This is a general version of this beautiful puzzle.
Place any number of standard chess pieces on a 8x8 chessboard, such that there is at least 1 empty square attacked by exactly 1 piece, at least 1 empty square attacked by exactly 2 pieces, ..., at least 1 empty square attacked by exactly $N$ pieces. What is the largest value of $N$ you can obtain? We have already seen that $N=6$ is possible, but can we do better?
Note I am also interested in the version of this puzzle where we include at least 1 empty square that is not attacked by any piece.