Lively music institute recalled in fury (7)
I believe the answer is:
ragtime
'lively music' is the definition.
'institute recalled in fury' is the wordplay.
'institute' becomes 'mit' (Massachusetts Institute of Technology).
'recalled' shows that the letters should be reversed in order.
'in' means one lot of letters goes inside another.
'fury' becomes 'rage' ('rage' can be a synonym of 'fury').
'mit' back-to-front is 'tim'.
'tim' put inside 'rage' is 'RAGTIME'.
(Other definitions for ragtime that I've seen before include "Kind of music" , "Form of jazz with a two-four rhythm" , "musical style" , "Scott Joplin genre" , "Migrate to find jazz music" .)