This is a nice instance of etymology taking as a starting point an inflected form: ἄχη is the Nom.Acc. plural of the neuter noun ἄχος, with contraction. Such forms are as a rule not contracted in Homer (ἄχεα), but that was not an obstacle for Greek scholars, even though μάχη has an [ē] which does not result from a contraction. The fact that a singular feminine (μάχη) is supposed to be derived from a neuter plural was not a problem either, that is, the morphological type was not a criterion. A variant of this etymology derives a feminine *ἄχη from ἄχομαι “to be grieved", and then μάχη. The etymology complies with the analogical principle of Alexandrian grammarians, and a parallel is adduced, that of ἴα / μία, feminine of "one", after which the adjunction of the initial [m] is justified. Semantically, "battle, fight" and "sorrow" can be associated as cause and consequence (which also justifies the plural ἄχη for a battle causes many sorrows). This explanation became standard and is repeated in Byzantine Etymologica, with different pairs exemplifying the adjunction of [m] (see Parallels)