The Tesla Primary usually consists of a spiral, or inverted conical coil of wire or tubing. Copper tubing is often used because at the frequency of operation of a Tesla Coil current tends to flow over the surface of conductors, so the inner core becomes redundant. Large diameter tubing or wire are used to keep the resistance of the coil low at the operating frequency.

My First Primary for this 1800 watt machine was a Flat spiral  and consisted of 14 turns soft copper  tubing with a diameter of 1/4"  and a spacing of 1/4"  between the conductor surfaces and a spacing of 1 1/2" between the primary and Secondary. This primary was for the 5 inch secondary. 

The Second Primary was wound in an inverted conical section of 10 degrees. This inverted conical primary gave beter power transfer. The spacing between primary and secondary is 2".  First two pictures are of the old primary for the five inch secondary. The last picture is the new primary with the six inch secondary. Tab on pictures to enlarge.