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It's Recado Bossa played as a swing.
cesar swing is so many different things : sometimes loosely resembling recado , sometimes practically the same tune but played as a swing.
The song was inspired by one of the drawings of a chimney sweep created by Mary Poppins' screenwriter, Don DaGradi. When asked about the drawing by the Sherman Brothers, DaGradi explained the ancient British folklore attributed to "sweeps" and how shaking hands with one could bring a person good luck. In their 1961 treatment, the Sherman Brothers had already amalgamated many of the P.L. Travers characters in the creation of "Bert". His theme music became "Chim Chim Cher-ee".
In addition to the "standard" version of the song which Bert sings to the children, he sings short snippets of the song to himself at various times, with different verses specific to an unfolding plot element.
The "Mentsch" music group notes some similarities with a Yiddish song written by Itsik Manger named "Vaylu". The song also shares some sonic similarities to the Yiddish folksong, Tumbalalaika.
The opening progression is so strong it's probably been around for as long as string instruments. The Waltz from the film seems to predate the Mobley by three years but who knows.
Thanks all! You're right Denis, pretty close though!
Here's Zoot Sims playing Recado Bossa Nova
Samy Daussat & Bruno Casties - Recado Bossa Nova - Guitar Transcription
Video of the transcription [I think]
Haha wow major third over the G minor at 0:19. Did he mean to do that? Looked pretty deliberate or maybe he was just distracted by some pretty lady walking by on the sidewalk outside the window :-).
I found this somewhere years ago: