Since the birth of the American musical in the 1930's, there have been many shows which demonstrate dichotomies between female characters, mainly those relating to femininity and sexuality. The shows that come to mind first, at least for me, are Anything Goes, Once On This Island, and Avenue Q. In each show, two female characters are portrayed as being vastly different in terms of their behavior and personality traits, and these differences are often highlighted by gaps in socio-economic status, race, and the pitch and range of their singing voices.
I'll start with Anything Goes, the oldest of the three; this analysis is going off what I remember of my experience with the 1962 and 1987 versions. The show it set in the 1930's, and two leading ladies in the show are Reno Sweeney, a night club singer with a shady past, but who now uses her act to evangelize with her troupe of "angels;" and Hope Harcourt, a wealthy young woman engaged to Lord Evelyn Oakleigh. The two women are connected through Billy Crocker, a friend of Reno and Hope's former lover. Reno expresses that she has feelings for Billy, but Billy only sees her as a friend - perhaps because she lacks dainty feminine qualities most men found desirable in that time period. Reno is, in fact, loud, has an attitude, and despite her evangelical activities is the sexual character in the show; whereas Hope is the virginal one, the innocent, feminine one, and the one destined for marriage. The two vocal styles are vastly different as well, Hope's voice is more traditional, almost operatic, and her range is much higher than
Reno's, who still hits some high notes but in a more modern, belting style (bear with me, I'm not a theater major).
These women represent a classic virgin/whore dichotomy, which is best illustrated in two duets: "It's DeLovely," sung by Billy and Hope; and "Let's Misbehave," sung by Reno and Evelyn. Reno is also used as a sexual pawn in order to break up Evelyn and Hope, so Billy can be with Hope. There is another related dichotomy, and that is between ladies who are marriage material (Hope), and women who are just seen as dating prospects or objects of sexual desire (Reno).
Fast forward to Once on This Island, a show that opened in the early 1990's, but it's a little unclear when the story is supposed to take place. Or where, for that matter. My guess would be the 1950's, and I'm fairly certain it takes place on the island of Haiti and the Dominican Republic. What is clear is the racial and economic tension between the two sides of the island: on one side, the people are poor and "black as night." On the other side, the people are rich and have lighter skin, as they descended from the French colonists and their slaves - one song, summarizing the history of the island, mentioned a rich white man who married a pale, blonde woman but had sex with the black peasant girls who served him.
The protagonist, Ti Moune, is a young, black girl on the poor side of the island. She is sweet, innocent, and naive, and longs to travel beyond her village. When Daniel, a rich man from the other side of the island, crashes in her village, she tends to him, keeping him alive, demonstrating that she is also very loving and selfless. When Daniel returns to his side of the island, Ti Moune decides to journey there herself to be with him. The two of them have a relationship, and people gossip about them, implying that she is a slut. While she is sweet and innocent, she is assumed to be inherently sexual by the townspeople, and it is implied that she is, in fact, having sex with Daniel. It is true that intersectionality theory is critical to the full analysis of gender within this show, because the racial differences skew the sexual dichotomies. While pure of heart, Ti Moune is arguably sexualized because of her race, and the virginal woman is Andrea, Daniel's wealthy and fairer-skinned fiancee. Before Ti Moune meets Andrea, Daniel sings to her about how wonderful and fascinating she is - and that "some girls you marry, some you love." Again we see the marriageable vs non-marriageable dichotomy illustrated in this musical. Daniel loves Ti Moune, but he must marry a certain "type" of woman.
Again we see class coming into play. Both Hope and Andrea are both the wealthy, asexual, and marriageable characters in their respective shows.
Finally, there's Avenue Q, the most modern of the three shows. This show is also a special case because of its satirical nature, and the fact that two vastly different female roles are played by the same actress - when the two characters are on stage she'll operate one of them, but do the dialogue for both. It's a little bizarre to see her switch between the two roles so rapidly. The female lead is a sweet kindergarten teacher named Kate Monster. Then there's Lucy the Slut, whose name says it all. While Kate Monster does have sex with Princeton (the male protagonist) once, she still mostly embodies the innocent side of the sexual spectrum, opposite Lucy. Lucy is not only easy, but she's sexually aggressive, whereas Kate is more passive - she's drunk the one time they have sex. Lucy is also kind of a bitch, who deliberately tries to keep Kate and Princeton apart during the second act.
In this we see more modern dichotomies, which may be in a satirical context, but are rooted in real ways in which we label women. There's the bitch/sweetheart dichotomy, which was sort of in Once on This Island to a lesser extent, and basically means that if a woman isn't a perfect angel, she must be pure evil. There is a virgin/whore dichotomy, though Kate embodies a less extreme part of the "pure" spectrum - she's not chaste and asexual, but again, she only has sex once, and she's very passive. We see another dichotomy, and that is where a woman is either extremely emotionally invested in a guy, or she's cold to him and just using him for her own benefit.
There's another reversal in Avenue Q: hair color. In shows such as Chicago and Grease, the more innocent character is typically blonde, whereas the more sexual character is brunette. We even see this in modern pop culture, such as Taylor Swift's "You Belong With Me" video (don't get me started on that one). In Avenue Q, Lucy the Slut is blonde and Kate Monster is brunette. The hair colors of these characters are used to instead highlight (no pun intended) Kate's down-to-Earth personality against Lucy's identity as the male fantasy.
Of course, the shows analyzed above are in no way the "best" examples or the most classic, they really were the first that came to mind and the relevant shows I happen to know best. I almost covered Wonderful Town as well, as the two women being sisters makes it another interesting case, but I figured 3 was a good number. I've only seen the movies for Chicago and Grease, both of which are somewhat different from the original Broadway shows, but those two definitely have virgin/whore, bitch/sweetheart, feminine/not-so-feminine, etc. dichotomies in them. Wicked is another great example, with Glinda being blonde, rich, feminine, desirable, "sweet," and of course Popular, but the complete opposite of Elphaba. If you look hard enough, you can even see the polarization of femininity between Grace and Miss Hannigan in Annie. You don't, however, see these dichotomies in every show, especially shows that focus on male characters and don't have a lot of female characters, such as the Book of Mormon; or shows with more then two leading ladies such as A Chorus Line, or even American Idiot, where there are multiple female archetypes.