Why would Melkor weave such a spiteful verse in the Ainuindale?

Question

Eru choreographed the first song and Melkor f'd that up, then Manwe was in charge of the second song and so again Melkor got his discord sown into the second verse. Why would he intentionally ruin the Music of the Ainur?

Answer 1

There are really two reasons

Melkor really wanted his own creation

This is Melkor's defining character trait, and motivates every decision he makes in the entire Silmarillion; he wants to be the Eru of his own creation. Unfortunately that's not actually an obtainable goal, because the power of true Creation wasn't given to him; only Eru can do that.

In the Ainulindalë, this desire hasn't quite driven Melkor to frothing madness yet; he's not willing to break everyone else's toys just because he can't make his own. Instead, he tries to inject his own ideas (or what he thinks are his own ideas; more on that soon) into the Creative Music; consider this passage (emphasis mine):

But now Ilúvatar sat and hearkened, and for a great while it seemed good to him, for in the music there were no flaws. But as the theme progressed, it came into the heart of Melkor to interweave matters of his own imagining that were not in accord with the theme of Ilúvatar, for he sought therein to increase the power and glory of the part assigned to himself. To Melkor among the Ainur had been given the greatest gifts of power and knowledge, and he had a share in all the gifts of his brethren. He had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own, and it seemed to him that Ilúvatar took no thought for the Void, and he was impatient of its emptiness. Yet he found not the Fire, for it is with Ilúvatar. But being alone he had begun to conceive thoughts of his own unlike those of his brethren.

Some of these thoughts he now wove into his music, and straightway discord arose about him

^{The Silmarillion I Ainulindalë}

This was part of Eru's plan

Despite how it may seem, Melkor isn't actually disrupting the Music. Rather, he's following Eru's plan for the world. Consider this passage (emphasis mine):

Then Ilúvatar spoke, and he said: 'Mighty are the Ainur, and mightiest among them is Melkor; but that he may know, and all the Ainur, that I am Ilúvatar, those things that ye have sung, I will show them forth, that ye may see what ye have done. And thou, Melkor, shalt see that no theme may be played that hath not its uttermost source in me, nor can any alter the music in my despite. For he that attempteth this shall prove but mine instrument in the devising of things more wonderful, which he himself hath not imagined.'

^{The Silmarillion I Ainulindalë}

There's a really great primary source quote (which I wish I could find; maybe a question in the works) where Tolkien very explicitly says that Eru created Melkor specifically to cause conflict within his world; conflict is what makes life interesting. This is hinted at in Ainulindalë itself, when Eru addresses Ulmo specifically:

Ilúvatar spoke to Ulmo, and said: 'Seest thou not how here in this little realm in the Deeps of Time Melkor hath made war upon thy province? He hath bethought him of bitter cold immoderate, and yet hath not destroyed the beauty of thy fountains, nor of my clear pools. Behold the snow, and the cunning work of frost! Melkor hath devised heats and fire without restraint, and hath not dried up thy desire nor utterly quelled the music of the sea. Behold rather the height and glory of the clouds, and the everchanging mists; and listen to the fall of rain upon the Earth! And in these clouds thou art drawn nearer to Manwë, thy friend, whom thou lovest.'

^{The Silmarillion I Ainulindalë}