MARK THE MUSIC The Life and Work of Marc Blitzstein. By Eric A. Gordon. Illustrated. 605 pp. New York: St. Martin's Press. $29.95.

The career of Marc Blitzstein is a curious chapter in the history of 20th-century music. As a composer he straddled the worlds of serious music and popular theater, writing art songs and symphonies as well as five Broadway shows and incidental music for plays by Shakespeare, Shaw and Lillian Hellman. Yet it is impossible to discuss his work without conjuring the names of others who surpassed him in talent and achievement. The influence of contemporary Berlin cabaret on his best-known work, the 1937 pro-union musical The Cradle Will Rock, is so indelibly marked that when it opened Kurt Weill went around New York saying to friends, ''Have you seen my new opera?'' And while Blitzstein served as a mentor and an inspiration to the young man who played piano for the Boston premiere of Cradle at Harvard University, Leonard Bernstein turned out to have the successful crossover career that would forever elude Blitzstein himself.

The heart of Blitzstein's story, as it emerges in Eric A. Gordon's thoroughly researched biography, is his political engagement as an American committed to Communism. As Mr. Gordon points out, during the Depression era those who felt most strongly about such social issues as civil rights, freedom of speech, unionization and Social Security were often Communists. ''Blitzstein himself serves as a good example in his own field: A talented composer who might have become wealthy and famous by going the commercial route, either in Hollywood or on Broadway, instead commits himself to a career of relative penury in order to write works that will influence his country in a progressive direction.'' But if politics was Blitzstein's passion, it was also his downfall, and not simply because of persecution (he was threatened but not destroyed by Senator Joseph McCarthy's anti-Red witch hunt in the 1950's). He spent months and sometimes years composing topical works that were already out-of-date by the time they were finished.

Ironically, while subsequent events have discredited the once admirable idealism of Communists, changing social attitudes have made openly gay public figures acceptable, even heroic. In Mark the Music, Mr. Gordon spares few details about his subject's unapologetic homosexuality. Blitzstein's first lover was the conductor Alexander Smallens. He performed the ''social gesture'' of marriage in 1933 at the age of 28 to the writer Eva Goldbeck, a brainy but masochistic, anorexic woman whose first words upon spying Blitzstein at the MacDowell artists' colony were ''Who's this fairy coming?'' Although they occasionally enjoyed ''moonlight'' (their code word for sex), the marriage was primarily intellectual and ultimately short-lived; sexually and professionally frustrated, Goldbeck starved herself to death at the age of 34. Blitzstein lived for a time with a bisexual Southerner he met in the Army during World War II, and he was murdered in 1964, at 58, by three sailors he picked up in a bar in Martinique.

A fascinating array of famous names and stories decorates the pages of Mark the Music. Eve Arden, Carol Channing and Bea Arthur made their first splashes in show business singing Blitzstein songs. The dramatic saga of the opening night of The Cradle Will Rock - when Orson Welles's Federal Theater cast, locked out of the theater on Government orders and forbidden to perform on any other stage, sang the score from their seats in a hastily rented auditorium - now overpowers the show's cartoon-simple sentiments. And the 1954 revival of The Threepenny Opera, instigated by Blitzstein's nimble and now standard English adaptation of Bertolt Brecht's lyrics, was a major turning point in the legitimization of Off Broadway. Sophisticated, intelligent and experimental as it was, however, Blitzstein's remained a talent that never quite lived up to its promise.

Sticking doggedly to a march-of-time chronology in this, the first full-length treatment of Blitzstein's life and the author's first book, Mr. Gordon has written a detailed but at times dull book that, erring on the side of inclusion rather than selection, spends equal numbers of pages on Blitzstein's major accomplishments and on shows he never completed or that flopped miserably. It is a mixed blessing to the reader that the biographer remains absolutely faithful to an unsatisfying life.

New York Times, July 16, 1989