Preheat the oven at 350-degrees. Line a baking sheet with parchment. Combine the first six ingredients; reserve a 1/2 cup in a separate bowl.
Beat the butter until soft. Add the bulk of the spice mix, continuing to beat until light and fluffy. Add the eggs and vanilla.
In a separate bowl, combine the flour, cocoa, cream of tartar, baking powder, and salt. Slowly work this into the butter mix.
Use a teaspoon scoop or spoon to form small balls. Roll them in the reserved sugar-spice mix. Set them spaced out on the baking sheet.
Bake for 10 to 12 minutes. Due to the cocoa, it is hard to gauge doneness by color, so look for them to be set with some light crackling. These don't crackle as much as standard snickerdoodles, though. Let them rest on the sheet for about 10 minutes, then transition them to a rack to completely cool.