Albright was born in Gary, Indiana, and began learning the piano at the age of five, and attended the Juilliard Preparatory Department (1959-62), the Eastman School of Music (1962-3) and the University of Michigan (1963-70). He then received a Fulbright scholarship to study with Olivier Messiaen in Paris for a year. In 1970 he was appointed to the faculty of the University of Michigan, where he taught until his death in Ann Arbor, Michigan in 1998.
His music combined elements of non-tonal classical music with American popular music. In particular, he was an enthusiast for ragtime, and was instrumental in helping to revive interest in composers such as Scott Joplin and Joseph Lamb.