Editing Serialism (section)

==Theory of twelve-tone serial music==
{{Main|Twelve-tone technique}}
Due to Babbitt's work, in the mid-20th century serialist thought became rooted in set theory and began to use a quasi-mathematical vocabulary for the manipulation of the basic sets. Musical [[Set theory (music)|set theory]] is often used to analyze and compose serial music, and is also sometimes used in tonal and nonserial atonal analysis.{{cn|date=December 2020}}

The basis for serial composition is Schoenberg's twelve-tone technique, where the 12 notes of the chromatic scale are organized into a row. This "basic" row is then used to create permutations, that is, rows derived from the basic set by reordering its elements. The row may be used to produce a set of intervals, or a composer may derive the row from a particular succession of intervals. A row that uses all of the intervals in their ascending form once is an [[Tone row#all-interval row|all-interval row]]. In addition to permutations, the basic row may have some set of notes derived from it, which is used to create a new row. These are ''derived sets''.{{cn|date=December 2020}}

Because there are tonal chord progressions that use all twelve notes, it is possible to create pitch rows with very strong tonal implications, and even to write tonal music using twelve-tone technique. Most tone rows contain subsets that can imply a [[tonic (music)|pitch center]]; a composer can create music centered on one or more of the row's constituent pitches by emphasizing or avoiding these subsets, respectively, as well as through other, more complex compositional devices.{{sfn|Newlin|1974}}{{sfn|Perle|1977}}

To serialize other elements of music, a system quantifying an identifiable element must be created or defined (this is called "[[Parameter|parametrization]]", after the term in mathematics). For example, if duration is serialized, a set of durations must be specified; if [[Timbre|tone colour]] (timbre) is serialized, a set of separate tone colours must be identified; and so on.{{cn|date=December 2020}}

The selected set or sets, their permutations and derived sets form the composer's basic material.{{cn|date=December 2020}}

Composition using twelve-tone serial methods focuses on each appearance of the collection of twelve chromatic notes, called an [[Total chromatic|aggregate]]. (Sets of more or fewer pitches, or of elements other than pitch, may be treated analogously.) One principle operative in some serial compositions is that no element of the aggregate should be reused in the same contrapuntal strand (statement of a series) until all the other members have been used, and each member must appear only in its place in the series. Yet, since most serial compositions have multiple (at least two, sometimes as many as a few dozen) series statements occurring concurrently, interwoven with each other in time, and feature repetitions of some of their pitches, this principle as stated is more a referential abstraction than a description of the concrete reality of a musical work that is termed "serial".{{Citation needed|date=September 2020}}

A series may be divided into subsets, and the members of the aggregate not part of a subset are said to be its ''complement''. A subset is ''self-complementing'' if it contains half of the set and its complement is also a permutation of the original subset. This is most commonly seen with ''hexachords'', six-note segments of a tone row. A hexachord that is self-complementing for a particular permutation is called ''prime combinatorial''. A hexachord that is self-complementing for all the canonic operations—[[Melodic inversion|inversion]], [[Retrograde (music)|retrograde]], and [[retrograde inversion]]—is called ''all-combinatorial''.{{cn|date=December 2020}}