I am asking for research to determine many class options and how I want to build a potential character. This concept has interested me for a while and I'd like to know. However, I could not help but notice that it doesn't say anything about it. As worded here:

Bladesong Starting at 2nd level, you can invoke a secret elven magic called the Bladesong, provided that you aren't wearing medium or heavy armor or using a shield. It graces you with supernatural speed, agility, and focus. You can use a bonus action to start the Bladesong, which lasts for 1 minute. It ends early if you are incapacitated, if you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon. You can also dismiss the Bladesong at any time you choose (no action required).

This would mean that it isn't exactly limited to melee weapons, but also certain ranged weapons such as the hand crossbow, the dart, and the sling. But, a problem I could not help but notice was that this ability is a bit hindering.

Song of Victory
Starting at 14th level, you add your Intelligence modifier (minimum of +1) to the damage of your melee weapon attacks while your Bladesong is active.

Thus, would it be viable for one to just use a hand crossbow, or would it be better to take thrown melee weapons?

A Hand Crossbow is not usable with Bladesong and does not synergize well with the Bladesinger class either

The only pertinent ending condition for Bladesong is the one you have listed.

It ends early if... you use two hands to make an attack with a weapon. (SCAG p142)

Hand crossbows only uses one hand to attack, as they lack the "two-handed" and "versatile" traits (PHB p147 and 149). Nothing else in the Bladesong feature precludes their use.

However, they are not a valid weapon choice as per the Bladesinger Styles descriptor, all of which are melee weapons. This question, while dealing with spells, has some great answers regarding the nature of flavour versus crunch in D&D 5e. It boils down to there is no flavour crunch within spell descriptions and that everything is important. Applying that logic to this section of the manual results in HCs not being a viable choice, because they are not listed as such.

Related to this, it's interesting to note that both High Elves and Wood Elves have racial proficiency with bows, both long and short. So even though these are weapons that literally every single Elf knows how to use effectively, they are not mentioned at all in the descriptor of different Bladesinger styles. This omission leads credence to the assertion that only melee weapons are valid choices for Bladesinging.

Negative Synergies

There are several other mechanical reasons why a Hand Crossbow is not a good choice as a weapon to utilize with this feature.

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  The Bladesinging Arcane Tradition and Wizard class don't grant proficiency with the Hand Crossbow (SCAG p142 and PHB p113). The Training in War and Song Feature grants proficiency with

  
  
  

  
  

  one type of one-handed melee weapon

  
  

  
  
  

  This can be overcome via multi-classing, but I've included it here to point that out. The other way you could get proficiency with the Hand Crossbow is by being a Drow Elf (PHB p24).

  
  



• You won't be able to use the Extra Attack feature because of the Hand Crossbow's Loading trait (PHB p147 and 149) unless you take the Crossbow Expert feat (PHB 165).



• The Song of Victory feature becomes useless, as you can only add your Intelligence modifier to the damage rolls of melee weapon attacks (SCAG p142).

Probably not : it's written for melee weapons (thrown should work)

Here is the text from Sword Coast Adventurer's Guide that indicates that this style of fighting is used with swords.

Under Bladesinging

They are elf wizards who master a school of sword fighting grounded in a tradition of arcane magic. (SCAG, p. 141)

Under the Green Box (Bladesinging Styles) on the next page:

From its inception, as a martial and magical art, Bladesinging has
been tied to the sword, more specifically the longsword. (SCAG, p. 142)

Other melee weapons can be used with the blade song.

When you read further into Bladesinging

Cat: Styles that employ a sword belong to this family...
Bird: Styles that focus on the use of a hafted weapon, uch as an axe or a hammer, have been grouped together as bird styles ... Eagle styles bladesingers use small hand axes, and many maneuvers in the style focus on fluid ways to throw the weapon and draw a new one.
Snake. Practitioners of these styles use a flail, chain, or whip... (SCAG, p. 142)

Nowhere does it mention hand crossbows, or any other bow. If they had intended to include one handed bows in this style, it would likely have been put into that green box with an animal name. Given the thrown axe style, a thrown dagger style would seem to fit, perhaps as a variation on the Cat fighting style.

