'balletically' originates from Italian, specifically the word 'balletto', where 'ballare' meant 'to dance'.

'balletically' changed from Italian 'balletto' into French 'ballet', which entered English as 'ballet'; from that root the adjective 'balletic' and then the adverb 'balletically' were formed.

Initially, it referred to a 'little dance' or the dance itself, but over time it evolved into the modern sense 'in the manner of ballet' (i.e., gracefully).