My daughter's 1849 house does not have roof soffits or gable overhangs. The house is in an area bordering on area 2 & 3 which requires R-39 rating. She has a loft area with a vaulted ceiling that was finished buy inserting pressed homasoat type sheets between the exposed 2x6 rafters. There is minimal insulation properties at all. I plan to rework the area. Since the house design has no provison for ventilation of the underside of the roof deck, What are my options for insulation methods. She wants to keep the look of exposed beams. 2x8s would rob too much ceiling height. I am thinking rigid between the rafters and 3/8 sheetrock as the finish. Total ceiling ares is approx 250sf.

You can get about R-27 or 28 with 5 1/2 inches of isocyanurate foam rigid boards inserted between the 2x6s. You can then add an inch of the same foam board across the bottom of the rafters, for another R-5, giving you R-32 or 33 total. That's about the best you can do, but really, the roof should be ventilated above the foam, ay least an inch of vent space. You can get another R-7.5 by screwing 2x2s to the bottom of the rafters, making them the depth of a 2x8, and adding another 1 1/2 of foam. That way you can approach R-40, but still not solving the ventilation problem. And you'll certainly lose the beam ceiling look.