Here are a couple of drafts using satin and basket weaves, which I first mentioned a few days ago. The correct term for combining satin with another weave is damassé (whereas combining a warp-faced satin with a weft-faced satin is damask).
The first draft is for a straight threading on 40 shafts.
The next draft is an advancing threading on 40 shafts conforming to a network of 8-end initial. The circles in the lift plan were elongated 3x to match the fact that a single repeat of the advancing threading is 3x as long as a single repeat of the straight threading. The result: the circles come out round, sort of.
And here's a version for 24 shafts, straight threading.