We're still catching up to Bartok. To play Bartok, either on the piano or on strings, is to enter a completely new universe of harmony - no rule is broken, but every rule is bent so far that rules no longer seem like rules.
Richard Feynman always liked to say that there are two types of genius: one is the type which seems like us, if only we were hundreds of times smarter than we are. If you look at the scores of some early 20th century composers like Mahler, or Schoenberg, or Ives, or Prokofiev, or Debussy, or Sibelius, or Shostakovich, you can almost hear their thought processes in your head - 'what would happen if I turn this music upside down?', 'what would it sound like if I combined this chord with that one?', 'what would happen if I included a folk tune here?', 'what would happen if I compress the form to its minimal function?'. Then there's the second type, the rarer type: like Bartok, or Janacek, or Stravinsky, or Messiaen, whose ideas are so fecund and so far beyond what we conceive that they seem to thinking beyond any plane we can understand. There is no way we will ever understand how they came up with the ideas they had. They simply hear sounds in their heads that change the curvature of the earth. In the music of composers like this, there are more things in the heavens and the earth than were dreamt of by composers who came by their modernism more self-consciously.
Then again, if you spent decades of your life obsessively collecting folk music, almost to the detriment of your composing, you'd have a completely unique perspective too. To an exponential level beyond any composer before him (and nearly any after), Bartok simply seems to be operating in a different frame of reference. There's no mistaking his music for anything but tonality (with a very few exceptions...), but the normal rules of tonal harmony do not apply here. In order to do what Bartok does, you need vast experience with alternate scales, alternate tunings, alternate rhythms, exactly the sort of training that hardly any classical musician gets. Generally speaking, if you play outside the realm of classical music, you're still playing jazz and rock music, which still operates in the world of Western harmony and rhythm. Bartok, on the other hand, was learning the music of the Balkans and their surrounding environes, a musical landscape dominated by the world of the Roma (or Gypsies as they're more commonly known). The Romani people originally hailed from Northern India, and it's impossible not to hear vague echoes of Indian ragas. Along the way to Eastern Europe, they had to pass through Arab countries and Caliphates, through the Ottoman Empire, to live for many centuries next to Jews, to migrate through Northern Africa and Spain. The larger Western World, with all its alternate perspectives, is bound up in the Romani experience, and therefore in its music.
Bookending the story of German music was Bach at the beginning, who coined the very harmonic language which we generally still use today. At the end comes the Second Viennese School: Schoenberg, Berg, Webern, who took Bach's monarchical organization of twelve notes and dethroned them so that no note means more to the music than any other.
In the shadow of Schoenberg's black hole, many musicians tried to restore light: the aforementioned four of Stravinsky, Janacek, Messiaen, and Bartok all had their solutions. So did Ives, and Hindemith, and Vaughan Williams, and Prokofiev, and Harry Partch, and Britten, and Miles Davis, and Thelonius Monk, and John Coltrane.
Of all these many names, only two truly made attempts to systematize them (not counting Kodaly, who unfortunately operated in Bartok's shadow). One was Hindemith, who was a kind of professional systematizer among composers. It says something not quite complimentary about the greatness of his music that he was as great a theorist as he was a composer. His music often sounds like schematics to prove his theories.
There is no such schematic in Bartok, the other composer who created a system. For both the piano and the violin, Bartok created a series of pieces for students, every piece increasingly complex than the last, increasingly difficult to play, increasingly impressive in how it suspends the normal rules of how music is composed.
Bach had his Well-Tempered Clavier, probably also written for student use, and which defined music for two-hundred years after its writing, and to this day defines the harmonic rules of music we generally use. But the history of music since Schoenberg has yet to be written. Perhaps it will come to be seen that with Bartok's 44 Duets for Violins, and even moreso his Mikrokosmos for piano, music begins anew even more than it does with Stravinsky's Rite of Spring. In the case of the violin pieces, 44 duets, none more than two-and-a-half minutes long, and some are over in thirty seconds. You could learn to play violin simply by playing these pieces, which I'm sure was the point of writing them. But more importantly, if you learned music theory from these pieces, you would have a completely different and more nuanced conception of harmony and counterpoint than you would if you learned it from Bach. At this new beginning for music, no longer is tonality a monarchy with the tonic as king and the dominant note as his viceroy. In Bartok, and to a lesser extent in a number of his peers, tonality becomes a democracy, with an infinitely greater number of harmonic permutations possible to contribute in every series of chords.