Composer, unknown, dropping down ranks of minor celebrity (5)
I believe the answer is:
liszt
'composer' is the definition.
(Franz Liszt)
I don't understand the rest of the clue.
(Other definitions for liszt that I've seen before include "Hungarian composer/pianist" , "Pianist and composer Cockneys associate with Brahms" , "Hungarian virtuoso composer" , "Franz ---, Hung. composer" , "Brahms and - is rhyming slang for drunk" .)