Pergola roof help please

[Capt W]

Just finished installing a pergola kit on my (smallish) back patio. I used a 4x4 kit from Amazon to create a perfectly square frame, which is roughly 7x11 and about 8 feet high. While I was originally going to add a "traditional" pergola-type roof, I now want to add a closed roof for sun and weather protection. Don't know yet if I'm going to go with a polycarbonate roof or plywood/shingle. My problem is I'm not sure of the best way to create slope; I want to jack the roof about 3-4 inches nearest the house to facilitate rainwater runoff into the yard.

Since the frame is square I need something across the rearmost 4x4 to raise the rafters (planning to use 2x4s). For the front side I'll make birdsmouth cuts and use common rafter ties. I just can't seem to find anything to neatly raise the rafters on the house side. The widely available rafter ties don't provide "lift."

Anybody have any ideas? I was thinking of adding another 4x4 across the long span to provide 3-1/2 inches of lift, but that's a heck of a lot of unnecessary weight. There has to be some kind of rafter bracket, but I just can't find it. Thanks in advance.

One caveat...in order to stay within local building codes (and to avoid an expensive permit), the patio cover and any parts thereof may not touch the main house. As I've set it up, the frame is a hair over 6 inches from the home.

 

[Number_35]

@Capt W, I'm unaware of an appropriate bracket, but propose this low-budget workaround. Cut five 6" lengths of 4x4 and install them, spaced equally, across the top of the back 4x4. Then install a 1x4 across the top. This will help avoid most of the extra weight.

I would bevel the 1x4 at about 3 degrees to accommodate the slight slope (4" over 84"). You could add a beveled 1x4 across the front 4x4 also, to allow the patio cover to sit flush at front and back.

Edit: I read your post more thoroughly, and see that you're planning to add rafters. If so, it would not be necessary to bevel the 1x4 at the back, and a 1x4 wouldn't be needed at the front.

 

[Capt W]

Thank you! This is a winner of an idea...exactly the simple solution I was looking for. Really appreciate the schematic as well. I'll post a pic when I get it done.