Increase default snap without snap to grid?

Hi,
is it possible to increase how the pencil or any other tools snaps? For instance, the pencil move each neighbour by a pixel no matter the size of the brush. Can I make it move each “x” pixels. Say I am working in a 1920 * 1080 canvas. But I want that my minimum brush and how it move to the neighbour is also 4 pixels.
The problem with snap to grid is that it snaps when you actually paint the pixel. I would like to be able to hover over the canvas so the cursor moves each 4 pixels. Do you know what I mean?

thanks