Not notice tradition moved by anthem (7)
I believe the answer is:
introit
'anthem' is the definition.
(thesaurus)
'not notice tradition moved' is the wordplay.
'not' suggests deleting specific letters.
'notice' becomes 'ad'.
'moved' indicates anagramming the letters.
'tradition' with 'ad' taken out is 'trition'.
'trition' is an anagram of 'INTROIT'.
'by' acts as a link.
(Other definitions for introit that I've seen before include "Opening hymn" , "musical piece" , "Rot in it for a church entry hymn" , "Church psalm" , "piece of church music" .)