The musician and inventor was born in Stockton and was instructed in music by his father, Robert, and as an articled apprentice by Thomas Ebdon. Wright succeeded his father as organist at Sedgefield c 1784, and in 1797 did the same at Stockton. In 1817 he was organist at Kirkleatham but he returned to Stockton and remained there as organist, teacher and composer until his death.
Wright married Elizabeth Foxton in 1794 and set her operetta Rusticity to music. Wright's several inventions - for regulating musical time, for transposing music and for improving keyed instruments - are described by him in the advertisement to his Concerto for Harpsichord and Pianoforte (London c 1795).