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Rhymes cannot be computed for badinage. Rhyming is a fact about pronunciation, not spelling, and badinage is not recorded in the CMU Pronouncing Dictionary — so there is no phoneme string to match against. AltF Lexicon will not derive one from the letters, because spelling is a poor guide to sound in English: cough, though and through share four letters and rhyme with none of each other.
Two words rhyme when their pronunciations agree on every phoneme from the last stressed vowel to the end of the word. Nothing about spelling is consulted — that is why eight rhymes with freight and straight, and why cough rhymes with none of though, through or bough.
Rhymes cannot be computed for badinage. It has no entry in the CMU Pronouncing Dictionary, so the corpus holds no phoneme string for it and there is nothing to match other words against. The syllable split shown on the word page was derived from spelling, which is good enough to break a word into parts but nowhere near good enough to decide what it rhymes with.
Roughly one entry in four across the corpus carries a real transcription; the rest are compounds, proper nouns, taxonomic names and rare vocabulary that the pronouncing dictionary never covered. A rhyme list guessed from letters would be confidently wrong, so none is printed.
Pronunciations and rhyme keys come from the CMU Pronouncing Dictionary; definitions and commonness come from WordNet and a corpus of everyday English. Full sources and licences.