The "New Thing" movement in jazz, for which Ayler was among the foremost exponents, can perhaps best be seen as the attempt to "jazzify" the ideas of Arnold Schoenberg, without whose example it would never in a million years have come into being. Unfortunately, there was nothing about the music of Schoenberg and his followers that was in the least bit improvisatory. Whether in its early "free atonality" phase or in its later "serial" one, the music of Schoenberg's school seems too intellectual, too complex, and too highly structured for jazz to have ever borrowed its ideas in anything more than a superficial way. In other words, improvised jazz, though perhaps more structured than the popular mind would care to acknowledge, has at the end of the day an unpremeditated immediacy that seems completely foreign to Schoenberg's ideas, and therefore attempts in jazz to rival its impact almost seem destined to failure.
All that said, if I had to pick just one recording from the New Thing-period of jazz history, Albert Ayler's "Live in Greenwhich Village" would probably be it. Unlike fellow New-Thingers -- such as Cecil Taylor, for instance, who tends to be overcerebral and borrows so much from post-Bartok classical music; and John Coltrane, whose atonal forays seem more like a regression, rather than a progression, after his earlier modal work; and Ornette Coleman who, in certain ways, was really just a sub-par bebop musician -- Ayler just seems to have paradoxically had the right sort of "primitivistic" grounding (in early New Orleans jazz and the Blues) which, along with a completely unaffected sincerity, helps to pull this sort of music off. Why the least cerebral amongst all the musicians named should tend to be the most successful at this specific style of music is a major mystery. Perhaps because his understanding of atonality was the most superficial?
Whichever way you slice it, Ayler's spin on the New Thing is great, essential, and unpretentious listening. Almost alone amongst the New Thingers, he miraculously found a way to make Schoenberg work in jazz, and work completely.