And here's the dulcimer
Here's a tune played on a dulcimer with diatonic/chromatic fretboard
Listen to Pastorale Michel Corrette by nickosullivan #np on #SoundCloud
It's not that uncommon to see that kind of mixed diatonic/chromatic fret arrangement on folk traditional or older dulcimer ancestor instruments- Swedish hummels, French epinettes... The practice goes far back. There are quite a few early mtn dulcimers with the arrangement, and J.J. Niles experimented with making such dulcimers as well. Lots of musicians like having the option for those odd extra notes, without having a completely chromatic fretboard.
It helps to be fretting with the fingers rather than a noter if you want to be more nimble in getting the 'far' non-diatonic frets. You can play tunes with any kind of accidentals in them, and you can play tunes that switch keys midstream without retuning... quite useful.
There's a few on this page: http://www.davidbeede.com/octavedulcimers.htm
Take a look at the center dulcimer in that picture at the top right. It looks like a royal pain to build and I see that David stopped offering that option. If memory serves, he used to call it an "evil half-breed" fretboard. I would love to have one just to see how people react!
I think the idea is that you can play the melody string without speed bumps, while accidentals are available on other strings if you need them. In DAA (or any 1-5-5 tuning) it would make perfect sense to a piano player: white keys on the near string, both white and black keys on the middle string. In DAd (1-5-1) you'd find the black keys shifted up a little higher on the fretboard but still the same idea.
Come to think of it, that fretboard looks a lot like a piano keyboard. Short frets are the black keys. It strikes me as a perfectly logical layout for someone who wants all the notes but isn't comfortable looking at (or sliding a noter across) a fully chromatic fretboard.