[rubberbandage]

This exists in a few synths, and is included in Apple’s Logic Pro for software (MIDI) instruments as “Hermode Tuning” (http://www.hermode.com/html/hermode-tuning_en.html)

It’s a really unusual experience using it, because the never-quite-tuned sound is part of a piano’s character. With Hermode tuning engaged, every chord has a ringing pure bell-like quality, especially with 7th/diminished chords (which sound immensely satisfying).

Since few instruments in the real world can tune so precisely, Logic amusingly has a slider to make the Hermode tuning “less perfect” if desired.