There is a possibility of addressing this question historically... My understanding (which is possibly apocryphal) is that
mf came before
mp, and originally meant "normal volume". To explain, "forte" has two meanings, in the same way that the English word "loud" can both
- Mean "high volume" — as in "play it loud" — and
- Refer to the concept of loudness more generally — as in "how loud is this note?", to which an acceptable answer could be "not very", "quiet", "moderately loud", etc.
mf was originally taken to mean "with medium loudness"; and
mp came about later on, as a sort of reinterpretation of what
mf meant. Do I have any proof for this? Nothing but sketchy recollections from a music history class.
Overall, I'd lean towards
mf being closer to "medium volume" than
mp, although of course what that means is subjective and will vary based on the piece, as others have indicated. I wouldn't think this usage is entirely set in stone, though. For example, Boulez used twelve dynamic markings for serialising dynamics (
ffff), while Mozart used six (
ff); as far as these and similar practices indicate, it doesn't seem like there's a reliable symbol for "exactly medium volume". Even MIDI velocity can't encode something exactly between 0 (minimum) and 127 (maximum), which would be 63.5.
But, this shouldn't worry us too much. If a composer wants to notate dynamics that precisely, they will probably make that clear in the score. Otherwise, it is open to interpretation...