Almost Magic I: Lopsided Diagonals

In this puzzle, an almost magic square is a 3-by-3 grid of positive integers whose 8 line sums (3 rows, 3 columns and 2 diagonals) differ from each other by at most 1. So every line sums to either S or S+1 for some S.

Fill the grid below with nine distinct positive integers so that:

1. the grid is an almost magic square, and
2. the two diagonals have different sums. (This means the square cannot be exactly magic.)

Question: What is the smallest possible total of the nine numbers?

Answer format: a single integer.