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The Ancient Greek word ἡμιόλιος means literally "one and a half", referring to the ratio 3:2 and the interval of a perfect fifth in music. I wonder how this word is composed of: is it ἡμι- (half) + ὅλος (whole) or ἡμι- (half) + ὀλίγος (few) ? It seems ἡμι + ὅλος in its sense but why then is it ἡμιόλιος and not ἡμιόλος?

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It is indeed ἡμι- 'half' + ὅλος 'whole', but there's a third component, -ιος, which forms adjectives, usually but not exclusively out of nouns, indicating the possession of a characteristic of the prototype word (as in e.g. ἀρχή 'beginning' → ἀρχαῖος 'primeval', θάλασσα 'sea' → θαλάσσιος 'maritime', Λέσβος 'Lesbos' → Λέσβιος 'Lesbian', &c.), as ἡμιόλιος is an adjective. If you see ἡμιόλιος being used as a noun, it's presumably agreeing with an elided λόγος, meaning, in this context, 'ratio'.

The connection with ὀλίγος doesn't make much semantic sense, and Greek isn't prone to losing intervocalic γ before a back vowel.

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