You are about to enter the Twilight Zone. I submit for your consideration an oddly named book lying on an ordinary desk: How Equal Temperament Ruined Harmony (and Why You Should Care), by professor Ross W. Duffin. This book was written by a madman. Or is he? You should understand: If Duffin is mad, he's not alone. And the spaces between the lines of his book are filled with the silent laughter of the gods.
The gods are laughing at their little joke on musicians. When it comes to the tuning of instruments, especially keyboards and fretted instruments, nature drops a giant hairball in our path.
From a review by Jan Swafford at Slate (Matt Yglesias gives the short version for non-musicians). I wonder whether Duffin writes about the normalization of jazz harmonies by English performers in Broadway-derived musicals, which can be very strange if you know the originals well.
You can play a keyboard instrument that isn’t tuned in equal temperament. What you usually can’t do easily is choose your tuning on the spur of the moment (which is possible playing string instruments like violins, and even playing wind instruments, but not the standard piano). An English musician named Geoff Smith, however, has invented a piano with “fluid tuning,” which you can see and listen to here. It’s expected to be very useful especially for non-Western genres, and certainly looks pretty neat.