Beat-matched automatic transitions

Given tempo (beats per minute) and musical key for both the outgoing and the incoming track — from a streaming source's metadata, from local analysis, from a tag — three things can be derived:

1. how long the blend should last,
2. how much to bend the outgoing track's tempo so both share a beat during the overlap,
3. how much to shift its pitch so the two keys do not clash.

Everything is applied to the outgoing track only and dies with it, so the incoming track plays at its natural tempo and pitch from its first solo moment.

base duration from current tempo   (slower track → longer blend)
  × tempo-gap factor               (bigger gap → longer, to hide it)
  × key-gap factor                 (harmonically distant → longer, to mask it)
  → snap to a whole number of beats → clamp to a sane range