To be honest, I do not do any exercises to work on my tone, and I’m not actually aware of any such exercises. This is not to say that there aren’t techniques to improve one’s tone quality on the instrument. Any good teacher will work on technique with a student, and my two primary teachers did just that with me.
The main idea is to approach the piano with care and love, to “draw” or “pull” the sound from the instrument, rather than force it. One of the primary tenets of the Russian School of pianism is to produce a warm, round, and rich sound, and achieving this requires more from the wrists, elbows, andy body than the fingers themselves.
One of the banes of being a pianist is that we are dealt a new and foreign instrument every time we go somewhere to rehearse or to perform. If we’re lucky, we have a few moments to get “familiar” with the instrument, to learn its indiosyncracies, its personality, and unfortunately, its flaws. While any other instrumentalist has the luxury of a lifetime with their instrument to learn these traits, a pianist might have only a few minutes. And discovering these “flaws” during a performance can be treacherous and will inevitably affect the performance. I’ve never really discussed this with anyone, but I imagine that every time I sit down at a new instrument, my brain is working overtime with a wealth of raw data going in, and there is some kind of immediate processing occurring, whereby I am being forced to react instantaneously to the information influx. Certain notes might pop out while others sound dull, and the brain has to remember where I might need to give less and be cautious, versus when I need to give a little more to project. It’s both tricky and frustrating, but I imagine to a student of neuroscience, it’s a fascinating process.
Sometimes, and one hopes MOST of the time, one can coax some beauty out of an instrument, but occasionally one is provided such a bad instrument, that very little can be done. The worst part of this situation is the fear that the listener cannot discern a bad instrument from a bad instrumentalist. Frankly, I’m surprised that I still run into audience members who are under the impression that I travel with my own instrument!