In The Music of the LOTR Films, Doug Adams identifies four themes for the One Ring (with one theme's name changing as it matures into a Fourth Age theme).

THE HISTORY OF THE RING - which I will continue to call the Ring Theme, is used "whenever the One Ring changes hands or marks significant progress in its journey"  (Doug Adams, CR-FOTR liner notes, page 7)

THE SEDUCTION OF THE RING - which I will continue to call the Seduction Theme, "represents the One Ring's overwhelming allure." (Doug Adams, CR-FOTR liner notes, page 8)

THE FATE OF THE RING - this theme appeared once as an "incomplete phrase" in the TTT. It reappeared in the ROTK as the Fourth Age Theme, The Destruction of the Ring.

THE DESTRUCTION OF THE RING, which develops out of the Fate of the Ring theme, is heard -- not while we see the destruction of the Ring -- but when we see the result of the Ring's destruction has on Barad-dûr and the lands surrounding it.