When I studied music in college, there was a term that was thrown around an awful lot: "intentional fallacy." The idea that the performance of a piece could be more authentic if you could understand the mind or intentions of the artist was very strongly resisted.
I'd agree with that about 70%. However, AS A PERFORMER, trying to figure out the mind of the composer is part of the fun of immersing myself in a piece.
I have one particular interpretation that is SUPER clear to me, but that not everybody has agreed with (in fact, nobody has!).
It's the beginning of Beethoven's 9th symphony. To me, it sounds like the instruments are tuning and rehearsing. The timbral density gets more intense, and the piece finally erupts in the famous unison melody.
This is not theme and countertheme. This is order erupting spontaneously out of chaos. If I were to conduct this piece, I'd actually make the orchestra's warm up merge directly into the performance of the piece-- no tap! tap! tap! and a few moments of silence to gather the orchestra's energy. No applause while the first violinist or conductor walk onto the stage. Just a few moments of eye contact, a count-in, and rehearsal gives way to performance without any artificial delineation of the moment.
If an undergrad theory student were to look at the piece through theory-on-paper eyes, they would probably say something like:
The piece begins with a pianissimo tremolo on the V, with the 3rd omitted. More instruments are added. Finally, in bar 17, the first theme in d minor begins.
How totally useless that analysis is in thinking about this great piece of music!
There's nothing new under the sun, and this is something very often said of the study of Western art music in particular. The notes of Beethoven's 9th will always be Beethoven's 9th. But by reading between the lines, or by gazing into the flames, or by peering into a crystal ball, eventually something will come out of the process, not of Beethoven, or even of the conscious you, but the universe of sound speaking THROUGH both of you.
My point is this (and I hope by now you've learned to look for the bold-face punchlines in any of my answer, Aaron!:
By playing what is unwritten, you may reveal what has been undiscovered. Whether the original composer intended it, or whether they'd even agree if they were alive, is irrelevant. There's no right or wrong (though uptight theory profs might argue that point). What there IS is music trying to burst forth from the chaos.