I know it’s hard to explain 3D objects in words, so I’m sorry this is so wordy.
The previous owner of my house had an outdoor patio cover/roof put in and to do so they cut the rafter tails, which were only about 10 inches long, with a horizontal cut from the bottom edge of the roof toward the exterior wall. So imagine these rafter tails cut horizontally from roof line back to the wall. https://youtu.be/Lm8Afm7He4U?t=147
The horizontal cut ends (leaves the rafter) right at the exterior wall, so that each rafter tail now looks like a small right triangle, with a skinny tip of the triangle being at the edge of the roof. (This may not have been up to code.)
The patio cover was later removed and since there is no gutter water just comes off the roof, and near our back door a bunch of rain spills. I want to put up a fascia board to hold gutters, so I’ve done some research and I plan to “sister” another 2x4, about 3-4 feet long, to the rafter. That is, I plan to attach another 2x4 to the rafter, parallel with it and touching it, with some of it sticking out of the exterior wall and some of it going up into the attic. It is similar to https://youtu.be/UttfbMXjzik?t=26 except the top of my exterior wall is not stucco. Instead, between the rafters (and at the top of the exterior wall) is a 2x4 piece of wood about 2 feet long. (Some of these have a vent hole with a screen.) So to make room for the new 2x4 I can cut 1 1/2 inches off of the end those pieces of wood between the rafters, to fit the new 2x4 (which is really 1 1/2 inches wide).
Here’s the problem though. The bottom of the rafter is going through a ~half inch deep notch in the ceiling joist (is that the right name?) that rafter is sitting on top of, just before the rafter goes through the exterior wall. So if I try to put the new 2x4 next to the rafter, the new 2x4 will be sitting higher up, since there is no notch in the joist. If make the notch in the joist wider to accommodate the new 2x4 the notch will be 3 inches wide (2 times 1 1/2). I believe that’s too wide, because I’m told (online research) that a notch’s width can’t be more than 1/3 the total depth of the ceiling joist. I don’t know for sure what the ceiling joist’s dimension are, but I would guess it’s less than 9 inches in depth. Is that a reasonable guess? I don’t know how to see the joist to figure it out. If I assume it’s less than 9 inches in depth, I can’t widen the notch, right?
FYI, to avoid taking off the roof I was going to slide the new 2x4 up into the roof without taking off the roof. So I can’t just put a (birdsmouth) notch in the new 2x4 and lower it down from above. The only solution I can think of is to cut the new 2x4 length-wise to make it a little skinnier for the entire length of the board that will be fitting into the attic. Then I could slide it into the attic from the outside, through the hole I cut in the board between the rafters. But then it will be sitting on just the corner edge of the ceiling joist. Is that okay if it’s just is going to be holding up a fascia board and gutters? Is there a better solution?