Bartok

One of the reasons Bartók was and is so influential to composers is that he developed his own theoretical approach to composition called polymodal chromaticism. Analogous to Schoenberg's 12 tone method, Bartók created a system in which he "juxtaposes different pitch collections, based on a common pitch, to derive extensions from the basic diatonic mode" (Braddell). In his Harvard Lectures, Bartók shows how he combined the Phrygian and Lydian modes into one mode containing all 12 tones. Unlike the chordal chromaticism of the late nineteenth century, Bartók developed what he called modal chromaticism. Bartók explains that he was inspired to create this system because he discovered the basis of it within the folk melodies of Eastern Europe.
Bartók was also obsessed with the idea of symmetry, and he was fascinated with the golden mean. An example of just how meticulous Bartók was when it came to symmetry is that the total number of beats in the fourth string quartet is equal to 2584. This is the 18th number in the Fibonacci series, a rather remarkable feat! So Bartók was interested in symmetry in the aesthetic sense of the word, (symmetry that gives the feeling of proportionality and balance but is not perfectly symmetrical) as well as the mathematical sense. One of the reasons the whole tone scale, augmented chords, and diminished seventh chords, were favored by Bartók, is because of their inherent symmetries.
Bartók's usage of canons in his fourth string quartet could be seen as another example of his inclination for symmetry. One of the first appearances of symmetry in all of music is with the canon. Songs such as frère Jacque and row your boat are simple canons or rounds. Canons are symmetrical because they derive from the same material, one original theme that is repeated and varied depending on the composition. There are many types of cannons. In the quartet, Bartók uses free canons and free imitation, where there is less restriction on individual parts. He also uses the mirror canon, in which the subsequent voices imitate the initial voice in inversion.