Why doesn't the level 2 feature say "melee weapon" like the 14th level feature does?

Good question. I will suggest to you that having written all of that about swords in the introduction of the Bladesinger class, and the animal named styles in the green box for some other melee weapons, the authors didn't feel the need to be redundant in their phrasing of the 2d level class feature that unlocks the Bladesinger school. Or, they overlooked it in the final edit and now rue the day that text miners discovered their oversight. ;) The evidence points to this being linked to melee weapons, not any kind of bow. (Interestingly, the errata does not address this).

Ask your DM

If your DM is willing to let you use a hand crossbow with the Bladesong, you'll need to work together on that, and deal with the 14th level class feature when you get there. Hopefully, your adventures will take you that deep into the campaign!

Probably Not Strictly RAW Compatible Due to Reloading

First off I must say that I highly recommend the latter half of Nesbitto's answer for explaining why this is a suboptimal choice. I would add only that this might not be relevant to a tier one campaign or to a multiclass dip into Bladesinger (probably not a bad option for an Arcane Trickster or Dex-based Eldrich Knight), and that little stops a character from fighting in one way at low levels and a totally different way when the reach higher levels.

General Restrictions

As per the rules you cited, the only things that can end a bladesong are wearing the wrong armor, using a shield, being incapacitated, or "[using] two hands to make an attack with a weapon". While the lore strongly implies having one melee weapon in one hand, there is no such requirement. If you look at this previous question you'll see the argument for them being able to dual-wield, despite this also not really matching the lore. Basically the ability as written only deals with giving limitations not listing valid options, which makes sense in a game where you can do anything.

Lore Based Mechanical Issues?

I hope I'm not being overly contentious, but several answers here seem to rule out the hand crossbow by relying on the fluff and lore of the Bladesinger, citing this question, involving spells not making a rules/fluff distinction, as a basis. This approach seems to be an overly broad reading of the rules on spells (where there is no set apart section with clear rules) into determining that every word of lore relating to character classes is a mandatory aspect of the character and game mechanic. I have never heard of a Warlock character being told she absolutely had to be "driven by an insatiable need for knowledge and power" or that a circle of the land Druid character had to actually spend any of his time meeting "within sacred circles of trees or standing stones to whisper primal secrets in Druidic" but perhaps some tables read such things directly into class mechanics, and presumably they would ban hand crossbow wielding Bladesingers. However I think such lore based limitations are usually viewed more as house rule decisions than a rules as written-based decisions.

For the record, I insisted that my own Bladesinger character had to not only use a sword but literally sing to his sword (the sword was called Caroline, so he sang a certain Neil Diamond hit at the beginning of every combat), but that was obviously very much a self-enforced rule.

But even if all lore is mechanically significant it makes no difference. Even if it is a firm rule that a Bladesinger specializes in and mostly uses a blade, whip, axe, etc., as heavily implied by lore, that in no indicator that they do not pull out a hand crossbow from time to time and that this interrupts their musical magicks.

Thus we are so far in the clear.

Probably an Issue: What does "use two hands to make an attack" Mean?

There is an issue though. As per the equipment entry on bolts: "Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon)". Thus if loading/reloading being "part of the attack" counts as part of "making an attack" for the purposes of Bladesong then the character has used two hands to make that attack.

While the wording is so close that it seems to strongly imply that this is the case, there is still some wiggle room based on the probable intent of the rule for reloading being to address needing a free hand during the attack action (which can be one of the hands holding a two-handed weapon), and the actual "making" of an attack not necessarily referring to the full attack action.

That said, reasonable minds could certainly differ, and I imagine most DMs would err on the side of handwaving. However under the most conservative reading of rules there does seem to be an issue.

Note: I believe the rules on ammunition were errataed to cover dual hand crossbow reloading confusion, so your Player's Handbook may read differently.