You can make a very effective 1/8" bias with fine silk neckties.  Cut 1/2" to 5/8" strips lengthwise from a tie (needs to be thin fabric) and use a 1/4" bias tape maker to make 1/4" bias strips.  Fold the strips in half lengthwise to make 1/8" bias.  Sew the double folded edge down first.  These make lovely stems for small flowers, as well as vines or tendrils.