Following the direction set by the refurbished building (and perhaps inspired by 8vo’s work of a few years previous), the Barbican identity uses only Futura. But not just any Futura. The brand guidelines (thoughtfully designed in 2004–07 by North) specify Futura SH from Scangraphic, a cut designed specifically for large sizes. It is quite tightly spaced and omits most of the ink traps and other compensations made in the text-optimized cuts. Also note that the round shapes remain circular in all weights, unlike some versions of Futura.
The brand guideines book really is a fine piece of work. Anyone building a new typographic identity — or just a brand book — can glean much from it. Download a PDF of the typographic section.