Two reasonable functional explanations are available using jazz harmony. First for most ♭VII7-I progressions, the ♭VII7 chord is interchangeable with the iiø7 chord. Hence, you can think of the | D7(♯11) | E∆7 | progression as being a iiø7-V7alt-I with a missing V7 chord: | F♯ø7 (B13♭9) | E∆7 |. *Second: you can alternatively think of the D7(♯11) chord as using D half-whole diminished, in which case the D7(♯11) is interchangeable with B7(♯11), the V7 chord for E∆7.
Standard ii-V-I Explanation
For your example, D9 (the ♭VII7 chord) is interchangeable with F♯ø7 (the iiø7 chord). Those two chords share the same notes:
D9 contains the notes: D-F♯-A-C-E
F♯ø7 contains the notes: F♯-A-C-E
This means that the | E∆7 | D7(♯11) | E∆7 | progression is effectively the same as | E∆7 | F♯ø7 (B13♭9) | E∆7 |. You can think of this as a I-iiø7-V7-I progression with a missing V7. In fact, iiø-V7alt licks in EMaj tend to sound great over the ♭VII7 chord.
If you replace the D7 chord with F♯ø7, you might think to improvise over F♯ Locrian. However, the D7 chord is actually D7(♯11), which means you need to play a G♯ rather than a G♮. This leads you to think of the iiø7 chord in terms of F♯ Locrian ♯2. The Locrian ♯2 scale is an advanced but an extremely common choice among modern jazz musicians for a iiø7 chord like F♯ø7. If you want more description of that scale, check out this outstanding answer. In fact, F♯ø7 and D7(♯11) are extremely similar: the F♯ Locrian ♯2 scale and the D Lydian Dominant scale are both formed from an A melodic minor parent scale.
Diminished Scale Explanation
Instead of thinking about the D7♯11 chord as using A melodic minor, we can think about this chord as using the D half-whole diminished scale (D E♭ F F♯ G♯ A B C). This scale fits D7(♯11) because it contains:
- the ♯9 (D, the root of the D7♯11 chord)
- the ♭9 (C, the ♭7 of the D7♯11 chord)
- the ♮13 (A, the ♯11 of the D7♯11 chord)
However, when dealing with a diminished scale, the dom7 chords are interchangeable as you move up/down minor thirds. For this example, D7(♯11) is interchangeable with B7, A♭(♯11), and F(♯11). So you can think of the progression | D7(♯11) | E∆7 |as being equivalent to | B7(♯11) |, the V7 chord to E∆7.