From a pure theoretical perspective, what would be the appropriate names for these two chords? I want to respect the specific degrees. In other words, not treating the double flat seventh as a sixth, nor the third as a ninth.
R, b3, 5, bb7
R, bb3, b5, bb7
For reference, theses chord happen naturally in the Double Harmonic scale, the third and seventh mode respectively.
Since music theory does not address the double-harmonic scale or its modes, you could really call those chords anything you like. These chords simply don't exist in standard music theory, because chords are considered to be combinations of minor and major thirds.
In this case, the naming convention would simply serve the purpose of expressing the intervals in the chords, so names like
would be sufficient.
Otherwise, a functional theory of the double harmonic would have to be invented, naming these chords according to their operation within the key. For example
The 'Chord Symbol' system of naming chords is based on a diatonic major scale. 'C' is a C major triad, we don't have to say 'C MAJOR'. 'Cadd2' is a C major triad plus a major 2nd. If we want a minor triad or a minor 2nd, we have to say so. (Yes,'7' is an exception, it means 'minor 7th'.) But basically the major triad and scale is taken as the norm, anything else needs to be named as a specific modification of it.
This naming system works for music in major, minor or simple modal tonalities. It falls down when confronted with the Double Harmonic scale. If you feel a triad-based naming system would be useful in this environment, you'll have to invent one. I think I'd just go with a simple description, like your initial suggestion: 'R, b3, 5, bb7', 'R, bb3, b5, bb7' etc.
(Note that even then, you're labelling the notes as modifications of the major scale! It's pretty well entrenched in our 'theory' syntax!)
