In my musical training I was taught that musicians figured it out in the west by trial error. Bach then wrote the Well Tempered Klavier to demonstrate pieces in all 12 major and 12 minor keys, in 1722.
There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century.
Notably, well-temperament is not equal-temperament. Also under-appreciated is that you can change the pitch of each strike of a key on a clavichord by striking harder/softer or pressing into the key. It uses a “hammer-on” like action, so if you press into the key after striking a note, you will increase the tension on the string and raise the pitch. For me, realizing this gave me a whole new appreciation for CPE Bach’s music. There’s probably a dissertation in that somewhere for somebody.
A klavier is not necessarily a clavichord, but an organ has some interesting tuning tricks up its sleeves with its various stops.
Organs are not really tuned with the stops, the stops engage or disengage banks ('ranks' in organ terminology) of related pipes. The only way this would affect tuning is if you have a secondary rank with an alternate tuning, you'd have to disable one rank and enable another.
Organs are tuned directly at the individual pipes (just like string instruments are tuned at the strings). Each pipe, depending on the kind of pipe used has a slide that goes in and out on one end to match the pipe length or, alternatively, a tiny tongue of metal that is rolled up or extended. Tuning (or 'voicing') an organ is super labor intensive and time consuming.
I am well aware. I have participated in the tuning of a (small) organ, as I am an organist. You can affect the perceived tuning of the instrument by engaging mutation stops / aliquots. Further, speaking from experience, in practice, stops have a tendency to go out of tune at different rates. Most organists know their instrument very well, have have an intuition for what keys will sound best with what stops at any given time on their instrument.
There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century.
https://en.wikipedia.org/wiki/Well_temperament#targetText=Or....