The standard cadence is the V chord landing on I. The
backdoor replaces the V with the ♭VII7 — a
dominant 7 chord rooted a whole step below the tonic (B♭7 → C,
E♭7 → F, etc.). The ♭7 of the ♭VII7 resolves down a half-step
to the 5 of I, same leading-tone gesture as V-I but coming from
the other side. The cadence sounds fresh — familiar in
its function, surprising in its bass motion.
The borrowed minor iv precedes the ♭VII7 — it's the related
ii of that dominant, and its ♭3 (A♭ in Fm7) is the very note that
gives the backdoor its colour. So the full backdoor formula is
iv → ♭VII7 → I. The bass sets up by a P4 (iv up to
♭VII7), then resolves up a whole step into the tonic —
that upward step is what makes the cadence sound like it came
through the side door.
Resolution
♭7 of ♭VII7 (A♭ in B♭7) → 5 of I (G in C), half-step down.
Root motion
iv → ♭VII7 (P4 up) → I (M2 up). Sets up, then resolves from a step below.
01
C major
02
F major
03
B♭ major
04
E♭ major
05
A♭ major
06
D♭ major
07
G♭ major
08
B major
09
E major
10
A major
11
D major
12
G major
Same resolution, different door. Cadence by side approach.