How to bore a bore? Recorder

How would I create a tapered bore for a homemade soprano or alto recorder from a block of wood?

Needs to be one piece at least 300mm long.
Is there a custom drill bit for this?
Do I drill then use a reamer for this?

I've seen videos online but if you were a backyard tinkerer, what would be the best method with available tools? (Or is this too specialised?)

It seems weird that from my research I can't find a tool that would make a narrow, tapered hole that at least equals a foot long. Something that I could borrow for my purpose.

Any gurus out there know the answer? I thank you in advance for your knowledge!
 
Super late reply.

"The Amateur Wind Instrument Maker" by Trevor Robinson has suggestions on making effective reamers. I have not tried this myself (just trying to make my life harder making square bore instruments), but this book has a lot of useful information on instrument building in general. The word amateur in the title may give the wrong impression.
 
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