What is the name of this paint texture?

[emm74]

OP, I would search "hand troweled wall texture" - that's giving me images that are fairly close to your photo. You'd have to discuss the specifics with the drywall contractor. However, as someone else mentioned, if you are doing new construction with a tract builder, it's unlikely they will customize to that degree.

[Pitt Chick]

emm, that post was over two years ago..I bet the house is finished already...