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I am playing an Aarakocra bard character that looks like a crow/raven in a D&D campaign.

What kind of instruments would he be proficient in and play? Would an Aarakocra be able to play a flute, given that he has a beak? Would he even need a flute if he can sing like a bird?

RAW, there is no limitation.

As usual, this is not a thing that the ruleset cares about, so Aarakocra can play any instrument just as well as anyone else.

If you're looking for a well fitting instrument for them, I would go for something small/light (i.e. not flight-hindering) and string based, like a small harp or lyre, since their talons would make some great guitar-picks.

If we go further, we might even think about altering the less fitting instruments to improve the aarakocran user experience. So just imagine a flute that would have a very specifically shaped mouthpiece and finger holes to accomodate the beak and talons.

As for the question about not needing a flute – I, as a human, can whistle and clap my hands and sing (not very well, but still) and yet various musical instruments were a fairly early invention of our history.

The rules gloss over the issue (but you don't have to).

You got bard levels? Yes? Great! Someone in your culture can inspire people with some kind of music, and that's what your bard does. Your society is full of bird people? That's fine; someone figured out the details.

Our real-world instruments and music were designed to be operated by your typical human with ten fingers, two feet, lips, etc. because that's the kind of person we have in abundance. Look, though, at Ravel's Piano Concerto for the Left Hand. One guy missing his right arm inspired a musician to accommodate that limitation. Surely a world full of beaked people would inspire even more ingenuity?

As a general rule, I'd suggest that, if you blow into an instrument, then the lips version and the beak version have different mouthpieces. I could get into the weeds with further speculation about how bird instruments might work, but what's the point? It's a fantasy game. We hand-wave the fact that we can reverse traumatic injuries with a Cure Wounds spell. I think you can invent as much or as little lore as you like to explain how your aarakocra plays the flute.

The rules as written angle have been handled, but the idea got me googling. Since you didn't actually specify rules as written vs lore or whatever, here's some info and thoughts on real world birds to give you somewhere to start. You can use this to justify or handwave your character, or for roleplaying ideas.

Real world bird respiration, it turns out, is pretty weird. They have relatively small lungs, then extra air sacs in various places. The lungs actually empty when inhaling, and fill when exhaling. I have no idea what effect this would have on woodwind instruments, and of course there's no indication Aarakocra work the same way.

In any case, from what I can google, birds can in fact breath through their noses. A bird-person could potentially play a human-style recorded by wrapping the mouthpiece in a strip of cloth to act like a gasket, cram the thing in one nostril, and plug the other.

Transverse flutes--the kind you hold to the side--are likely a different story. They're played by shaping the lips to blow at a particular angle over an opening. While the corners of birds' mouths can stretch, as evidenced by the various comical gull-eating-a-starfish photos strewn across the internet, I can't find any indication of whether birds have muscles that can shape these, and even if they can it's hard to imagine the rest of the beak seals well enough for it to matter. It's possible they could hold the flute at a funny angle and get the airflow right, but it'd likely be very difficult to play. Again, real world anatomy may not apply, but it could make for some interesting roleplay.

As for your last question, not all birds can produce musical vocalizations. You'd be hard-pressed to caw an aria or quack a ballad. This can be used to justify needing or not needing an instrument at your discretion. Even if they can sing musically, various implements could be used to alter or amplify the sound, like how the musician provides the buzz for a trumpet.