Inexpensive paintable pergola beams?
The redwood and cedar that is being cut now is much younger than that wood that was cut back in the day and lasted 20yrs. Now, the wood doesn't last near as long.
You want something that lasts in the weather and that's going to be good old pressure treated pine. In the case of a pergola, saving some wood and replacing the other, almost always costs just as much as a completely new structure.
A 4X12 would need to be a double 2X12. I haven't seen a 4by in pressure treated. My price would be 15 per ft, so a 12'X16' pergola would be 2880.00, and for that price, I would even paint it, too.