It is possible. What you describe is what mathematicians call a deconvolution filter...you apply a filter to the signal so that what you get back is only the original signal.
Think of what you hear as the original signal convolved with the response function of the room. You just need to design a time series filter to extract the orginal (remove all reflections of the room response).
Unforunately noise will interfere with the exact process and furthermore it only works mathematically for ONE precise position in the room to less than an inch (as the room response function changes with your relative position and rapidly so at higher frequencies)
It is simpler just to remove the room response by putting on a pair of headphones but then it sounds unnatural....this suggests that the goal is not necessarily to remove the room response entirely, as room reverberation is pleasing and helps us hear more in the music.
So it is really a very complex issue altogether which nobody has solved yet.
A TACT or PEQ can compensate for some of the worst room modal offences at the very lowest frequencies. This is practical because LF frequencies have a large effect in a room over a wide area.
Since we are sensitive to timbre and the way signals decay it not strictly ideal to compensate for room modes by attenuating the signal as in a PEQ: Room Acoustic Treatment is really the best solution, including some trial and error, as Professor Sabin discorvered. (Unlike a PEQ, room acoustic treatment tackles the reverberation tail in the signal but leaves the primary signal unadulterated)
Designing great music halls is an art just as much as it is an engineering feat.
