What is solfège?
Solfege, in the broadest sense, refers to a method of teaching Western music that includes pitch recognition, sight reading, and sight singing. It can also mean, in a narrower sense, sight-singing. Sight-singing is that the first time you see a sheet music you've never heard of, and sing it at the correct pitch using "do, re, mi..." (sol-fa syllables) or training to be able to do that.
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Sol-fa syllables derives from a Latin hymn (Ut queant laxis), and they have been modified little by little, and are still used today to express scale-degrees or names of notes. There are different types of solfege, depending on how to handle note names, scale-degree names, and chromatic variations. See "Types of Solfege" for more details.