It seems to be such a modern creation compared to other woodwinds, and also very unlike other instruments, yet is extremely popular.
Was there a certain purpose it was created for? Is it based off of some earlier designs I'm just unfamiliar with? When did it truly become accepted as a proper instrument and not just some odd invention?
It was created by a man named Adolphe Sax. Long story short, it was created as a bridge between brass and woodwind instruments in french military bands in the 1800s. Sax started at the lower end with bass saxes, and he then went on to create tenor, alto, soprano, and all other crazy sizes!
If you're interested, there's a book called "Devil's Horn" (I'm sorry I don't remember the author! ) about the history of the sax and it's place in music. It's fasinating!
Not sure if this'll help, but someone over in /r/musictheory might be able to help you out.