Look what the score is doing: you have an oscillation rising from F♯ in the initial RH part, moving up to C♯, but artfully dodging A♯. At the same time, you have an accompaniment that consists largely of the fifth D-A alternating with the auxiliary notes E-G. The entire section is acting like an elaboration of a D major chord in a kind of quasi-Lydian setting. The RH is working with F♯, G♯, B and C♯; the LH, with D, E, G and A.
From there, the RH holds a G♯ slightly overlapping the A which commences the melody in the LH. (Leading tone, anyone?) The LH melody oscillates around A using the same set of tones as the initial accompaniment while RH rocks between F♯-C♯ fifth and its auxiliaries, G♯-B, so you have an elaboration of F♯ minor, but with a stressed A. However, as the melody progresses, it starts to stress E and, later, D against the auxiliary chord (i.e., E7).
At m.17, both voices start a two-part counterpoint. These two bars are slightly ambiguous. There is a suggestion of F minor to E diminished to G major, but the real point seems to be oblique movement out from F to B, with a wedge motion in the LH through E to G - not so much a progression as a contrapuntal movement towards a goal. This is the first time that F♮ and A♯ make an appearance, and they are both acting as stressed upper and lower leading tones.
G leads to the D-A dyad, which leads to A held under an F♯-C♯ broken figuration, with the A "resolving" to the initial E-G third under C♯. (The A is held slightly under the C# before dropping into the third, i.e., there is a strong suggestion of A7.) From here, both sets of accompanying dyads - E-G and D-A in the LH, G♯-B and F♯-C♯ in the RH - settling into an alternation of D M7 and E split-third, finally touching on F♮-A♯ which leads into a wedge back through the E split-third to the final D M7 chord. This piece is very definitely in D.
The thing to note here is that the wedge to the final D M7 is possible due to the fact that the segments in each hand are related by inversion. F and A♯ give the missing semitone in each segment, which is why they get special handling.
- You have strong contrapuntal movement in a wedge to D M7 due to the inversional relationship between each hand's scale segments.
- With F and A♯ being omitted until the crux of the piece, and the limitation of each hand to a range of a fifth, you have scale segments that essentially define a quasi-Lydian D.
- The harmonies that are stressed tend to play up a quasi-Lydian D: D, F# minor, with E7, G, and A7 strongly suggested.