How far can you span a pergola?
Considering this, how far can a 2x12 span for a pergola?
There's not quite enough information here to give a definitive answer, but for decks, a rule of thumb is when supporting joists that span 12 feet, a double ply beam can span in feet a value equal to its depth in inches. Thus, a double 2x12 beam could span 12 feet - with 20' beams, posts at 10' might look the best.
Regarding this, how far can a 2x10 span for a pergola?
A 4x4 does make a good knee brace when 6x6 posts are used. Also a 2x10 can span 13'9" when being used as a deck joist so I am not sure why it couldn't easily carry 14 feet on a pergola which is going to halve 75 percent less load on it.
Re: 20 foot clearspan beam size In that case, you need something like a 12-16" GLULAM or LVL to span the 20' and can use simple 2x8-10 dimensional lumber 16"OC as floor joists.