!"
I think your best bet is to take a long look at the spells available and choose ones that stick around and have you use your BA. Bigby's Hand was mentioned already, but as a 5th level spell it's out of your reach at the moment. Flaming Sphere is something you should be able to get now, though. Find Familiar, while not quite the same, does allow for a variety of (lower case) actions that you can do during a round. And, while not very glamorous, Expeditious Retreat is a BA and allows you to continuously use your BA for a Dash Action as long as it's up. Unseen Servant isn't very useful in combat, but it also has BA usage.
This reddit post has a good list from 2014 and other sources. I'm not sure what you have access to, though.
Also, in legacy edition - that I am playing with the wizard -, you can only cast a cantrip with a leveled spell, when one of them has bonus action casting time. But I am not talking about pure damage output, but simply more variety.
And as various posters keep pointing out there are lots of ways to do more than cast one levelled spell as a wizard even in legacy D&D.
1) Take Animate Dead, get yourself 4 zombie / skeleton servants, now you can cast a levelled spell and make 4 attacks each turn.
2) Cast Flaming Sphere, now you can cast a levelled spell, and hit something with the sphere and position the sphere to force enemies to move.
3) Take and cast Summon Shadowspawn (or Summon Fey, or Summon Undead), now you can cast a levelled spell, and move & attack at least once with your summon, and use a BA/special abililty of your summon.
4) Cast Minute Meteors, now you can cast a levelled spell and throw up to two meteors at your favourite targets.
5) Take the Telekinetic Feat, now you can cast a levelled spell and shove a creature into danger, or pull a creature out of danger on each turn.
6) Pick up Find Familiar, now you can cast a levelled spell and any of the above, plus you can move around your familiar and give someone advantage on an attack.
7) Take Dust Devil, now you can potentially throw around enemies, and move your Dust Devil, and cast a levelled spell.
8) Take Darkness and cast it on a pebble, and have your Mage Hand hold that pebble, now you can cast a levelled spell and move around an area of darkness to help your allies or hinder the enemies every round.
9) Take Gust of Wind and cast it, now you can blow around enemies and cast a levelled spell on your turn.
10) Take and cast Shadowblade, now you can make 2 attacks on your turn.
And that's just your choices at level 6. As you level up there are even more ways to do a bunch of stuff on your turn - more summoning spells, Animate Objects, Bigby's hand, Dominate spells, watery sphere, etc...
With the right combination of feats, spells and subclasses it is easy for the Wizard to have the most things to do on their turn. E.g.
Order of Scribes Wizard at 6th level, in one round can:
Move and take an Action with their Familiar. Move and attack with 4 Skeletons Ride around on their Phantom Steed. Move your awakened Mind spellbook to get a new perspective on the battlefield (or use Telekinetic to shove and enemy into the perfect position). Move and attack and BA teleport with a Summoned Fey. Cast a levelled spell.
At 7th level they can add a Faithful Hound which attacks one creature within reach.
But I am not talking about pure damage output, but simply more variety.
The point I tried to make before is that you are way to stuck on "variety" only meaning "cast more spells every turn" when there is so much more that you can do.
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This reminds me of Overchannel, and I really like this idea! Thank you!
"Everything that has a beginning will come to an end!"
Oh sorry, then definitely just allow your Wizard to cast two spells with every action at 5th level, and when you get to 11th level and some martials get 3 attacks per action, allow the wizard to cast 3 spells in one action. It definitely isn't OP at all. But still if you are worried, then just restrict the level of the spells cast to be no higher that the Wizard's proficiency bonus divided by 2 (rounded down). So at 5th level the wizard can cast 2x 1st level spells or 1x2nd or 3rd level spell per Action, paying the same number of spell slots as per normal. At level 9 they can cast 2x 1st or 2nd level spells or 1x 3rd or 4th level spell, and at level 17 they can cast 3x 3rd, 2nd, or 1st level spells or 1x 4th-9th level spell.
Your restriction suggestion is actually a MUST if more than 1 spell casting is allowed. Something that Focoe wrote to use CON save to see if you can even do it, or having a feat that is similar to second wind so you can only do it 3 times per short/long rest and maybe even to limit the spells or spell levels it affects.
