Final answer:
The nickname for the 5 record companies that dominated the music industry is known as the 'Big Five'. They were a major part of the discussions about censorship in music during the 1980s.
Step-by-step explanation:
The nickname for the 5 record companies that dominated the musical recording market is called the Big Five. This term refers to the major record companies which had a substantial influence over the industry and were often involved in debates surrounding censorship, such as the introduction of warning labels on records featuring 'explicit lyrics'.
Interestingly, in the 1980s, discussions about censorship in music led to a cultural shift, with artists like Twisted Sister's Dee Snider and folk musician John Denver testifying against such labels, advocating for artistic freedom in the face of congressional hearings.