  We characterise when a simple Happel-Reiten-Smalo tilt of a length heart is
again a length heart in terms of approximation theory and the existence of a
stability condition with a phase gap. We apply simple-minded reduction to
provide a sufficient condition for infinite iterability of simple-minded
mutation/simple tilting. We use simple-minded mutation pairs to provide a
common framework to show that mutation of simple-minded collections (resp.
$w$-simple-minded systems, for $w \geq 1$) gives simple-minded collections
(resp. $w$-simple-minded systems) under mild conditions, in the process
providing a unified proof of results of A. Dugas and P. Jorgensen. Finally, we
show that under mild conditions, mutation of simple-minded collections is
compatible with mutation of $w$-simple-minded systems via a singularity
category construction due to H. Jin.
