Can a wizard learn a spell from a bard, assuming the spell is on both class lists? This is related to Can a warlock teach a spell to a wizard? but is not exactly the same, since warlocks don't really understand wizards' magic, and bard is a different class.
Our party is only level 2 and we're trying to allay costs for transcribing spells a bit. Would this work out?
Your plan will not work for 2 reasons:
The Player's Handbook outlines the ways that a Wizard can learn a new spell via Wizard class features:
By leveling up (research)
The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse.
Now, thematically, you might say that, as part of the Wizard's research, the Bard was able to help them learn and master the spell. However, this would not grant you any additional spells beyond what the Wizard would normally be able to learn on the occasion of their leveling up. So this would be a flavor-only thing and not mechanically helpful which it seems is what you want.
By finding spells on scrolls or in spellbooks and copying them into their spellbook
They can also learn them from other wizards, from ancient tomes or inscriptions, and from ancient creatures (such as the fey) that are steeped in magic. (PHB 112)
In general, the way a Wizard learns spells is by deciphering some written form of the spell and then laboriously copying it into his own spell book.
Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. (PHB 114)
Note how this quote emphasises that the material they learn from is generally assumed to be 1) written and 2) written by a wizard specifically.
So there is no mechanical way for a Bard to teach a Wizard a spell.
Not only does it not work mechanically, it also doesn't make sense thematically since Bards and Wizards have very different ways of casting and their techniques are just not transferable.
"We're trying to allay costs for transcribing spells a bit"
This part deserves some attention as well. Aside from leveling, using any other method in the Wizard class to add spells to your spellbook require you to take the time and pay the cost associated with adding that spell to your spellbook.
Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it.
So even if a Bard were to teach the spell, you would still have to transcribe it into the Wizard's spellbook and thus take the time and costs specified in the book.
As always, the DM may allow this to happen in some form, but just know that that decision is neither supported by the rules nor the flavor of the Wizard and does not change what the rules say about it.
Not directly; according to the rules, a wizard needs a written source to study and copy from, to add to his spellbook. So the bard would need to craft a scroll with the spell on it, for the wizard to use (assuming that PCs may create scrolls in their downtime at the table in question). This will not allay your costs.
The answers to the related question that is cited in this question, about wizards learning spells from warlocks, has all the information needed to answer this question. The only difference is that warlocks have a couple of possible workarounds that bards do not. The point is that, for the same reasons, adding any spells to a wizard spellbook, other than the two he gets for going up a level, requires a written source, according to the RAW and all of the examples and methods given.
Here are some of the other questions on this topic that may provide you with additional material: Can a warlock teach a spell to a wizard? , Can Wizards, Tome Warlocks, and those with the Ritual Caster feat use spells from each other's books? , Is there a difference between “spell scrolls” and “spells on a scroll” for copying purposes?
Pertinent text for adding spells to your spell book PHB pg. 94:
The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse.
Note that there are no restrictions on how you come about this knowledge. Most examples in this section cite finding written sources, however any research you conduct on your own could also qualify. Examples of research include practicing with other spellcasters, experimenting on your downtime with random components, studying books, talking to other casters, debates, etc.
Basically, it's up to you and the DM to come up with a way for your Wizard to add spells. The typical method is scrolls and books, but you aren't limited to just that. Since you are capable of adding spells through research, which is not limited to scrolls and books, then speaking with a Bard who has spells you do not, and having them show you how to do the spell, can indeed qualify as a method of adding new spells to your book.
Further supporting evidence can be found on PHB pg. 203 with reference to the schools of magic.
The Schools of Magic
Academies of magic group spells into eight categories called schools of magic. Scholars, particularly wizards, apply these categories to all spells, believing that all magic functions in essentially the same way, whether it derives from rigorous study or is bestowed by a deity.
The inference here is fairly clear. A wizard doesn't see a Bard spell functioning any differently from a Wizard spell. This is also clearly highlighted under the Wizard text oh PHB pg. 112, Scholars of the Arcane (emphasis mine):
Wizards live and die by their spells. Everything else is secondary. They learn new spells as they experiment and grow in experience. They can also learn them from other wizards, from ancient tomes or inscriptions, and from ancient creatures (such as the fey) that are steeped in magic.
So clearly a written version of a spell is not required in order to scribe one. Nevermind the very obvious inference from reducibility, in that the first wizard would have had no written references to scribe their spells from. This means that spells can and are able to be scribed from sources other than written mediums.
Lastly, the wizard itself serves as an example of something able to scribe a spell without having a written scroll or book to reference it from. If a wizard loses their spellbook, they can create a new one without having a written reference:
Replacing the Book. You can copy a spell from your own spellbook into another book—for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell. You need spend only 1 hour and 10 gp for each level of the copied spell.
If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spellbook.
This last bit is important because you aren't using anything written yet you are scribing a spell anyways.
However you want to look at it, your Wizard does not REQUIRE a written form of a spell in order to scribe it. The wizard simply needs the spell to be on their spell list, they need some way to learn it, most commonly a written form, but it could be an instructor capable of casting a spell on their class list, or it could be simple experimentation on their own using a DC set by their DM to reflect the difficulty of the spell.
The only thing the rules state you have to do is:
This is all covered under PHB pg. 114, Copying a Spell in the Book:
Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.
None of this says anything about you having to have a written version of the spell to copy it at all. It says you have to decipher the unique notation of the wizard who wrote it? Great. What if a wizard didn't write it at all? Step done, no deciphering needed, next step. Practice hand gestures? No somatic component. Step done. Practice verbal? Good, verbal component only spell. Nice and easy. Practice away my friend.
