Similar to how to set musgrave to get something periodic?, you can map your texture radially around a cylinder using a material such as the following :
Here the coordinates are constructed based on the Radial coordinate (which varies between 0.0 and 1.0 for around the circumference) and the Z coordinate (for along the cylinder length).
The radial gradient (0.0 to 1.0) is multiplied by 2*pi
and the X and Y coordinates generated using Sine and Cosine functions - such that the 0.0 to 1.0 gradient represents a full rotation, so that it matches up at the seam (ie, 0 degrees matches to 360 degrees around the circumference).