"Equal temperament is almost universal nowadays and gives almost perfect fifths but somewhat nasty thirds."

Hmm. In the case of pianos, tuning to strict equal temperament across all the octaves doesn't work. Octaves are "stretched," and the stretching required depends on the type of piano being tuned. It works best on concert grands. I'll let someone else deal with the technicalities...

In the world of harmonicas, and presumably other free-reed instruments, fine-tuning is a variable feast. As a harmonica player I should restrict myself to talking just about those.   Chromatic harmonicas (the ones with a button on one end) are generally tuned to 12T equal temperament. When it comes to diatonic harmonicas, such as 10-hole blues harps and tremolo harps, the story is mixed. Some are tuned to equal (Tombo tremolos and Hohner Golden Melodies, Lee Oskars and Suzuki harps for example), whereas many others, such as many Hohner blues harps and tremolos, are tuned to a compromise between Just and equal. There are even occasional harps tuned to Just, e.g., the Seydel 1847. It's all about making chord-playing not sounding too horrible ;-). Some of us don't rely on playing chords very much, including me (my metier is Irish tunes), so I retune my blues harps to something close to equal. I can't bear that flat third that comes with Just intonation when I'm playing with other people.

As an addendum, every harmonica I've ever bought over the last fifty years has been tuned to at least A441+ or even sharper than that. It makes us sound brighter in the mix but you'd have to have an amazing ear to think that we sound sharp!

Theory is all very well... Coming up, my rant against metronomes... ;-)