Try DAC. I don't know which style you're playing in, but with the "melody" string tuned to G, the #1 note of the scale is on the 4th fret. That puts you in Dorian mode. It is kind of a minor, but not the "real" one. The "real" one, the natural minor (aka the Aeolian mode) starts at the first fret. And at the 8th fret. DAC tuning will give you that scale with nice drones. 

@picklechickens I am a straight by-ear player.  One alternative might be to use a different tuning to get Samanthra to lay out right on the fretboard.  (I tune all over the place in order to get the sound which hits my ears best.)  The only caution is to not make huge tuning leaps and break strings.  It really can be fun to try a variety of tunings.  

Hello and thank you to everyone who commented on my profile page! Nice to encounter such warmth and friendliness before I've even posted anything!

As I wrote there, I inherited a dulcimer last year and have just recently started trying to play it. I'm especially interested in shape-note tunes and have been trying to learn Samanthra. I was advised to tune the dulcimer to DAG and play it in d minor. This seemed to work well until the end of the third phrase. There's a low b-flat there, but the second fret is a b natural. Does anyone have other suggestions for how to make this tune work?