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The way I understand it is that the difference between the keys is related to the root. If you tune perfect fifths and accurate major thirds to C, it'll have an effect on pieces in F# minor.
No, there should be no difference at all between harmonic ratios for each key in equal temperament. In effect you get exactly the same scale with exactly the same relationships, only it can start on any frequency. Conventionally its tuned to A440, but there's no reason why you couldn't tune it half a semitone above that - around A450 - and in theory the scale should still have the same mood.
Of course if people really have a kind of rudimentary perfect pitch that won't be true. It would be an interesting thing to experiment with.
Synthesizers are usually tuned with equal temperament because even when you get access to the tuning tables, hardly anyone knows how to calculate the ratios needed to use them. Apparently non-Western scales are popular in Arab countries, and synths are regularly tuned to non-Western scales there. But in Europe and the US, equal temperament is usually the default aznd it's rarely changed.
Harpsichords, I don't know - I suspect not. It probably depends how early-music the player is trying to be. There are various tunings with names like Kirnberger and Valloti which are all variations on not-quite-equal temperament, and I'd expect those to sound different away from the home key.
Perfectly tuned keys sound very strange to Western ears. Terry Riley did an album called Shri Camel in perfect intonation and it's not easy to listen to, because all the intervals sound out of tune - at least until you get used to them.
The other thing about equal is that introduces movement. There's a theory that because the intervals all beat slightly, there's a subconscious tendency to want to move on, which means that music speeded up and became busier as temperament moved away from perfect intervals.
I'm not sure how true that is, but it's an interesting idea.
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