"Pediatric Reflection" by Ogden Nash
Many an infant that screams like a calliope
Could be soothed by a little attention to its diope.
What does "diope" mean here?
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Ogden Nash was fond of the humorous effect created by employing a rhyme that only works if you distort the pronunciation of one of the words, an effect that he signalled by a deliberate mis-spelling. Here are three examples, with glosses for the mis-spelled words:
Better yet, if called by a panther,
Ogden Nash (1941). ‘The Panther’. In The Face is Familiar, p. 64. New York: Garden City.
A girl who is bespectacled,
She may not get her nectacled.†
But safety pins and bassinets
Await the girl who fascinets.‡
† neck tickled ‡ fascinates.
A bit of talcum
Is always walcum.†
‘Reflection on Babies’. In Nash, p. 323.
So in the couplet quoted in the question, “diope” is a mis-spelling of “diaper” that makes it rhyme with “calliope”.
A calliope is a steam-driven organ, named after the ancient Greek muse Calliope, and notorious for its volume: Wikipedia says, “some small calliopes are audible for miles”. The word should be pronounced /kəˈlaɪəpi/ with four syllables, not /ˈkæliˌoʊp/ with three, for Nash’s joke to work.
1874 poster: “Calliope, the wonderful operonicon or steam car of the muses”. Library of Congress, digital ID pga.01335.