I'm having trouble figuring the v and vii chords in the double harmonic scale. Is it correct to assume the v is the supertonic half-diminished 7th in its 2nd inversion and the vii is the supertonic dominant 7th in its 3rd inversion? Can anyone confirm and what would the correct notation be?
As I see it, there really is no standard notation for these, since there are not the standard triads or seventh chords used in Roman numeral notation.
The C double harmonic minor scale is
C Df E F G Af B C.
Thus the "V" chord is
G B Df and the "VII" chord is
B Df F. Neither one of these are one of the four standard types of triads (major, minor, diminished, augmented).
Your assumptions (stated in the original post) are incorrect because you're assuming that the "V" chord has all of the same pitches as the "ii" chord, just with the fifth scale degree in the bass. Since this isn't the case, the "V" and "VII" chords will not be the "ii" chord in various inversions.
But there are interesting aspects to these "V" and VII" chords: both are subsets of the whole-tone collection, and as such they can have dominant function. Furthermore, they're both subsets of the same whole-tone collection
Df Ef F G A B, so they can both resolve to the tonic of C.
Another way to view their dominant function is to view both chords as subsets of the V7f5 (in this case,
G B Df F).
There are more things to say; the "VII" is an Italian augmented sixth chord (though in a funny inversion) and the "V" is a subset of the French augmented sixth. But that starts to get us a bit far off-base, I think!
Edit: Related to the "V" chord
G B Df, I just came across Matt L.'s answer in whats the triad formed root +maj 3rd +maj 2nd called?, where he says: