ok. up 'til now, i've focused on the steps you'd take to make sure all your pieces around the ring matched the pattern exactly. it's quite a bit of work and will be slow going until you get used to the steps and can do them without thinking so hard the whole time.
BUT ... who says they pieces have to match the pattern exactly? who says they have to be anything or any shape at all in particular? the main reason for using the paper piecing pattern is to make sure you have the right size and shape for the big "chunks" of the ring you'll be sewing to the background fabric.
you could just as easily do this the same way you would a string quilt or a crazy quilt. just add pieces of any size and shape as you go along the ring. lay them down in any way that pleases you or the shape of the scraps.
once they're all on, and trimmed to the pattern, you'll have something spontaneous and fun to build your rings with. and not nearly as big a headache. :lol: