To begin the quest, talk to the Composer and choose Deer Drum. He will tell you the following:
Are you enjoying the lute? Well I do certainly hope so.
Ah well, I suppose we can't stop there can we? There's more to music than stringed instruments. Now, we go for the foundation of melody. The rhythm!
We're going to need to build you a drum. This time we need a high quality deer hide. The skin must be so perfectly carved that the drum has the right resonance and tonality to strike you in the chest and set your ears ablaze!
But first you're going to need a proper skinning knife. Hunters worth their salt often use these knives so check hunting shacks to see if you can find one. Once you've got it, you'll need to hunt down a deer. The hide must be immaculate!
Bring it here and we'll drum something up for you...oh ha ha! I am the very soul of wit.
First, you need to acquire the Hunter's Skinning Knife. It can be found easily in Hunting Shack crates. When you have the knife in your inventory, you have a small chance to loot an Immaculate Deer Hide every time you kill a deer. If you manage to get one, a message will tell you so and the skinning knife will break. Return the hide to the Composer and choose the same dialogue option as before. He will say:
Turn in Immaculate Deer Hide
It is...well it is immaculate! I must say, well done. You're still a work in progress but this is good news, very good news indeed.
Alright here is your drum. Enjoy yourself but not too much.
You will receive the Composer's Drum upon finishing the quest.