Wow- I was poking around in the forum and found this excellent question. Really set me to thinking.

I had a number of years of formal musical theory, been playing music for decades, taught math for over 10 years and I find it challenging to find a concise answer to this thoughtful question. I'll keep thinking on a good answer but here's a few thoughts about the "key" of a piece of music. My opinions of course, but a cloudy channeling of my teachers.

'Disclaimer: there are a lot of people making excellent music that don't worry a whit about music theory. Ear and experience drop them in the groove.

Formally, key is determined by the "key signature". A major scale has a specific progression: start at the root note, then whole step, whole step,half step, followed by 3 more whole step and 1 more half step. For C scale (and its relative A minor) there are no sharp or flat notes along the way. On a piano, that usually means no "black" notes on the keyboard. All of the other keys will include some sharps and flats.

Many classical pieces are written in a particular key because, to many, each key has a particular feel. Beethoven deliberately chose the key of F major for his Pastorial symphony because of the "feel" of the tones in that key-" inspiring, heavenly" to use the words of others. You can transpose and play it in a different key and it sounds ok but it just isn't right. Other pieces transpose just fine but I believe in the "key can define/support" feel thing.

Now away from my teachers and onto the experience thing.

Sometimes you'll change keys to make it easier to play. Irving Berlin famously had a piano that physically moved the keyboard around because he could only write and play in the key of C major! He used this piano when playing with others (horns/strings) that needed to transpose out of C to make the tunes playable on their instruments.

Sometimes you'll choose a key because everyone is familiar with it (12 bar blues in A or E, anyone?) and can jam the shit out of it. This is really fun! And, the reason I'm personally here at GT. For a horn guy, for example, it may be Bb but a good guitarist just changes position and jams on.

As a keyboard player over the years, I usually had to change keys to fit the vocal range of a singer. I envied the guitar players who were always so cool about it while I pitched a fit about transposing on my keyboard. I was such a baby about it. Most of these guitarists weren't really thinking about theory. They just moved to a different position on the neck. Easy as changing a hat and always sounded great.

Keep digging into it and stay curious. Realize that while you can play and jam without theory, actually having an idea about what's going makes it so much more rewarding.

And, surprisingly, inspire your creativity.