A character can use a spell-trigger item (such as a wand) if the spell it casts appears on his or her spell list. From the SRD:
Anyone with a spell on his or her spell list knows how to use a spell trigger item that stores that spell. (This is the case even for a character who can’t actually cast spells, such as a 3rd-level paladin.)
The term "spell list" isn't defined in the PHB glossary, but it's used in the class feature descriptions of spellcasting classes. For instance, the Sorcerer says:
A sorcerer casts arcane spells which are drawn primarily from the sorcerer/wizard spell list.
Unfortunately, the Sublime Chord's "Spells per Day" class feature (Complete Arcane) does not exactly use this nomenclature. It says:
A sublime chord has the ability to cast a small number of arcane spells, all of 4th level or higher. To cast a sublime chord spell, a character must have a Charisma score of at least 10 + the spell’s level, so if she has a Charisma of 13 or lower, she cannot cast any sublime chord spells. Bonus spells are based on Charisma, and saving throws against these spells have a DC of 10 + spell level + the sublime chord’s Cha modifier. A sublime chord can choose spells from the sorcerer/wizard spell list or the bard spell list; if a spell appears on both lists at different levels, she uses the bard version of the spell.
Question: Can a Sublime Chord use a wand of a low-level Sorcerer/Wizard spell that does not appear on the Bard list (e.g., Magic Missile), without using UMD?
Argument for: Nowhere in the rules is a "Sublime Chord spell list" mentioned. Instead, Sublime Chords have (somewhat restricted) access to both the Bard spell list, and the Sorcerer/Wizard spell list (both of which are mentioned in the rules). Thus, Magic Missile appears on Sublime Chord's spell list (or rather, one of the two spell lists it has access to), so the Sublime Chord should be able to use a wand of it without UMD.
Argument against: The Sublime Chord's spell list should be understood to mean "the list of spells from which it can select spells known," and is thus effectively "the union of the Bard and Sorcerer/Wizard spell lists from levels 4-9, using the Bard version where they overlap." Thus, Magic Missile is not on the Sublime Chord's spell list, and a Sublime Chord would have to use UMD to activate a wand of it.