The saxophonist, who wrote the theme tune for The Avengers, Tomorrow's World and Modesty Blaise, died on Saturday, his management told the BBC. He had been unwell for several months.
Sir John made many TV and stage appearances during his career, also working with some of the biggest jazz names around, including Charlie Parker, Ella Fitzgerald and Nat King Cole.
He was married to jazz singer Dame Cleo Laine.
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