> Western music notation already has one: the grand staff has a gap in the middle. Simply put a gap with a single ledger line between each octave and you get a system that looks an awful lot like the current system.

Except that's not how a multi-staff partition works. Each staff has a clef that determines it's offset; they can be the same or wildly different. They also usually signify different "voices" or instruments, further encoding musical nuance beyond pitch and tone.