Just as Carol once said, and as has been inferred throughout history, alchemy and songs share a deep connection to each other.
The song that was sung by the alchemists is actually a technique which consumes life for transmutation into energy.
In episode 1 the alchemists used it upon their selected sacrifices and took those lives to transmute energy, which they then used to complete Youaltepuztli.
However, in episode 12 it was used upon their very own lives, 3 forms of perfect beauty.
By making the ultimate sacrifice they were able to transmute an even higher level of energy
that was strong enough to completely neutralize the eternal pollution of the land that would have resulted from use of the nuclear weapon.
A number of mechanisms in this process are nearly identical to that of the "Climax Song," the last resort of the Symphogear adaptors — a song that burns life.