fter the Music of the Ainur ceased, Ilúvatar spoke to them, telling them they were all very powerful, especially Melkor, but not even Melkor could destroy the vision of Ilúvatar, for no one was more powerful than he. "Ilúvatar arose in splendor, and he went forth from the fair regions that he had made for the Ainur; and the Ainur followed him. But when they were come into the Void, Ilúvatar said to them: 'Behold your Music!'" 5 For the first time, the Ainur had sight where before there had only been hearing, and they saw the World born before them. As they watched, the World grew and changed, a living thing before their eyes. They saw the coming of the Children of Ilúvatar: Elves and Men, the Firstborn and the Followers. Though they had been involved in the preparation of their world, they had not thought that the world had any purpose other than its own beauty. The Children of Ilúvatar came with the third theme of Music and was of Ilúvatar's conception only. The Ainur had nothing to do with their creation, and because of this, they loved the Children even more.
After seeing and loving the Children of Ilúvatar, the Ainur were infused with a deep desire to remove to the World and prepare the world for the coming of the Children. As Ilúvatar spoke to the Ainur about each of the gifts with which they endowed the World, the Ainur rejoiced at all they had done. But even as he spoke, and the Ainur were gazing at this vision, Ilúvatar took away that vision and hid it from them, and they perceived Darkness, something they had never before known. Then there was unrest among the Ainur, but Ilúvatar knew their thoughts and desires, and said "'Eä! Let these things Be!...and the World shall Be; and those of you that will may go down into it.' And suddenly the Ainur saw afar off a light, as it were a cloud with a living heart of flame;
and they knew that this was no vision only, but that Ilúvatar had made a new thing: Eä, the World that Is" 6 that the Elves call Arda.
After that, some of the Ainur remained with Ilúvatar beyond the confines of the world, but many of the greatest went into the World, on one condition that all of their power would be forever contained within the world and they would remain in it forever, becoming its life and it becoming theirs. Those who descended into Eä became the Valar, the Powers of the World.
When the Valar entered the World, they were astounded because it seemed to them that all that they had seen had never been—as if the World had been unmade. They realized that the Music had been a flowering and growth, and the Vision had only been a showing of what would be, but they had now entered at the beginning of Time and they had to achieve the Music and the Vision.
From this point on, the Valar began their great labors to prepare Eä for the coming of the Children of Ilúvatar until it was finally ready. In this work, Manwë, Aulë, and Ulmo did the most work; but Melkor was there from the beginning and he meddled in all that was done, destroying the beauty and the work of the others of the Valar. But Manwë called to him many other spirits to hinder the destruction of Melkor, so the World could flower and grow. Though the fruits of their efforts were never as they originally intended, eventually, Eä was ordered and made firm. "And thus was the habitation of the Children of Ilúvatar established at the last in the Deeps of Time and amidst the the innumerable stars." 7
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The Creation of the World is the second part of The Music of the Ainur, completing the exposition before The Silmarillion proper. In many ways, it is similar to most other creation myths: something out of nothing, creating order from chaos, and so forth. Even the familiar Biblical line "let it be" is used in this story. In the same way that G-d, said let there be light, and there was light, Eru Ilúvatar says "Eä! Let these things Be!", creating the world where before there was only Void.
What I see as a major difference, is that, instead of one Supreme Power ordering the world, the Supreme Power gives individual powers to beings of his thoughts, and these beings go into the World to realize the vision of that Supreme Power. Another difference is Melkor. I personally do not know of any other creation myth in which an evil power caused destruction to the vision of the Supreme Power. Where G-d created the world in seven days with no problems and everything He said was realized, the Valar had to work very hard to make the World into what they wanted it to be. Their original vision could never be realized because of the destruction of Melkor, but they still created a world of beauty for the Children of Ilúvatar.
Though the Valar created the World to the best of their ability, the war with Melkor continued to rage. How the Valar continued in the World and what happened next is related in The Two Trees of Valinor.