Man, this is a beautiful piece. Sounds almost like a round.
That's one thing I think of whenever I hear medieval music -- a round. Blasted
.
From what I hear in this piece, it's mainly harmonies that are fluctuating between a I and V chord (shoot, you could simply say it's, forgive my rusty and probably unreliable psuedo-music theory lingo, shifting tonal centers underneath a "drone" of a I chord).
Medieval music works WELL with chords. When you start getting into polyphony, you start to see many medieval pieces being accompanied by chordal instruments (lutes come to mind). Prior to that, at least for written forms, it was mainly monophonic, switching over to interchangable lines. I'm thinking back to Gregorian chant, though.
My idea would be to try sticking to simple I V | I IV V basslines, and having melodic lines fluctuate around those. Then slowly start to branch out, chordally. I'd stick with your normal major/minor chords. No Mm7's, no flat 2nds -- keep everything as 'pure' as possible. Little dissonance.
Heck, just keep emulating. When I try to figure out music like this, I recreate first, and ask questions second, haha.
Listening through that playlist might give you some ideas, too!
Aha, all fantastic points!
Depending on the locale of your medieval (Euro vs. Middle-Eastern, etc), also playing around with 2nds & 7ths and modes like Phyrigian could add some beautiful spice.