Rules for forming adjectives in -αιος from nouns ending in -ος, -α, -η and the resulting accentual changes (perispasmos/proparoxytone/paroxytone), with examples and exceptions (e.g. ἀρχαῖος, Πειραιεύς, Ἐπιπολαῖος); also notes on feminine-derived forms and initial i- forms.
Πᾶν εἰς 'ος' καθαρὸν ὑπὲρ δύο συλλαβὰς τῇ 'αι' διφθόγγῳ παραληγόμενον, εἰ ἀπὸ ὀξυτονουμένου θηλυκοῦ γένοιτο, προπερισπᾶται, ἀρχή ἀρχαῖος, κορυφή κορυφαῖος, πηγή πηγαῖος, ἀμοιβή ἀμοιβαῖος, σπουδή σπουδαῖος, τροπή τροπαῖος, εὐνή εὐναῖος. ἔστι δὲ καὶ Εὐναῖος ποταμὸς Καρίας. τὸ μέντοι ἐπιπόλαιος οὐκ ἀπὸ τοῦ ἐπιπολῆς παρῆκται, ἀλλὰ παρὰ τὸ ἐπιπολάζω· πλὴν οἱ πλείους αὐτὸ περισπῶσιν. Ἐπιπολαῖος δὲ τὸ ἐθνικόν. * Τὰ ἀπὸ τῶν εἰς 'α' ληγόντων μακρᾷ παραληγομένων παραγόμενα διὰ τοῦ 'αιος' προπερισπᾶσθαι θέλει, μοῦσα Μουσαῖος, γενναῖος, Πισαῖος, ἑρσαῖος, Ἀσαῖος· παρὰ γὰρ τὸ αἶσα γέγονεν καὶ ἐξέστη πρότερον τὸ 'ι'. οὐ μάχεται τὸ Ἥραιος προπαροξυνόμενον· πρὸς γὰρ ἀντιδιαστολὴν τοῦ προπερισπωμένου ἐπιθετικοῦ ἐγένετο· ὅθεν Ἥραιον πόλις Θρᾴκης. Ἡρόδοτος τετάρτῃ (c. 90), τὸ δὲ τεῖχος Ἡραῖον. τὸ μέντοι Πείραιος προπαροξυνόμενον παρὰ ῥῆμα τὸ πειρῶ ὡς παρὰ τὸ φιλῶ Φίλαιος καὶ παρὰ τὸ βέβηκα βέβαιος. τὸ δὲ Πειραιός—οὕτως ἐκαλεῖτο ὁ λιμὴν τῆς Ἀττικῆς καὶ τὸ ἐθνικὸν ὁμοφώνως. ὕστερον δὲ Πειραιεύς. ἔστι δὲ καὶ τῆς Κορινθίας λιμήν—πρὸς ἀντιδιαστολὴν ὀξύνεται. * Καὶ καθόλου τὰ ἀπὸ τῶν εἰς 'α' ἢ 'η' ἢ εἰς 'ος' παραγόμενα προπερισπᾶται, ὥρα ὡραῖος, θύρα θυραῖος, Πέλλα Πελλαῖος, Γάζα Γαζαῖος, ὕλη ὑλαῖος, πύλη πυλαῖος, Δίκτη Δικταῖος, Οἴτη Οἰταῖος. Σοφοκλῆς «ἐγὼ μὲν ὦ γένεθλον Οἰταίου πατρός» (Phil. 453), ἀγέλη ἀγελαῖος. τὸ δὲ Ἀγέλαιος κύριον προπαροξύνεται, δρόμος δρομαῖος, χέρσος χερσαῖος, ἤπειρος ἠπειραῖος, νησαῖος, Ἄμυμνος Ἀμυμναῖος, Ἄμυρος Ἀμυραῖος, Ἄπρος Ἀπραῖος. τὸ δὲ φύλαιος καὶ δείλαιος προπαροξύνονται. καὶ τὸ δίκαιος δὲ καὶ βίαιος προπαροξύνονται, ἐπειδὴ τὰ ἀπὸ θηλυκῶν παραγόμενα καὶ τὸ 'ι' ἐν τῇ ἀρχῇ ἔχοντα προπαροξύνεσθαι θέλει. Τὰ παρώνυμα παρ´ οὐδετέρων γινόμενα ὀξύνεται, γῆρας γηραιός, γέρας γεραιός, κράτος κραταιός, εἰ μὴ ἔχοι κτητικὴν σημασίαν ὡς
Every pure noun in -ος of more than two syllables, with the diphthong αι in the penult, if it is formed from an oxytone feminine, is accented with a circumflex on the antepenult: ἀρχή ἀρχαῖος, κορυφή κορυφαῖος, πηγή πηγαῖος, ἀμοιβή ἀμοιβαῖος, σπουδή σπουδαῖος, τροπή τροπαῖος, εὐνή εὐναῖος. There is also a river of Caria called Εὐναῖος. The word ἐπιπόλαιος, however, is not derived from ἐπιπολῆς, but from ἐπιπολάζω; nevertheless most people accent it with a circumflex. Ἐπιπολαῖος is the ethnic adjective.
Those formed from words ending in -α, with a long penult, by means of -αιος, ought to be accented with a circumflex on the antepenult: μοῦσα Μουσαῖος, γενναῖος, Πισαῖος, ἑρσαῖος, Ἀσαῖος; for it has come from αἶσα, and earlier the ι has dropped out. The proparoxytone Ἥραιος is not in conflict with this; for it arose by way of contrast with the proparoxytone adjectival form; hence Ἥραιον, a city of Thrace. Herodotus in the fourth book (c. 90): τὸ δὲ τεῖχος Ἡραῖον. But Πείραιος, which is proparoxytone, is from the verb πειρῶ, just as from φιλῶ comes Φίλαιος and from βέβηκα comes βέβαιος. And Πειραιός—thus the harbor of Attica was called, and the ethnic adjective likewise with the same sound. Later it was Πειραιεύς. There is also a harbor of Corinthia—and by way of contrast it is accented with an acute.
And in general, those formed from words ending in -α or -η or -ος are accented with a circumflex on the antepenult: ὥρα ὡραῖος, θύρα θυραῖος, Πέλλα Πελλαῖος, Γάζα Γαζαῖος, ὕλη ὑλαῖος, πύλη πυλαῖος, Δίκτη Δικταῖος, Οἴτη Οἰταῖος. Sophocles: «ἐγὼ μὲν ὦ γένεθλον Οἰταίου πατρός» (Phil. 453), ἀγέλη ἀγελαῖος. But Ἀγέλαιος as a proper name is proparoxytone; δρόμος δρομαῖος, χέρσος χερσαῖος, ἤπειρος ἠπειραῖος, νησαῖος, Ἄμυμνος Ἀμυμναῖος, Ἄμυρος Ἀμυραῖος, Ἄπρος Ἀπραῖος. But φύλαιος and δείλαιος are proparoxytone. And δίκαιος and βίαιος too are proparoxytone, because those derived from feminines and having ι at the beginning tend to be proparoxytone. Paronyms formed from neuters are oxytone: γῆρας γηραιός, γέρας γεραιός, κράτος κραταιός, unless it has a possessive sense, as…