Hi,  Perhaps I can help.  Can you send me a picture?  Does the cylinder
play one tune when it turns one-half turn, and then another tune when
it turns the rest of the way around?

Is the cylinder about 2-1/4 inches in diameter or more like 3 inches
in diameter?  It's the diameter, the number of teeth, and the number
of tunes per turn that makes it an overture box.  A 3-inch diameter
cylinder that plays two tunes each time it rotates once is called a
"Two-per-turn" music box.

Also, an overture box wouldn't play 10 or 12 tunes.  It would play
fewer tunes and have lots of fine teeth.

Regards,
Craig Smith