after its collapse at the end of World War II. After Schoenberg's death, Igor Stravinsky used the twelve-tone technique. Iannis Xenakis generated pitch sets from mathematical formulae, and also saw the expansion of tonal possibilities as part of a synthesis between the hierarchical principle and the theory of numbers, principles which have dominated music since at least the time of Parmenides. Free atonality The twelve-tone technique was preceded by Schoenberg's freely atonal pieces of 1908 to 1923, which, though free, often have as an "integrative element...a minute intervallic cell" that in addition to expansion may be transformed as with a tone row, and in which individual notes may "function as pivotal elements, to permit overlapping statements of a basic cell or the linking of two or more basic cells". The twelve-tone technique was also preceded by nondodecaphonic serial composition used independently in the works of Alexander Scriabin, Igor Stravinsky, Béla Bartók, Carl Ruggles, and others. "Essentially, Schoenberg and Hauer systematized and defined for their own dodecaphonic purposes a pervasive technical feature of 'modern' musical practice, the ostinato." Composing atonal music Setting out to compose atonal music may seem complicated because of both the vagueness and generality of the term. Additionally George Perle explains that, "the 'free' atonality that preceded dodecaphony precludes by definition the possibility of self-consistent, generally applicable compositional procedures". However, he provides one example as a way to compose atonal pieces, a pre-twelve-tone technique piece by Anton Webern, which rigorously avoids anything that suggests tonality, to choose pitches that do not imply tonality. In other words, reverse the rules of the common practice period so that what was not allowed is required and what was required is not allowed. This is what was done by Charles Seeger in his explanation of dissonant counterpoint, which is a way to write atonal counterpoint. Kostka and Payne list four procedures as operational in the atonal music of Schoenberg, all of which may be taken as negative rules. Avoidance of melodic or harmonic octaves, avoidance of traditional pitch collections such as major or minor triads, avoidance of more than three successive pitches from the same diatonic scale, and use of disjunct melodies (avoidance of conjunct melodies). Further, Perle agrees with Oster and Katz that, "the abandonment of the concept of a root-generator of the individual chord is a radical development that renders futile any attempt at a systematic formulation of chord structure