Please let me correct me. Of course MIDI Control Change exits to do pitch tuning.
The basic procedure for altering a parameter value is to first send the Registered or Non-Registered ParameterNumber corresponding to the parameter to be modified, followed by the Data Entry, Data Increment, or Data Decrement value to be applied to the parameter.
Thanks JJC! I completely forgot about that one... even though I'm actually using it in a project of mine! :-) Using the fine tune RPN is the much more suitable way to affect the tuning of all notes simultaneously, thanks for the hint.
Nadal, if you can't send those control change messages via MIDI, there is also a way to do it in the Fluidsynth shell. Please write again if you need help with that.