Beyond some of the more obvious ones like Blue Moon, Moonlight in Vermont (Bob had that one at #76), and probably Sinatra's most popular moon song Bart Howard's Fly Me To The Moon, Frank Sinatra did quite a few songs with a lunar theme.
Frank does a fabulous version of Glenn Miller and Mitchell Parish's Moonlight Serenade (Bob also had this at #91):
Jimmy Van-Heusen and Johnny Burke wrote Polkadots and Moonbeams and Moonlight Becomes You. Polkadots and Moonbeams was one of Frank's first big hits with the Tommy Dorsey Orchestra.
This is also a Van-Heusen Burke song:
And here is Frank Sinatra singing and Billy May orchestrating Moonlight on the Ganges:
Pundette: Gershwin's Foggy Day and They Can't Take That Away From Me (both Pundette and Mark went with Gershwin last week and she had Billy May's orchestrated Baubles, Bangles and Beads at #48 on her list and Mark has it as his song of the century this week) and Pundette had Old Devil Moon at #61
Bob Belvedere: Sinatra's Best 25 through 23 At Long Last Love, That's Life, Angel Eyes, and My Funny Valentine (a Rogers and Hart song like my last week's pick Blue Moon) and Honorable Mentions 6 (which has Roger and Hart's I Didn't Know What Time It Was) and Bob had Old Devil Moon at #93
Mark Steyn: Baubles, Bangles and Beads, Arlen and Gershwin's The Gal That Got Away, Young at Heart and I've Got the World on a String and Mark had East of the Sun (and West of the Moon) at #19
EBL: Blue Moon, The Nearness of You, More Sinatra Summer, Things We Did Last Summer, and Sinatra at the Beach
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It’s a swingin’ world.