Ken, I mostly play the J-40 because of its standard scale length. It is my "short scale" baritone, but it has lived tuned down for years (mostly D-D with mediums). As I mentioned before, the 416 baritone with its 27" scale is getting tough for my fretting hand, which is why it is for sale. I suppose I could keep it and capo up a couple of frets to shorten the effective scale length, but that seems pointless when I have other guitars available to down-tune.
I play C tunings for some slack-key tunes [CGEGCE and CFCGCE and CGDGBE] and typically use a bluegrass set, or Elixir Medium-Lights. Tuning down below D on the sixth / bass string really wants a medium gauge 56-59 to keep enough tension and avoid floppiness. My J-40 is currently wearing heavies, and either gets tuned C-C (standard minus two steps) or sometimes it gets tuned D-D if I'm playing along with others. I prefer to keep the capo below the fourth fret, or else the neck gets pretty short before running into the body. Even heavies at D-D pitch are equivalent to mediums at normal E-E tuning. That tension chart is real handy when you are experimenting in this realm. I even put together a spreadsheet to test different combos of string gauge and tuned pitch.