Right now, each primary piece (/KQBNR/) has a unique tone, but it's hard to tell each of them apart. And there's no distinguishment between white pieces and black pieces. I think a "better" musical scheme would be to have some way to audiably know which capital piece was being moved and by which side.

Maybe a different instrument for each piece, and a different octave for each player? 8 notes to an octave, and the board is 8x8... So, for example, an opening move of b1-c3 could be four notes (octave 1, note 2; o2,n1; o1,n3; o2,n3) of a flute, while black's first move might be d7-d5 (o3,n4; o4,n7; o3,n4; o4,n5) played on a pennywhistle. Or some such.
x-axis, white = octave 1, notes 1-8
y-axis, white = octave 2, notes 1-8
x-axis, black = octave 3, notes 1-8
y-axis, black = octave 4, notes 1-8
pawn = pennywhistle
rook = tuba
knight = flute
bishop = harpsichord
queen = mandolin
king = trumpet