But take a trip to Los Angeles and you will find this son of Glasgow has a star on the Hollywood Walk of Fame.
Primrose - who lived from 1904 to 1982 - was described in his obituary as "the greatest violist of his time" and the man who made the viola popular.
But In the city of his birth he is virtually unheard of outside certain musical circles.
It seems strange partly because he died only 32 years ago and partly because he was a star in Hollywood.
A plaque commemorating Primrose is situated on the wall of the house he was born in - a first floor flat at 18 Wilton Drive - an area locals call North Kelvinside and the rest of us call Maryhill.
The plaque shows a hand caressing a viola with the words: "One of the greatest musicians of all time the viola player William Primrose was born here 1904."
Primrose's father John, a violinist with the Scottish Symphony Orchestra, started teaching him how to play the violin at the age of four.
Wilton Drive at that time had such a proliferation of musicians it was known as Harmony Row and young William was soon giving local solo concerts, including one at the St Andrew's Halls.
Despite Primrose preferring the larger viola, his father insisted he stick with his violin lessons and it was not until he was a 21-year old studying in London that he was able to concentrate on viola playing.
Ditching the violin was risky but it was the best move Primrose ever made. In classical music circles, his superb viola playing saw him rise to superstar status.
He was the soloist on the first ever recording of Harold in Italy, a work by Hector Berlioz and he gave the world premiere performance of Viola Concerto by Bela Bartok in 1949.
Benjamin Britten wrote a piece of work specially for him and he became a CBE in 1953.
It is perhaps bizarrely Scottish that a man who was so highly regarded throughout the world can be so unrecognised in his native city.
Primrose spent his later years in the USA. He had three wives, a passion for cricket and chess, and a lifelong love affair with the viola.
The plaque at his childhood home was designed and crafted by Glasgow artist Ian Ramsay, and funded by the Friends of Glasgow School of Art.
The great Yehudi Menuhin once said that "Bill Primrose was certainly the first star of the viola".
High praise indeed and perhaps it is high time more is known locally about this unheralded Glaswegian.