Collins, Derek. 2004. Master of the Game: Competition and Performance in Greek Poetry. Hellenic Studies Series 7. Washington, DC: Center for Hellenic Studies. http://nrs.harvard.edu/urn-3:hul.ebook:CHS_CollinsD.Master_of_the_Game.2004.
Part III. Epic Competition in Performance: Homer and Rhapsodes
Muses love singing by turns
16. The Amoebaean Muses
And the Muses, who sang ameibomenai opi kalêi.
All nine Muses ameibomenai opi kalêi sang a dirge.
This line with the two that follow caught the attention of at least one scholiast who thought it inappropriate for the Muses to sing a lament.  But even so, it is their fixed manner of singing that interests us, which we can see once more represented in the Homeric Hymn to Apollo (189):
And all the Muses together ameibomenai opi kalêi
[The Muses] themselves while Apollo played the cithara were singing by relay and by turns.
Then to them [the judges] leapt forth, and by turns made a judgment
καὶ ξενίῃ ἀγαθῇ· ἡ γὰρ θέμις, ὅς τις ὑπάρξῃ.
After repaying you well with gifts he would have sent you away
also with good hospitality; for this is right for whoever has first given.
Gift-giving in Homeric poetry, even as compensation for initial hospitality, is never a matter of “equality,” but of the establishment of mutually recognized obligation. By repaying the disguised Odysseus well, Laertes hypothetically emphasizes the largesse of the real Odysseus. Similarly, in the context of warfare when Hector urges Alexander to join the battle, Alexander replies that νίκη δ᾽ ἐπαμείβεται ἄνδρας “victory comes in turn to men (Iliad 6.339).” His point is that victory requites men successively, and now that he has been cajoled by his wife, it seems better to him to stop grieving and arm for battle (6.338–9). Victory, however, at any one moment is an absolute value in Homeric poetry.  As a consequence, the notion of turn-taking or exchange in ἐπαμείβεσθαι extends beyond compensation—that is, it is not a matter of returning to a state of equilibrium, but of tipping the scales back in favor of the victor against his adversary. We find this sense especially in the compound ἀνταμείβεσθαι, which is not Homeric, but appears in other archaic poets specifically in the context of repayment for injustice.  In this way restitution actually demands overcompensation that subordinates one party to another. This meaning is detectable in a form such as ἀμοιβή ‘requital’, which is not found in the Iliad but only in the Odyssey in the context of gift-exchange (1.318) and sacrificial compensation (3.58). In Hesiod the juridical sense of ἀμοιβή is already brought out clearly when we are told that for a series of infractions, such as mistreating suppliants, orphans, the elderly, or sleeping with a brother’s wife, “in return for unjust deeds [Zeus] imposes a harsh compensation” (ἔργων ἀντ᾽ ἀδίκων χαλεπῆν ἐπέθηκεν ἀμοιβήν, Works and Days 334). In Hesiod’s view, Zeus’ punishment is not an equivalent recompense worked out in economic terms for a wrong committed, like the Old High German concept of Wergild, but a continual reminder that men remain subordinate to the judgment of Zeus. The harshness of Zeus’ punishment must therefore surpass the initial offense. A similarly punitive sense of ἀμοιβή, though not expressly in the context of theodicy, can be found in other archaic poets as well. 
ὀρχείσθην δὴ ἔπειτα ποτὶ χθονὶ πουλυβοτείρῃ
ταρφέ᾽ άμειβουένω· κοῦροι δ᾽ ἐπελήκεον ἄλλοι
ἑσταότες κατ᾽ ἀγῶνα, πολὺς δ᾽ ὑπὸ κόμπος ὀρώρει.
But when they had made trial of the ball up in the air,
the two then danced on the much-nourishing earth
rapidly taking turns: the other young men standing about
the gathering place stamped out the time, and a great noise rose up.
The precise details of this performance elude us, as does the aspect of ἀμειβομένω, which can be both temporal and instrumental. This ambiguity means that the turn-taking involved refers on the one hand to the opportunity each has quickly to dance his part (temporal). Although we are given no further information about the performance, each brother’s dance is aimed at outdoing the other, and we can expect that they are judged individually, as is usual for participants in Homeric contests. On the other hand the rapid turn-taking is also a basis for their competitive dancing (instrumental), and it is the contest in general for which everyone is gathered and tapping out a rhythm. We have, then, a situation in which the use of ἀμείβεσθαι highlights both Halius and Laodamas’ individual contributions to the dance contest, while at the same time hinting at the collective product that their competitive performance creates. Odysseus acknowledges the brothers’ collective talent, in fact, when he praises them as the best dancers (βητάρμονας … ἀρίστους, 8.383).