Overtone flute scale
The overtone flute scale is almost the same that harmonic series of an individual note. Overtone scale can be heard also e.g. in throat singing or mouth harp music. Nordic overtone flute melodies are usually moving around 8th overtone. In theory, the overtone scale could be played only by varying the strength of the air stream without closing the flute end, but then e.g. 8th overtone would sound unpleasantly high and loud. When played with closed flute end only every other overtone are played. By combining open and closed flute end it’s possible to get an overtone scale pleasant to human ear. The scale can be extended by closing the flute end only partly e.g. 1/2 or 3/4.
Overtone flute scale for D-tuned flute
Because the overtone flute scale is based on the overtone scale it does not have the same intonation than e.g. piano with equal temperament. Even if the lower register of the flute would be tuned in same intonation the higher notes would not be in the same tuning (in equal temperament). Of course you can accompany overtone flute music with piano if you accept the variations in tuning during the tunes. Overtone flute has different timbre than piano and may not sound unpleasant even if the tunings would not the exactly the same. In the end it’s only a matter of taste. You can test how accompanying sounds for your ears with the songs found from this website.
Joutsenlahti, R-L. 2000.
Yvert, J-P. 1996.