All-trichord hexachord

In music, the all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is {012478} and its Forte number is 6-Z17. The following is an example of the all-trichord hexachord.

${ \override Score.TimeSignature #'stencil = ##f \relative c' { \time 24/4 \set Score.tempoHideNote = ##t \tempo 1 = 160 c1 cis d e g aes } }$

Its complement is 6-Z43 and they share the interval vector of <3,2,2,3,3,2>.

It appears in pieces by Robert Morris and Elliott Carter. Carter uses all-interval twelve-tone sets consisting of all-trichord hexachords in his Symphonia: sum fluxae pretium spei. In 1992, composer and Carter scholar John Link generated a list of every all-interval twelve-tone row from which an all-trichord hexachord could be derived, and offered it to Carter to help with his work.

References

1 2 Schiff, David (1998). The Music of Elliott Carter, p. 34. ISBN 0-8014-3612-5.
↑ Whittall, Arnold (2008). The Cambridge Introduction to Serialism, p. 271. ISBN 978-0-521-68200-8.
↑ Alegant, Brian (2010). The Twelve-Tone Music of Luigi Dallapiccola, p. 307n4. ISBN 1-58046-325-8.
↑ Schiff (1998), p. 41.
↑ "John Link - The "Link Chords"". johnlinkmusic.com. Retrieved 2026-09-13.

Boland, Marguerite (1999). "The All-trichord Hexachord: Compositional Strategies in Elliott Carter's Con leggerezza pensosa and Gra and a Folio of Original Compositions". MA Thesis, La Trobe University (Alegant 2010, p. 307n4). OCLC 224102530.
Boland, Marguerite (2006). "'Linking' and 'Morphing': Harmonic Flow in Elliott Carter's Con Leggerezza Pensosa". Tempo 60, no. 237 (July): 33–43. doi:10.1017/S0040298206000209.
Boros, James (1990). "Some Properties of the All-Trichord Hexachord", In Theory Only 11/6: 19–41.
Capuzzo, Guy (2004). "The Complement Union Property in the Music of Elliott Carter". Journal of Music Theory 48, no. 1 (Spring): 1–24. doi:10.1215/00222909-48-1-1.
Capuzzo, Guy (2007). "Registral Constraints on All-Interval Rows in Elliott Carter's Changes". Intégral 21:79–108. JSTOR 40214036.
Morris, Robert D. (1990). "Pitch-Class Complementation and Its Generalizations". Journal of Music Theory 34, no. 2 (Autumn): 175–245. JSTOR 843837.
Morris, Robert (1995). "Compositional Spaces and Other Territories". Perspectives of New Music 33, nos. 1 & 2 (Winter–Summer):328–358. JSTOR 833710.
Ravenscroft, Brenda (2003). "Setting the Pace: The Role of Speeds in Elliott Carter's A Mirror on Which to Dwell". Music Analysis 22, no. 3 (October): 253–282. doi:10.1111/j.0262-5245.2003.00186.x. JSTOR 3700435.
Roeder, John (2009). "A Transformational Space for Elliott Carter's Recent Complement-union Music". In Mathematics and Computation in Music, edited by Timour Klouche and Thomas Noll, 303–310. Communications in Computer and Information Science 37. New York: Springer. pp. 303–310. ISBN 978-3-642-04579-0, 978-3-642-04578-3.doi:10.1007/978-3-642-04579-0_29.
Sallmen, Mark (2007). "Listening to the Music Itself: Breaking Through the Shell of Elliott Carter's 'In Genesis'", Music Theory Online 13/3 (Alegant 2010, p. 307n4). (Accessed 31 March 2014)

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[schiff-1] 1 2 Schiff, David (1998). The Music of Elliott Carter, p. 34. ISBN 0-8014-3612-5.

[2] Whittall, Arnold (2008). The Cambridge Introduction to Serialism, p. 271. ISBN 978-0-521-68200-8.

[3] Alegant, Brian (2010). The Twelve-Tone Music of Luigi Dallapiccola, p. 307n4. ISBN 1-58046-325-8.

[4] Schiff (1998), p. 41.

[5] "John Link - The "Link Chords"". johnlinkmusic.com. Retrieved 2026-09-13.

v t e Musical set theory
All-interval tetrachord All-trichord hexachord Complement Forte number Identity Interval class Interval vector Multiplication Permutation Pitch class Pitch interval Pitch-interval class Set List Similarity relation Transformation Z-relation
Diatonic set theory	Bisector Cardinality equals variety Common tone (Deep scale property) Diatonic scale Generated collection Generic and specific intervals (Myhill's property) Maximal evenness Rothenberg propriety Structure implies multiplicity

v t e Twelve-tone technique and serialism
Fundamentals	Combinatoriality Complementation Derivation Hexachord Interval class Invariance Partition Cross partition Tone row Aggregate List
Permutations	Retrograde Inversion Retrograde inversion Multiplication
Notable composers	Milton Babbitt Pierre Boulez Josef Matthias Hauer Second Viennese School Alban Berg Arnold Schoenberg Anton Webern Charles Wuorinen ...more...
Related articles	All-interval twelve-tone row All-trichord hexachord Atonality Chromatic scale Duration series Equivalence Formula composition Modernism (music) Punctualism Semitone Set theory Time point Trope
List of dodecaphonic and serial compositions

All-trichord hexachord

See also

References

Further reading