I always thought of the 3940 and the 3970 as siblings, complementing each other. Same basic style and era but differently executed. Both have a given place in a diverse Patek collection. You could argue though, in the name of diversity, one of them should...
I've also been thinking about mixing it up with my straps and favor a more casual look. I have a blue Hodinkee shell cordovan strap on my 5040J, which plays nicely off the moon disc and dresses it down a bit, and a very casual medium brown calfskin strap ...