Dark and dischordant music was played into the creation of Middle Earth by Morgoth, so as long as Middle-Earth exists, evil will exist, according to Eru.
The Music of the Ainur is not the creation of Arda. The Music just generated a vision before the Ainur, it granted them a model to strive for (it's like Eru showed them a picture of Michelangelo's David and ordered his Ainur to reproduce it from a rock of marbled stone, which he provided for them). After seeing this model, the Valar (including Melkor) created the world on the sketchbook Eru provided for them (Eš), each following their own interpretation of the vision they saw :
For the Great Music had been but the growth and flowering of thought in the Tuneless Halls, and the Vision only a foreshowing; but now they had entered in at the beginning of Time, and the Valar perceived that the World had been but foreshadowed and foresung, and they must achieve it.
(from the AinulindalŽ)
During the Music of the Ainur, evil didn't yet exist; only the origins of evil were present : the longing to have more power and glory than others, and to dominate the others. Nothing is evil in the beginning, as evil cannot originate from Eru.
Not even during the first war between Melkor and the Valar, during their labours on Arda, evil existed, as Melkor was chased out of Arda by Tulkas Astaldo and Almaren was completed - the Spring of the Earth began.
Only when Melkor returned and built Utumno, evil and hatred began to (literally) leak out of him and irreparably spoiled Arda for all time, at least until its breaking :
Now Melkor began the delving and building of a vast fortress, deep under Earth, beneath dark mountains where the beams of Illuin were cold and dim. That stronghold was named Utumno. And though the Valar knew naught of it as yet, nonetheless the evil of Melkor and the blight of his hatred flowed out thence, and the Spring of Arda was marred. Green things fell sick and rotted, and rivers were choked with weeds and slime, and fens were made, rank and poisonous, the breeding place of flies; and forests grew dark and perilous, the haunts of fear; and beasts became monsters of horn and ivory and dyed the earth with blood. Then the Valar knew indeed that Melkor was at work again, and they sought for his hiding place. But Melkor, trusting in the strength of Utumno and the might of his servants, came forth suddenly to war, and struck the first blow, ere the Valar were prepared; and he assailed the lights of Illuin and Ormal, and cast down their pillars and broke their lamps. In the overthrow of the mighty pillars lands were broken and seas arose in tumult; and when the lamps were spilled destroying flame was poured out over the Earth. And the shape of Arda and the symmetry of its waters and its lands was marred in that time, so that the first designs of the Valar were never after restored.
(from chapter 1 of the Quenta Silmarillion)
After the breaking of Almaren only Valinor, the Blessed Realm, the Undying Lands, on the continent of Aman was impregnable and unspoiled by Melkor's evil, as it was blessed by the Valar.