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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.^[2] The prime form of this set class is {012478}^[1] and its Forte number is 6-Z17. 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.^[3] Carter uses all-interval twelve-tone sets consisting of all-trichord hexachords in his Symphonia: sum fluxae pretium spei.^[4]

Sources

^ ^a ^b 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.

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] 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.

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[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-[[Image:All-trichord hexachord.png|thumb|300px|All-trichord hexachord ({{audio|All-trichord hexachord.mid|Play}}).]]
+[[File:All-trichord hexachord.png|thumb|upright=1.2|All-trichord hexachord[[File:All-trichord hexachord.mid]]]]
-[[Image:All-interval hexachord triads.png|thumb|350px|All-trichord hexachord trichords.<ref name=schiff>Schiff, David (1998). ''The Music of Elliott Carter'', p.34. {{ISBN|0-8014-3612-5}}.</ref> ({{audio|All-interval hexachord triads.mid|Play}})]]
+[[File:All-interval hexachord triads.png|thumb|upright=1.4|All-trichord hexachord trichords.<ref name=schiff>[[David Schiff|Schiff, David]] (1998). ''The Music of Elliott Carter'', p. 34. {{ISBN|0-8014-3612-5}}.</ref>[[File:All-interval hexachord triads.mid]]]]
-In [[music]], the '''all-trichord hexachord''' is a unique [[hexachord]] that contains all twelve [[trichord]]s, or from which all twelve possible trichords may be derived.<ref>Whittall, Arnold (2008). ''The Cambridge Introduction to Serialism'', p. 271. {{ISBN|978-0-521-68200-8}}.</ref> The [[prime form (music)|prime form]] of this set class is {012478}<ref name=schiff/> and its [[Forte number]] is '''6-Z17'''. Its [[Complement (music)#Aggregate complementation|complement]] is '''6-Z43''' and they share the [[interval vector]] of <3,2,2,3,3,2>.
+In [[music]], the '''all-trichord hexachord''' is a unique [[hexachord]] that contains all twelve [[trichord]]s, or from which all twelve possible trichords may be derived.<ref>[[Arnold Whittall|Whittall, Arnold]] (2008). ''The Cambridge Introduction to Serialism'', p. 271. {{ISBN|978-0-521-68200-8}}.</ref> The [[prime form (music)|prime form]] of this set class is {012478}<ref name=schiff /> and its [[Forte number]] is '''6-Z17'''. Its [[Complement (music)#Aggregate complementation|complement]] is '''6-Z43''' and they share the [[interval vector]] of <3,2,2,3,3,2>.
-It appears in pieces by [[Robert Morris (composer)|Robert Morris]] and [[Elliott Carter]].<ref name="Alegant">Alegant, Brian (2010). ''The Twelve-Tone Music of Luigi Dallapiccola'', p.307n4. {{ISBN|1-58046-325-8}}.</ref> Carter uses [[all-interval twelve-tone set]]s consisting of all-trichord hexachords in his ''[[Symphonia: sum fluxae pretium spei]]''.<ref>Schiff (1998), p.41.</ref>
+It appears in pieces by [[Robert Morris (composer)|Robert Morris]] and [[Elliott Carter]].<ref>Alegant, Brian (2010). ''The Twelve-Tone Music of [[Luigi Dallapiccola]]'', p. 307n4. {{ISBN|1-58046-325-8}}.</ref> Carter uses [[all-interval twelve-tone set]]s consisting of all-trichord hexachords in his ''[[Symphonia: sum fluxae pretium spei]]''.<ref>Schiff (1998), p. 41.</ref>
 ==See also==
@@ Line 12: / Line 12: @@
 ==Sources==
-<references/>
+<references />
 ==Further reading==
+{{div col|colwidth=45em}}
-*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}}.
+*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 (journal)|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}}.
+*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}}.
+*Capuzzo, Guy (2007). "Registral Constraints on All-Interval Rows in Elliott Carter's Changes". ''[[Intégral: The Journal of Applied Musical Thought|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}}.
+*[[Robert Morris (composer)|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):328–58. {{jstor|833710}}.
+*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): 253–82. {{doi|10.1111/j.0262-5245.2003.00186.x}}. {{jstor|3700435}}.
+*Ravenscroft, Brenda (2003). "Setting the Pace: The Role of Speeds in Elliott Carter's ''A Mirror on Which to Dwell''". ''[[Music Analysis (journal)|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 303–10. {{ISBN|978-3-642-04579-0}};  {{ISBN|978-3-642-04578-3}}. {{doi|10.1007/978-3-642-04579-0_29}}.
+*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). "[http://www.mtosmt.org/issues/mto.07.13.3/mto.07.13.3.sallmen.html 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)
+*Sallmen, Mark (2007). "[http://www.mtosmt.org/issues/mto.07.13.3/mto.07.13.3.sallmen.html 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)
+{{div col end}}
+{{Pitch segments}}
+{{Set theory (music)}}
+{{Twelve-tone technique}}
 [[Category:Chords]]
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 [[Category:Musical set theory]]
-{{Pitch segments}}
-{{Set theory (music)}}
-{{Twelve-tone technique}}
 {{music-theory-stub}}

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	Prime row 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