Unlike what we see at I.02.484, Ι.11.218, I.14.508, I.16.112, where the Muses are invoked as plural goddesses, the Muse here at I.02.761 is invoked as a singular goddess. And the Muse is of course singular also at the beginning of the Iliad, I.01.001, and at the beginning of the Odyssey, O.01.001. There are also two other cases where a singular Muse is invoked:
A) In the First Song of Demodokos, O.08.73–82, which is featured as a proto-Iliad (see the comment at O.08.073-082), a singular Muse inspires the singer of tales at the beginning of his performance, at O.08.073. I think that the self-awareness at I.02.761 in invoking a singular Muse has to do with the singularity of the subject at I.02.760–770, since the subject in this case is Achilles. The Muse is asked for an answer to the Iliadic question: who is the ‘best of the Achaeans’? The answer of the Muse is that Achilles is the best. He is the singularity of the Iliad as epic, just as Odysseus is the singularity of the Odyssey as epic. That is why, I propose, the singular Muse here is the goddess Calliope. She is the perfect singularity of a Muse for these notionally singular heroes of two singularly important epics. After all, Calliope is the Muse of Epic. I refer here to my relevant arguments in Homer the Preclassic E[pilegomena] §109 (p. 345) about Calliope as the Muse of kings (Hesiod Theogony 79–93). Similarly, as I argue there, Orpheus was once the singular poet of kings, but his status was degraded in the Athenian phase of Homeric reception.
B) In the Third Song of Demodokos, O.08.499–533, when the singer of tales marks the beginning of his performance at O.08.499, the anonymous ‘divinity’ that he invokes at that point is a theos, in the singular. Short-term, this theos ‘divinity’ can be understood to be either Apollo or ‘the Muse’, as the disguised Odysseus himself remarks at O.08.488. Long-term, however, Apollo and the Muses are surrogates here for Zeus himself, who at O.13.025 is finally identified as the transcendent source of inspiration for the singing of Demodokos. The figuring of Zeus as such a transcendent source was traditionally considered to be a signature, as it were, of ‘Homer’ himself, as we read in the reference at Pindar Nemean 2.1–3 to the Homēridai, a guild of singers from Chios who claimed, as ‘descendants of Homer’, to be the legitimate transmitters of ‘Homer’ as their poetic ancestor. An example of such a reference is the wording at the very beginning of Pindar Nemean 2.1–3: ὅθεν περ καὶ Ὁμηρίδαι | ῥαπτῶν ἐπέων τὰ πόλλ᾿ ἀοιδοί | ἄρχονται, Διὸς ἐκ προοιμίου ‘(starting) from the point where [hothen] the Homēridai, singers, most of the time [ta polla] begin [arkhesthai] their stitched-together words, from the prelude [prooimion] of Zeus …’.