Serialism in the context of "Permutation (music)"

Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about the early 20th century to the present day, where a hierarchy of harmonies focusing on a single, central triad is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized European classical music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".

The term is also occasionally used to describe music that is neither tonal nor serial, especially the pre-twelve-tone music of the Second Viennese School, principally Alban Berg, Arnold Schoenberg, and Anton Webern. However, "as a categorical label, 'atonal' generally means only that the piece is in the Western tradition and is not 'tonal'", although there are longer periods, e.g., medieval, renaissance, and modern modal music to which this definition does not apply. "Serialism arose partly as a means of organizing more coherently the relations used in the pre-serial 'free atonal' music. ... Thus, many useful and crucial insights about even strictly serial music depend only on such basic atonal theory".

The Second Viennese School (German: Zweite Wiener Schule, Neue Wiener Schule) was a group of composers that comprised Arnold Schoenberg and his pupils, particularly Alban Berg and Anton Webern, and close associates in early 20th-century Vienna. Their music was initially characterized by late-Romantic expanded tonality and later, a totally chromatic expressionism without a firm tonal centre, often referred to as atonality; and later still, Schoenberg's serial twelve-tone technique. Using this technique when composing, Schoenberg employed all 12 tones present in Western music's chromatic scale when forming a melody, this melody being the "prime series". This method would later be enhanced in compositions by Schoenberg and his followers through permutations such as "retrogrades", "inversions", "transformations" etc. Later, composers such as Pierre Boulez drew influence from Schoenberg's technique in furthering serialization; serializing not only pitch but rhythm, articulation, and dynamics as well. Theodor Adorno said that the twelve-tone method, when it had evolved into maturity, was a "veritable message in a bottle", addressed to an unknown and uncertain future. Though this common development took place, it neither followed a common time-line nor a cooperative path. Likewise, it was not a direct result of Schoenberg's teaching—which, as his various published textbooks demonstrate, was highly traditional and conservative. Schoenberg's textbooks also reveal that the Second Viennese School spawned not from the development of his serial method, but rather from the influence of his creative example.

In music, a hexachord (also hexachordon) is a six-note series, as exhibited in a scale (hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial theory. The word is taken from the Greek: ἑξάχορδος, compounded from ἕξ (hex, six) and χορδή (chordē, string [of the lyre], whence "note"), and was also the term used in music theory up to the 18th century for the interval of a sixth ("hexachord major" being the major sixth and "hexachord minor" the minor sixth).

Karlheinz Stockhausen (German: [kaʁlˈhaɪnts ˈʃtɔkhaʊzn̩] ; 22 August 1928 – 5 December 2007) was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. He is known for his groundbreaking work in electronic music, having been called the "father of electronic music", for introducing controlled chance (aleatory techniques) into serial composition, and for musical spatialization.

Stockhausen was educated at the Hochschule für Musik Köln and the University of Cologne, later studying with Olivier Messiaen in Paris and with Werner Meyer-Eppler at the University of Bonn. As one of the leading figures of the Darmstadt School, his compositions and theories were and remain widely influential, not only on composers of art music, but also on jazz and popular music. His works, composed over a period of nearly sixty years, eschew traditional forms. In addition to electronic music – both with and without live performers – they range from miniatures for musical boxes through works for solo instruments, songs, chamber music, choral and orchestral music, to a cycle of seven full-length operas. His theoretical and other writings comprise ten large volumes. He received numerous prizes and distinctions for his compositions, recordings, and for the scores produced by his publishing company.

20th-century classical music is Western art music that was written between 1901 and 2000, inclusive. Musical style diverged during the 20th century as it never had previously, so this century was without a dominant style. Modernism, impressionism, and post-romanticism can all be traced to the decades before the turn of the 20th century, but can be included because they evolved beyond the musical boundaries of the 19th-century styles that were part of the earlier common practice period. Neoclassicism and expressionism came mostly after 1900. Minimalism started later in the century and can be seen as a change from the modern to postmodern era, although some date postmodernism from as early as about 1930. Aleatory, atonality, serialism, musique concrète, and electronic music were all developed during the century. Jazz and ethnic folk music became important influences on many composers during this century.

Free improvisation (also known as free form music or free music) is a form of improvised music centered on a commitment to non-idiomatic musical expression. It is inspired by free jazz, serialism and indeterminacy, and is characterized by a general rejection of formal music theory and tonality, instead following the intuition of its performers. The term can refer to both a technique—employed by any musician in any genre—and as a recognizable genre of experimental music in its own right.

Free improvisation, as a genre of music, developed primarily in the U.K. as well as the U.S. and Europe in the mid to late 1960s, largely as an outgrowth of free jazz and contemporary classical music. Exponents of free improvised music include saxophonists Evan Parker, Anthony Braxton, Peter Brötzmann, and John Zorn, composer Pauline Oliveros, trombonist George E. Lewis, guitarists Derek Bailey, Henry Kaiser and Fred Frith, bassists Damon Smith and Jair-Rohm Parker Wells and the improvising groups Spontaneous Music Ensemble and AMM.

