Alessi played with the San Francisco Symphony, the Metropolitan Opera orchestra, and the Radio City Music Hall Orchestra, among other ensembles.
Maria Leone Alessi, his wife, is a former Metropolitan Opera soprano. His son Joseph is the principal trombonist with the New York Philharmonic, and his son Ralph is a jazz trumpeter.
The Brooklyn-born Alessi studied at Juilliard, and played in the Navy band in World War II. After the war, he continued his studies at the Manhattan School of Music. With William Vacchiano, one of his former teachers, Alessi created the straight mute, which allows a trumpet to reach very low notes that it would not otherwise be able to play.
Alessi moved west with his family at the end of the 1950s, and spent a year playing in Las Vegas before moving to San Francisco and performing with the San Francisco Symphony and the San Francisco Opera's orchestra. He also began teaching, which he continued to do for many years.
At a funeral mass held today in San Rafael, California, Joseph Alessi Jr. played Puccini's "O babbino caro," and Ralph Alessi played "Amazing Grace" with saxophonist Ravi Coltrane. Maria Leone Alessi sang Oscar Rasbach's "Trees."