GavestonPS said: "ETA: THE POST THAT FOLLOWS CONCERNS A DIFFERENT SONG, "There Won't Be Trumpets." I misread the original post and several people kindly corrected me. It goes without saying that I regret the error.
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No. It followed soon after Nurse Apple's big Act I monologue. (Forgive me, but I'm too lazy to go and look up what she says, but it's a long and very funny rant about the treatment of her "Cookies".)
In SONDEIM & CO., Sondheim says Lee Remick was getting applause at the end of the monologue, but was struggling with the song. Having both seemed to kill the momentum of the scene, anyway, so they cut the number.
ETA *I* believe--this is me talking now, not Sondheim or anyone else with the production--"There Won't Be Trumpets" was intended as a sort of anti-"I want" song in the R&H (as defined by Lehman Engel) tradition.
Cutting the song has always seemed problematic to me, in theory, since the entire piece ends with (SPOILER) the blowing of a trumpet. But now I can't recall whether they put the song back in the one time I saw the show (at the York Theater in the early 1980s). I knew the song from SIDE BY SIDE..., but didn't know ACW well enough to have missed it."
A production at the Jermyn Street Theatre in London 10 years ago included There Won't Be Trumpets but not There's Always a Woman. Both Fay's monologue and Trumpets worked together as a great character introduction (and were performed excellently by Rosalie Craig). Plus Hapgood was introduced with trumpets, so the song also gave him extra context and helped to keep you questioning.
I've wondered how There's Always a Woman fit into the show. From what I remember of the show following the Cookie Chase, a song like that probably would have destroyed the urgency of that part of the story. And as entertaining as the song is, there wasn't any significant conflict set up between Fay and Cora. It was all indirect and ideological, whereas the fun of the song is the personal rivalry.