But the other spells where you control summons would also cause issues, because with this things could go out of hand, so these restrictions should also be taken into consideration on those spells. Maybe it requires a complete rework of how magic is used. I think it is better not to touch it! :)
Thank you everyone for your valuable feedbacks!
"Everything that has a beginning will come to an end!"
I would point out here too, that at 5th level, Wizards get Fireball. A spell that is by far better than hitting one target twice with a sword. D&D is about resource management, and for the Wizards it's about managing spell slots.
You can cast a Cantrip and a Levelled Spell in one turn. At level 5, your cantrips increase in power, generally from say 1d8 to 2d8. A sword does not level up in power at level 5. They just get to swing that 1d8 damaging sword... twice... for 2d8 total damage.
What you're really talking about is "I'd like to roll dice multiple times with my magic attack" but I think you're missing how powerful magic users get above martials, starting at level 5.
I think you also missed my intention with the ability to do more in one round. As I wrote this requires lowering damage and spell DC, and might also require a complete rethinking how magic usage is done in a fight. Maybe a better system would be to use action points instead of spell slots, so a Fireball would cost you 5 AP, while a single target spell would cast 2AP, then you could add more to a round of fight than just 1 spell and then move on to the next character.
Also, in legacy edition - that I am playing with the wizard -, you can only cast a cantrip with a leveled spell, when one of them has bonus action casting time. But I am not talking about pure damage output, but simply more variety.
"Everything that has a beginning will come to an end!"
I think your best bet is to take a long look at the spells available and choose ones that stick around and have you use your BA. Bigby's Hand was mentioned already, but as a 5th level spell it's out of your reach at the moment. Flaming Sphere is something you should be able to get now, though. Find Familiar, while not quite the same, does allow for a variety of (lower case) actions that you can do during a round. And, while not very glamorous, Expeditious Retreat is a BA and allows you to continuously use your BA for a Dash Action as long as it's up. Unseen Servant isn't very useful in combat, but it also has BA usage.
This reddit post has a good list from 2014 and other sources. I'm not sure what you have access to, though.
And as various posters keep pointing out there are lots of ways to do more than cast one levelled spell as a wizard even in legacy D&D.
1) Take Animate Dead, get yourself 4 zombie / skeleton servants, now you can cast a levelled spell and make 4 attacks each turn.
2) Cast Flaming Sphere, now you can cast a levelled spell, and hit something with the sphere and position the sphere to force enemies to move.
3) Take and cast Summon Shadowspawn (or Summon Fey, or Summon Undead), now you can cast a levelled spell, and move & attack at least once with your summon, and use a BA/special abililty of your summon.
4) Cast Minute Meteors, now you can cast a levelled spell and throw up to two meteors at your favourite targets.
5) Take the Telekinetic Feat, now you can cast a levelled spell and shove a creature into danger, or pull a creature out of danger on each turn.
6) Pick up Find Familiar, now you can cast a levelled spell and any of the above, plus you can move around your familiar and give someone advantage on an attack.
7) Take Dust Devil, now you can potentially throw around enemies, and move your Dust Devil, and cast a levelled spell.
8) Take Darkness and cast it on a pebble, and have your Mage Hand hold that pebble, now you can cast a levelled spell and move around an area of darkness to help your allies or hinder the enemies every round.
9) Take Gust of Wind and cast it, now you can blow around enemies and cast a levelled spell on your turn.
10) Take and cast Shadowblade, now you can make 2 attacks on your turn.
And that's just your choices at level 6. As you level up there are even more ways to do a bunch of stuff on your turn - more summoning spells, Animate Objects, Bigby's hand, Dominate spells, watery sphere, etc...
With the right combination of feats, spells and subclasses it is easy for the Wizard to have the most things to do on their turn. E.g.
Order of Scribes Wizard at 6th level, in one round can:
Move and take an Action with their Familiar.
Move and attack with 4 Skeletons
Ride around on their Phantom Steed.
Move your awakened Mind spellbook to get a new perspective on the battlefield (or use Telekinetic to shove and enemy into the perfect position).
Move and attack and BA teleport with a Summoned Fey.
Cast a levelled spell.
At 7th level they can add a Faithful Hound which attacks one creature within reach.
The point I tried to make before is that you are way to stuck on "variety" only meaning "cast more spells every turn" when there is so much more that you can do.