Another excellent piece by Rachmaninoff. Or is it? The original theme of the entire work is from Paganini's 24th Caprice (a staple in the online classical music circles). Rachmaninoff took the theme and made 24 variations from it.
The variation here (15:27), 18, is a drastic variation of the original theme. Instead of being in A minor, it's now in D-flat major, as well as being inverted. Rachmaninoff created a beautiful, hopeful theme from it. The 18th variation is the most popular of the set for this stark contrast.
There are, of course, pertinent Bible verses to this. The first is from the incredibly Messianic 61st chapter of Isaiah:
"[Adonai] has sent me [Yeshua] to [...] to comfort all who mourn to console those who mourn in Zion, to give them beauty for ashes, the oil of joy for mourning, the garment of praise for the spirit of heaviness." (Isaiah 61:1-3 TLV)
This is also found in Jeremiah 31:
"Then will the virgin rejoice in the dance,
both young men and old men together.
For I will turn their mourning into joy,
and I will comfort them, and make them rejoice out of their sorrow." (Jeremiah 31:12 TLV)
Rachmaninoff, perhaps, didn't know that he was acting out Messianic prophecies. The original theme from Paganini can't exactly be called mournful or heavy. However, in the other variations, ones preserving the original thematic feel of the piece, Rachmaninoff often makes them more imposing than before.
In multiple spots, he includes Dies Irae, literally "day of wrath," to give the piece a far more sinister feel. The effect created is now certainly going from mourning to joy.
There are multiple instances in classical music where key changes, modulations, or inversions are used to present a mourning-to-joy situation. Consider the final chord of Lacrimosa from Mozart's Requiem. The piece, also about the day of judgement, is in D minor. It ends, however, on a D major chord, representing eternal joy with G-d in heaven.
Anytime you hear a particularly moving progression from sorrowful to joyful, remember the promise made by G-d that the same would happen with us. He will take the sorrows of our life and transform them into joys everlasting.