The photos on this page show some experimental clip designs. All of the designs here are essentially wedge-shaped, and a hammer is used to wedge them against the spring cord:
One of the clip designs shown above actually fairly usable: the long, half-round wedge made from a durable and hard-to-split hardwood (e.g. elm) works fairly well unless the spring cord is pretensioned a lot, in which case the cord has a tendency to slip. This problem can be solved to a degree by using an additional, well-polished steel wedge of a rectangular cross-section. This auxiliary wedge not only focuses the pressure on the spring cord being tensioned, but also slows down the wearing down of the primary, wooden wedge. The back of the primary, wooden wedge should have a curvature matching that of the washer core.
All this said, it's better to use the clothespin-style clip described here. Apparently something like that was what the ancients used, as Marsden mentions in his Technical Treatises.