I'd like to know if it's possible to modify the tuning of every single note with a given frequency on a keyboard and store it in a specific "scale set". Is that possible? The idea would be to play temperaments in a live execution without too much effort (ideally press a button and you switch temperament).
It sounds like you want Scala:
Scala is a freeware software application with versions supporting Windows, OS X, and Linux. It allows users to create and archive musical scales, analyze and transform them with built-in theoretical tools, play them with an on-screen keyboard or from an external MIDI keyboard, and export them to hardware and software synthesizers. (Wikipedia)
Scala is a powerful software tool for experimentation with musical tunings, such as just intonation scales, equal and historical temperaments, microtonal and macrotonal scales, and non-Western scales. It supports scale creation, editing, comparison, analysis, storage, tuning of electronic instruments, and MIDI file generation and tuning conversion. All this is integrated into a single application with a wide variety of mathematical routines and scale creation methods. Scala is ideal for the exploration of tunings and becoming familiar with the concepts involved. In addition, a very large library of scales is freely available for Scala and can be used for analysis or music creation. (Scala Homepage)
You might also find useful information in the following posts on this site:
- Software for microtonal MIDI?
- Is there any support for microtonality and temperaments in (or beyond) the MIDI specification?
- Is there a public domain collection of common musical tunings?
- Tuning a keyboard
- Software that allows playing in different temperaments
- Software for trying out 19-TET (and others)
And a bit indirectly...