I sometimes listen to incredibly moving music on YouTube and find myself regarding the music as both causing "rapture" and "anguish" in my soul at (almost) the same time. Is there a word that would embody "rapture" and "anguish" in this sense? I know they are antonyms but I'm wondering if there's such a word to describe this feeling.
It's a term frequently applied to the works of Ralph Vaughan Williams. There's an excellent description of his 1922 Third Symphony in this article from the Guardian, which may address the feelings you describe (although the word transcendent does not appear there).
There's a partial overview of Vaughan Williams's work at www.classical-music.com, including a link to a YouTube video of the Berlin Philharmonic performing the Fantasia on a Theme by Thomas Tallis. However, it's music that you can appreciate better with your eyes closed.
The two words you use, rapture and anguish are perhaps better identified in a context from anguish to rapture in order to maintain proper function based off of the words definitions and understanding their function. To describe the effects or affects of the music you're listing to of which cause rapture from anguish. Sometimes music or tones can cause more anguish of which one might desire more rapture from their state of emotion. This requires a more comprehensive explanation or understanding. By virtue we may come to the words comprehensive or comprehensively, or even exhaustive or exhaustively. Though that too would depend a lot on your own context. Though one could use the word catharthis depending on the context. For example: This piece of music is a great catharthis.