music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

, the chromatic hexachord is the

hexachord 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 theor ...

consisting of a consecutive six-

note Note, notes, or NOTE may refer to: Music and entertainment * Musical note, a pitched sound (or a symbol for a sound) in music * ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) shortened version ...

segment of the

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...

. It is the first hexachord as ordered by

Forte number In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in ''The Structure of Atonal Music'' (1973, ). The first number indicates the number of pitch cla ...

, and its

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

is the chromatic hexachord at the

tritone In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...

. For example, zero through five and six through eleven. On C: *C, C, D, D, E, F and *F, G, G, A, A, B. This is the first of the six hexachords identified by Milton Babbitt as all-combinatorial source sets, a "source set" being "a set considered only in terms of the content of its hexachords, and whose combinatorial characteristics are independent of the ordering imposed on this content" . In the larger context of thirty-five source hexachords catalogued by

Donald Martino Donald James Martino (May 16, 1931 – December 8, 2005) was a Pulitzer Prize winning American composer. Biography Born in Plainfield, New Jersey, Martino attended Plainfield High School. He began as a clarinetist, playing jazz for fun and p ...

, it is designated "Type A" . Applying the circle of fifths transformation to the chromatic hexachord produces the

diatonic hexachord The diatonic, Guidonian, or major hexachord (6-32) is a hexachord consisting of six consecutive pitches from the diatonic scale that are also a consecutive segment of the circle of fifths: F C G D A E = C D E F G A = "do-re-mi-fa-sol-la". It i ...

. As with the diatonic scale, the chromatic hexachord is, "hierarchical in interval makeup," and may also be produced by, or contains, 3-1, 3-2, 3-3, 3-6, and 3-7 . Serial compositions including

Karlheinz Stockhausen Karlheinz Stockhausen (; 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 groun ...

's ''

Kreuzspiel (Crossplay) is a composition by Karlheinz Stockhausen written for oboe, bass clarinet, piano and four percussionists in 1951 (it was later revised for just three percussionists, along with other changes). It is assigned the number 1/7 in the comp ...

'' and '' Klavierstück I'' feature the chromatic hexachord in permuted orderings, as do certain pieces composed by

Milton Babbitt Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, mathematician, and teacher. He is particularly noted for his Serialism, serial and electronic music. Biography Babbitt was born in Philadelphia t ...

Alban Berg Alban Maria Johannes Berg ( , ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sma ...

Ernst Krenek Ernst Heinrich Krenek (, 23 August 1900 – 22 December 1991) was an Austrian, later American, composer of Czech origin. He explored atonality and other modern styles and wrote a number of books, including ''Music Here and Now'' (1939), a study ...

Luigi Nono Luigi Nono (; 29 January 1924 – 8 May 1990) was an Italian avant-garde composer of classical music. Biography Early years Nono, born in Venice, was a member of a wealthy artistic family; his grandfather was a notable painter. Nono beg ...

, Karlheinz Stockhausen,

Igor Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...

, and

Anton Webern Anton Friedrich Wilhelm von Webern (3 December 188315 September 1945), better known as Anton Webern (), was an Austrian composer and conductor whose music was among the most radical of its milieu in its sheer concision, even aphorism, and stea ...

in various fixed-order

derivations Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

(

twelve-tone row In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets ar ...

s and

arrays An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...

). Babbitt's Second Quartet and ''Reflections'' for piano and tape feature the hexachord . The retrograde-symmetrical all-interval series employed by Luigi Nono for the first time in ''Canti per tredeci'' in 1955, also used in his ''

Il canto sospeso ''Il canto sospeso'' (''The Suspended Song'') is a cantata for vocal soloists, choir, and orchestra by the Italian composer Luigi Nono, written in 1955–56. It is one of the most admired examples of serial composition from the 1950s, but has al ...

'' and nearly all subsequent works up to ''Composizione per orchestra n. 2: Diario polacco ’58'' in 1959, is built from two chromatic hexachords . Webern - Piano Variations op

Stefan Wolpe Stefan Wolpe (25 August 1902, Berlin – 4 April 1972, New York City) was a German-Jewish-American composer. He was associated with interdisciplinary modernism, with affiliations ranging from the Bauhaus, Berlin agitprop theater and the kibbutz mo ...

's ''Suite in Hexachord'' (1936) begins with a chromatic hexachord on G, introducing the complementary hexachord in the final movement, while

Elliott Carter Elliott Cook Carter Jr. (December 11, 1908 – November 5, 2012) was an American modernist composer. One of the most respected composers of the second half of the 20th century, he combined elements of European modernism and American "ultra- ...

calls his own piece, "Inner Song" for solo oboe—the second movement of ''Trilogy'' for oboe and harp (1992)—"some thoughts about Wolpe's hexachord" .

Sources

* * * * * * * * Chromaticism Hexachords Musical set theory {{music-theory-stub

See also

Sources