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CGardner - what are the "rules" for framing? I imagine each photo is different, but is there a general rule like matching dark frames with dark photos and vice versa?
There aren't any "rules". You just need to start with a goal and then find the most effective strategy and tatics to meet it.
The goal in composing a photo is to pull the viewer to one or more focal points. Most composition strategies do this by creating contrast of: tone, sharpness, color, size, shape, etc. But one of the most powerful is tonal contrast.
In terms of cause and effect I try to use the tone of the mat to "push" the viewer toward the contrasting focal point of the photo, and avoid the opposite; having the mat tone tug at the attention of the viewer and pull the eye out of the photo.
On closely cropped shots such as the mule the negative space of the mat can be used to provide compositional balance that would otherwise be lacking. I cropped out the body of the mule that was a distraction and replaced it with the non-distracting, framing negative space of the mat in a similar gray tone. The mat is lighter in tone than the dark focal points of eye and nose in the corners of the image. A black mat would have competed with them, a white one would have also. So in that case something in the middle was ideal.
In your second shot the photo itself created a natural frame. But if I were to mat it I would use a very dark gray one, just a shade lighter than the edge of the photo.
In landscapes the contrast of the sky tends to pull the eye up and out the top of the photo. Surrounding the photo with a darker mat act like the bumper on a billiards table to literally bounce the eye back down to take another look at the darker foreground. The black mat also makes the foreground seem to have more detail by comparison. It's just an optical illusion, but so is seeing 3D in a 2D photo.
When the background of a photo is light it will be the darker and more colorful areas of the photo that become the natural focal point, such as the autumn leaf in your grate shot where the color contrast of the leaf on the neutral tone of the grate pulls the eye to it. To help "push" the eye inward to find it I'll use a lighter mat. For the grate shot I used the eye dropper to sample the gray color from the steel of the grate then adjusted it by eye.
The width of the mat and tone/color are something I adjust by trial and error until I find the balance I'm looking for: one that frames the content of the photo and invites the viewer to dive into the photo rather than tugging their eyes out of it. Start with that goal and then the rest should make sense in the same way you spotlight and object and darken the background around it. or keep the focal point sharp and blur the background. The mat is just an extension of the background, or in the case of a tightly cropped shot like the mule a proxy for the background that has been intentionally omitted with cropping to put more emphasis on the focal point.
When matting photos for hanging on a wall you need to take into account the tone/color of the wall and the viewing distance when selecting the mat. Consider what will be in the field of view when the photo is seen, which will vary with viewing distance. Again keep the goal of pushing/pulling the eye toward the focal point and preventing it from being pulled away from it.