Some 00 or 000 steel wool will give the band a brushed finish and 'mute' the highly reflective look. Start with 00. Rub the steel wool along the length of the band, back and fourth to accentuate a brushed look. Use light pressure and check your progress frequently.
If the resulting look is too dull, you can use 0000 or 00000 steel wool to 'brighten' it up. A jeweler or hobbyist could use jeweler's rough to polish, but that wouldn't be your goal. Be sure to blow out any leftover debris.