Hauptmann was born at Dresden, and studied violin under Scholz, piano under Franz Lanska, composition under Grosse and Francesco Morlacchi, the rival of Carl Maria von Weber. Afterwards, he completed his education as a violinist and composer under Louis Spohr, and till 1821 held various appointments in private families, varying his musical occupations with mathematical and other studies bearing chiefly on acoustics and kindred subjects.
For a time also Hauptmann was employed as an architect, but all other pursuits gave place to music, and a grand tragic opera, Mathilde belongs to the period just referred to. In 1822 he entered the orchestra of Kassel, again under Spohr's direction, and it was there that he first taught composition and musical theory to such men as Ferdinand David, Friedrich BurgmĂŒller, Kid and others.
In 1842 Hauptmann obtained the position of the Kantor of the Thomanerchor at the Thomaskirche in Leipzig (a post long previously occupied by Johann Sebastian Bach) together with that of professor of music theory at the conservatoire founded by Felix Mendelssohn, and it was in this capacity that his unique gift as a teacher developed itself and was acknowledged by a crowd of enthusiastic and more or less distinguished pupils.
Hauptmann's compositions are marked by symmetry and workmanship rather than by spontaneous invention. Among his vocal compositions may he mentioned two masses, choral songs for mixed voices (Op. 32, 47), and numerous part songs.
The results of his scientific research were embodied in his book Die Natur der Harmonik und Metrik (The Nature of Harmony and Meter, 1853), in which a philosophic explanation of the forms of music is attempted.