ASTM A228 says Rc41-60, which is a really broad range. Other specs for hard-drawn spring wire (not technically "music wire") might be Rc31-55, slightly less hard, but still a broad range of hardness. Generally, I assume the smaller diameters are toward the high side on hardness. Since it's a common fractional size, I'd try a HiRoc drill and use a jig/fixture with drill bushings. If not HiRoc, then some other carbide drill intended for high hardness, high carbon steel. Preferably, I'd grind a flat on the starting position for the hole.
When you say "in the end", do you mean an axial hole or a cross hole? If axial, you are talking about some fairly large diameter music wire, and possibly the core will be somewhat less hard than the surface. If cross hole, then a drill bushing and/or starting flat may be critical to getting your holes started without tearing up expensive drills.