Although the above score looks pretty scary in terms of chromatic notes, I used only major and minor chords to harmonize this tune. The result - a little like French modal writing of the 20th century organ composers. The key was not to use Tonic and Dominant chords in this setting. Instead lots of other chordal relationships of a tritone, major and minor thirds are pretty colorful.
So today I would like to share with you this harmonization in four parts. Here is the PDF and MIDI files if you want to play it.
If you would like to know how to write or improvise in this style on any tune you want, please let me know.