The Guildhall School of Music & Drama
is based in the Barbican. It holds regular free events before London Symphony Orchestra concerts, and next month there are events on the 1st, the 3rd and the 20th at 6pm, 5.30pm and 6pm respectively. Full details can be found on the School's website, currently on the What's On: About The School
There are also free concerts in the City of Westminster at St Martin-in-the-Fields
Free concerts are not unique to London; there are free events next month
in Birmingham, for example. Nor are they unique to the UK; here is a listing
from San Francisco.
Check out your local events calendar for free live music and other events.