You could shift every note in a song up by a quarter tone (or any similar small frequency) and still have the same song, yes. Western music doesn’t use quarter tones almost ever, so most instruments would need to be retuned to produce it (this could range from a few second slide adjustment for some instruments, to wrenching on hundreds of strings for an acoustic piano). It wouldn’t have a different letter name because (again) western music doesn’t have names for intervals that small; you would just wind up calling it C or C# depending on how you tuned the instruments, but with a different frequency than the normal C or C#.
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