Shanet was born in Brooklyn in 1918. He initially studied cello and received two degrees from Columbia University: a bachelor's in 1939 and a master's in musicology in 1941.
After serving in the Pacific during World War II, Shanet studied composition with Bohuslav Martinu and Aaron Copland and conducting with Serge Koussevitzky and Fritz Stiedry. He was a conducting assistant to Leonard Bernstein at the New York City Symphony in the early 1950's and program annotator for the New York Philharmonic in 1959 and 1960, according to the Times.
He appeared as guest conductor of the New York Philharmonic, the Boston Symphony Orchestra at Tanglewood, the CBS Symphony and orchestras in Holland and Israel.
According to the Times, Shanet was drawn to new music and seldom-played repertoire. He composed music for orchestra, string quartet and band.
Shanet was the author of two books, Learn to Read Music, published in 1956, and Philharmonic: A History of New York's Orchestra, published in 1975.
In 1953 Shanet joined the Columbia faculty as a professor of music and a conductor of the university's orchestra. He was chairman of Columbia's music department from 1972 to 1978, then a professor emeritus.