In the 16th century a completely new set of meanings of air arrived in English: ‘appearance’ or ‘demeanour’. The first known instance comes in Shakespeare’s 1 Henry IV, IV, i: ‘The quality and air of our attempt brooks no division’ (1596). This air was borrowed from French, where it probably represents an earlier, Old French, aire ‘nature, quality’, whose original literal meaning ‘place of origin’ (reflected in another derivative, eyrie) takes it back to Latin ager ‘place, field’, source of English agriculture and related to acre. (The final syllable of English debonair [13] came from Old French aire, incidentally; the phrase de bon aire meant ‘of good disposition’.) The final strand in modern English air comes via the Italian descendant of Latin āēr, aria.
This had absorbed the ‘nature, quality’ meanings of Old French aire, and developed them further to ‘melody’ (perhaps on the model of German weise, which means both ‘way, manner’ and ‘tune’ – its English cognate wise, as in ‘in no wise’, meant ‘song’ from the 11th to the 13th centuries). It seems likely that English air in the sense ‘tune’ is a direct translation of the Italian.
Here again, Shakespeare got in with it first – in A Midsummer Night’s Dream, I, i: ‘Your tongue’s sweet air more tunable than lark to shepherd’s ear’ (1590). (Aria itself became an English word in the 18th century.)