The chapter that follows is "The Coming of the Valar and the Building of Valinor".
Before the Music of the Ainur is in fact narrated, by Rúmil to Eriol (who had recently taken residence at the Cottage of Lost Play), the chapter begins after Christopher Tolkien's introduction with what he names the "Link" between the prior chapter and tale and this one. Eriol speaks with Lindo at the Cottage, sleeps and has a dream, and on the next day takes a long stroll outside listening to the birds, and comes across Rúmil. Learning that Rúmil is very old and knowledgeable, Eriol eagerly questions him of many things concerning the history of Valinor and Arda.
Following this, Christopher Tolkien comments on this Link, and then the narration of the Music of the Ainur begins, immediately from where Eriol and Rumil had left off. The narrated story holds few differences from the final Ainulindalë, such as the absence of Eru's proclamation of Eä! Let these things Be!, and of mentions to "the Secret Fire".