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A round in music is a song which can be sung by two or more groups of people. One group starts off and the next group start to sing the same song a bit later. It should sound nice together. When a group gets to the end of the song they start again. They can go round and round, singing it several times.
Usually rounds are in 2, 3 or 4 parts (meaning that they are for 2, 3 or 4 groups of people). When a group of people sing or play a round they usually sing it an agreed number of times. This means that the group which starts last also finishes last.
Frère Jacques is a famous French round. London's Burning is an English round. Both those rounds can be sung in 4 parts.
The oldest round that we know is Sumer is icumen in. This is a very famous piece of music because it is the oldest piece of music we know that was written down. It was written around 1250 by a monk. It can be sung in 4 parts with 2 bass drones (held notes).
Writing music by making one part imitate another is called canonic writing. Rounds are a kind of canon.