George Henry Crumb Jr. (24 October 1929 – 6 February 2022) was an American composer of avant-garde contemporary classical music. Early in his life he rejected the widespread modernist usage of serialism, developing a highly personal musical language which "range[s] in mood from peaceful to nightmarish". Crumb's compositions are known for pushing the limits of technical prowess by way of frequent use of extended techniques. The unusual timbres he employs evoke a surrealist atmosphere which portray emotions of considerable intensity with vast and sometimes haunting soundscapes. His few large-scale works include Echoes of Time and the River (1967), which won the 1968 Pulitzer Prize for Music, and Star-Child (1977), which won the 2001 Grammy Award for Best Contemporary Classical Composition; however, his output consists of mostly music for chamber ensembles or solo instrumentalists. Among his best known compositions are Black Angels (1970), a striking commentary on the Vietnam War for electric string quartet; Ancient Voices of Children (1970) for a mixed chamber ensemble; and Vox Balaenae (1971), a musical evocation of the humpback whale, for electric flute, electric cello, and amplified piano.

Born to a musical family, Crumb was acquainted with classical music at an early age and his affinity for Classical and Romantic composers in particular would stay throughout his life. He was especially influenced by composers such as Mahler, Debussy and Bartók; Crumb wrote his four-volume piano set Makrokosmos (1972–1979) in response to Bartók's earlier piano set Mikrokosmos. His compositions often contain musical quotations from wide range of composers including Bach, Chopin, Schubert, Strauss, and the jazz pianist and composer Thelonious Monk. The use of pastiche is also found in his music, as is text by Federico García Lorca, whose poetry Crumb set eleven times. Elements of theatricality appear in numerous compositions, inspiring choreographies from contemporary dance groups. To convey his unorthodox and complex musical style, Crumb's musical scores are facsimile manuscripts, using special notation "distinguished by astonishing clarity, precision and elegance, and by arresting graphic symbols in which staves are bent into arches, circles and other pictorial devices." Among his students were the composers Osvaldo Golijov, Jennifer Higdon, Christopher Rouse and Melinda Wagner.

Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, mathematician, and teacher. He was a Pulitzer Prize and MacArthur Fellowship recipient, recognized for his serial and electronic music.

Louis Joseph Andriessen (Dutch: [luˈʋi ˈɑndrisə(n)]; 6 June 1939 – 1 July 2021) was a Dutch composer, pianist and academic teacher. Considered the most influential Dutch composer of his generation, he was a central proponent of The Hague school of composition. Although his music was initially dominated by neoclassicism and serialism, his style gradually shifted to a synthesis of American minimalism, big band jazz and the expressionism of Igor Stravinsky.

Born in Utrecht into a musical family, Andriessen studied with his father, the composer Hendrik Andriessen as well as composers Kees van Baaren and Luciano Berio. Andriessen taught at the Royal Conservatory of The Hague from 1974 to 2012, influencing notable composers. His opera La Commedia, based on Dante's Divine Comedy, won the 2011 Grawemeyer Award for Music Composition and was selected in 2019 by critics at The Guardian as one of the most outstanding compositions of the 21st century.

Henryk Mikołaj Górecki (/ɡəˈrɛtski/ gə-RET-skee; Polish: [ˈxɛnrɨk miˈkɔwaj ɡuˈrɛt͡skʲi] ; 6 December 1933 – 12 November 2010) was a Polish composer of contemporary classical music. According to critic Alex Ross, no recent classical composer has had as much commercial success as Górecki. He became a leading figure of the Polish avant-garde during the post-Stalin cultural thaw. His Anton Webern-influenced serialist works of the 1950s and 1960s were characterized by adherence to dissonant modernism and influenced by Luigi Nono, Karlheinz Stockhausen, Krzysztof Penderecki and Kazimierz Serocki. He continued in this direction throughout the 1960s, but by the mid-1970s had changed to a less complex sacred minimalist sound, exemplified by the transitional Symphony No. 2 and the Symphony No. 3 (Symphony of Sorrowful Songs). This later style developed through several other distinct phases, from such works as his 1979 Beatus Vir, to the 1981 choral hymn Miserere, the 1993 Kleines Requiem für eine Polka and his requiem Good Night.

Górecki was largely unknown outside Poland until the late 1980s. In 1992, 15 years after it was composed, a recording of his Symphony of Sorrowful Songs with soprano Dawn Upshaw and conductor David Zinman became a worldwide commercial and critical success, selling more than a million copies and vastly exceeding the typical lifetime sales of a recording of symphonic music by a 20th-century composer. Commenting on its popularity, Górecki said, "Perhaps people find something they need in this piece of music ... somehow I hit the right note, something they were missing. Something somewhere had been lost to them. I feel that I instinctively knew what they needed." This popular acclaim did not generate wide interest in Górecki's other works, and he pointedly resisted the temptation to repeat earlier success, or compose for commercial reward.

René Leibowitz (/ˈliːbəwɪts/; French: [ʁəne lɛbɔwits]; 17 February 1913 – 29 August 1972) was a Polish and French composer, conductor, music theorist and teacher. He was historically significant in promoting the music of the Second Viennese School in Paris after the Second World War, and teaching a new generation of serialist composers.

Leibowitz remained firmly committed to the musical aesthetic of Arnold Schoenberg, and was to some extent sidelined among the French avant-garde in the 1950s, when, under the influence of Leibowitz's former student, Pierre Boulez and others, the music of Schoenberg's pupil Anton Webern was adopted as the orthodox model by younger composers.

Humphrey Searle (26 August 1915 – 12 May 1982) was an English composer and writer on music. His music combines aspects of late Romanticism and modernist serialism, particularly reminiscent of his primary influences, Franz Liszt, Arnold Schoenberg and Anton Webern, who was briefly his teacher. As a writer on music, Searle published texts on numerous topics; he was an authority on the music of Franz Liszt, and created the initial cataloguing system for his works.