Most steppers do 1.8 degrees / step, or 200 steps / revolution. Microstepping drives can get you to 0.9 degrees (1/2 step), 0.45 degrees (1/4 step, and so on. If you want exact degrees (360 / rev) you may want to use reduction gearing/belt/worm to get you there. I might suggest worm gear drive for your application. Worm drive would just about preclude you from accidently bumping the stepper out of position.
108:1 worm would give you one minute steps (60 / degree) in full step.
Sounds like a great project. Diamonds ??