ABANDONED: “Bad training” probably helped Scheinman become an adept composer and improviser in multiple genres.

When I quote to her from an on-line review that one of her pieces sounds “equally Polish, Spanish, and Italian,” she says, “I always find that kind of description distressing. I understand how people hear it that way, but it’s not intentional. If you’re a saxophone player, the gigs you get are generally jazz. As a violin player, I’ve learned — just to make a living — klezmer and all this other kind of stuff, and I’m not an expert in any of it, really. It’s a sort of total gypsy existence of crossing borders constantly and learning a little bit of this and a little bit of that, and I’m sure that comes out in the music. But there was a certain point where I stopped having to take every single gig, and I tried to get away from learning so much in crash courses in different kinds of folk music.”

That said, “a little bit of this and a little bit of that” (geez, echo of Fiddler on the Roof?) has found its way into all of her albums. The 2002 The Rabbi’s Lover was commissioned by John Zorn for the “Radical Jewish Culture” series of his Tzadik label, intended, she says on her Web site, “to express something Jewish either about me, or music, or something.” (Her non-Jewish grandfather, Telford Taylor, was lead prosecutor in the Nürnburg trials.) Her most recent album, though, last year’s 12 Songs (Cryptogramophone), most consistently delves into “the music that I come from, which is basically American music.” She says she’s fully aware that “there’s all these different types of American music and it’s an immigrant land. But it is different from learning Swiss music or all these things I’ve learned. I don’t intend to mix things. If it happens, it’s just from what I’ve absorbed unintentionally. I don’t have any agenda of creating some kind of new world music.”

To be sure, there are international folk flavors on 12 Songs. “Suza” is based on the Brazilian choriño rhythms she jammed on backstage with percussionists on her last tour of Brazil with Cantuária. The fetching “Little Calypso” is something she says she would imagine hearing “in a border cantina circa 1900.” But American folk melodies — Appalachia, Celtic fiddle, hymn tunes, Stephen Foster, parlor-music ditties — lace the entire album. And there’s also a particular American-ness about all Scheinman’s albums in her choice of physical space as subject matter for more-abstract pieces — as in 12 Songs’ “The Buoy Song,” an attempt “to write a song that was totally devoid of human feeling, just a landscape, a buoy in the mist. I read somewhere that past a certain point of magnification, all portraits become landscapes. That’s what going on here.” A self-portrait of the artist as isolated in a vast American landscape?

The focus on melody and song structure, and songs that seem to cry out for lyrics, is part of the directness that Scheinman hopes sets 12 Songs apart from her previous work. Although she doesn’t sing any of the songs — there are no lyrics for them — she has been singing other songs in her regular gig at Barbe’s in Brooklyn. The album is part of her exploration of the relationship between “a lyric and a melody on the violin.” She was further inspired by the sessions with Williams. “I think she’s one of the great American songwriters — take out the lyrics and they’d still be fucking incredible.”

On 12 Songs, guitar (Frisell), trumpet, clarinet, drums, bass, and various keyboards including claviola and accordion are deployed orchestrally, the voicings of each melodic line as important as any solo. Those lines overlap, emerge, and recede. Only a couple of Scheinman solo improvisations come readily to mind, variations on the sweet-toned melody from “The Frog Threw His Head Back and Laughed,” and the varied textures in the two-beat march “Moe Hawk,” with her low-toned double stops thickening before the line thins out into silvery runs. But she’s everywhere else, backing solos with melodies, rhythmic fillips, piquant harmonies. On disc, at least, her talent is for fitting in rather than standing out. Her own solo projects are often written with others in mind: Shalagaster (Tzadik, 2004) for the pianist Myra Melford and her upcoming Newport dates and a succeeding gig at New York’s Jazz Standard for pianist Jason Moran. In her press notes for 12 Songs she jokes, “One night I woke in a sweat worrying that I had actually forgotten to play.”