Has any other musical genre ever been defined primarily by what it's *not*?

Not to get hung up too much on semantics, but I don't think the "folk" being discussed is a "genre." That aside…

"Classical," "Popular," and "Folk" are three (large scale) musical "spheres" (the word I'd use -- I hope it doesn't sound pretentious) distinguished in Western society. Each is partially defined by what it is not. That is, they are defined against one another.

Yes, there is no air-tight definition of any—which is why I object to using "folk" as a category of scholarly analysis.

Roughly speaking, "Classical" became distinguished as the music of elites in the 19th century, in distinction from vernacular or "popular" music.

German-language discussants considered "Volk" to be that which was particular to people of a "nation" (not a nation-state, but an ethnically-defined nation).

English-language discussants came along and said, "Ah, music of a nation! We like that way of circumscribing music like that! But we have to get the true spirit of the Volk. That is not found in new-fangled and cosmopolitan [under the influence of interactions with others] Classical/art music. And we've got this popular music too, but look: it's been corrupted by industry and media."

So, "Folk" becomes a sort of "pure music of an ethnic group"—"pure" because it is uninfluenced by others or by education or by media or by industry. This generates an aesthetic of authenticity, and a proportional valuing of older as better and regional as better, etc.

The idea of "Folk" goes on to influence the thinking of elites (ironically?) in many places, and these places come up with somewhat different connotations. I've mentioned before that from my observations in India, *regionality* (and being sung in "regional" languages) is one of the very important characteristics of what people think of as "folk"—more so than oral transmission, since "Classical" music of India is all oral transmission, and old! This leads to mix up such as Anglophones in the West maybe thinking Classical Indian music of some sort is "folk"—it's being perceived as "of a people (essentially)," old, and oral leads to this judgement that an Indian would never make.

So what the English-speaking discussants first developed as a "Folk" concept—their reactionary split off from popular music that emphasized purity and race—wasn't scientific because 1) It wasn't universal and 2) the operating principle of definition was really subjective -- something to do with perceived authenticity. They tried to make it more scientific (C. Sharp et. al.) by attempting to quantify authenticity with the idea of "folk process" -- and idea that brings together both of the values of orality and age. But much of music in the world that wouldn't be described as "folk" undergoes that "process". It's just subjective again, a matter of degree-- a degree that satisfies the seeker of the folk aesthetic.

So something is "folk" when it satisfies, to sufficient degree, the aesthetic of the person holding a concept of folk. In the Anglosphere that concept includes the ideas that great age, orality, simplicity, corporality (bare feet?), etc etc (among a host of connotations) are indicators of authenticity. Because authenticity is the ultimate goal, however, the indicators of authenticity have been able to shift with times. A performance might be "authentic," not because it is old or orally produced, but because the performer "kept it simple" or "sang in an untrained voice."

On a more basic level of musical style though, something is "folk" for Westerners if it signifies any combination of (perceived) "untrained/unrefined," "ethnically-specific," "old," "provincial," and "tradition."