My little posting on Nico Muhly's "I like the crotch on the idea that …" has elicited a flood of e-mail suggesting that crotch is just a malapropism of some kind — an eggcorn, perhaps — for crux. I've dismissed this proposal, in part because of the preposition on in Muhly's sentence, in part because R. Kelly's song "I like the crotch on you" seemed to me to be an obvious model, and in part because I'm disinclined to jump to simple error as the explanation for examples I find remarkable at first (especially in the writing of people who take some care about their style).
Crotch and crux are indeed phonologically similar and at least vaguely similar semantically, and crux is rarer and more learnèd, so an eggcornish replacement of crux by crotch would not be surprising. In fact, it seems to be attested, at least in the crotch of the matter for the crux of the matter. There are not a great many Google hits for the crotch of the matter, and nearly all of them are obvious plays on words, but the expression does appear in a list of "Farberisms", mangled sayings attributed to University of Delaware computer science professor David Farber ("All the lemmings are coming/going home to roost", "And I take the blunt of it!"). The list is very long, however, and might just be too good to be believed. But I'll concede that crotch might occur as an error for crux.
BUT… crux seems to occur with on only in the context of rock climbing, where the word is defined in a climbers' dictionary as:
Crux – The most crucial, difficult part of the climb.
Otherwise, crux occurs with of.
On the other hand, crotch can occur with on, as body-part names in general can ("Look at the mouth on that alligator!"); that's the use in the R. Kelly song.
So I concluded that Muhly's use was an allusion to R. Kelly's (literal) use, extending the meaning of the word metonymically.
Although I didn't rule out the possibility of error, when I'm faced with an example that seems remarkable to me at first and doesn't fit error types that are familiar to me, I start by treating it as intended and pursue other accounts. I'll look for parallel examples, check the literature, consider possible explanations, consult the source of the example, and so on. (I haven't so far found a way to get in touch with Muhly.) I do this because of the long experience I've had in dealing with things I wouldn't myself say or write; they so often turn out to be variants (acceptable for some people though not for me), or creative uses of language, and not inadvertent errors. Many other linguists have had similar experiences, many times.
People who haven't had such experiences are inclined to dismiss examples they find remarkable as just errors; see some discussion in the latest (the fifth) installment in my series of postings on "the thin line between error and mere variation". So a dozen or so Language Log readers wrote to suggest that Muhly just made a mistake, while I credited him with creative language use. (Josh Kamensky has suggested that Muhly's use was both an allusion to the R. Kelly song and also a deliberate play on the crux of the matter. So even more creative than I gave him credit for. [Added 29 August: Muhly now confirms (both parts of) Kamensky's understanding, adding that he uses crotch specifically "for objects that have no specific physical center".])
[Etymological digression. Several correspondents noted a possible etymological connection between crotch and crux. The OED goes to some length to argue that the two words have different sources: crotch going back to a late Latin word meaning 'crook, hook', crux (and cross) going back, of course, to the Latin word for 'cross'. But the OED notes that crotch and cross eventually became associated with one another. Etymology isn't always so neat.]