This thread is useless without pitchers.
People with music degrees can probably understand this much better than me. I’m a musician, I play or sing the note by the pitch apparent to me. I do understand that the math of music does not always agree with the sound of pitches that music comes from. I think the article explains the way around the inexact math of music. I don’t worry about it. When I play the piano I play the notes the tuner gave me and don’t have to think about what it would sound like in a different key. Different keys however do make the same musical piece sound different, that is mostly because of the compromises made with musical math. I play an organ that has a transposer on it. The organ uses a computer to come up with frequencies of each note and can be tuned (programmed) to different pitches. The organ is different than the piano, that is an electronic organ in that the computer recomputes notes based on the frequency of only one octave. Pipe organs on the other hand are individually tuned, each pipe in a pipe organ of perhaps thousands of pipes must be tuned to sound right with the pipes around it. Sounds like a simple task but it isn’t so simple, tuning an organ is a much more complicated task than tuning a piano.
Tuning forks used to be used to tune pianos now a little computer listens to each string and tells you when it is in tune, the better instruments even help you tune the multiple strings of a piano note slightly off of each other to make them blend without beats. Doing this takes care of the same piece played in different keys, for the most part anyway and makes it sound more natural and correct while to some very good ears may make the piano sound out of tune. It can be a real science. I’m glad I just have to play.