music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, the all-trichord hexachord is a unique

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 ...

that contains all twelve

trichord In music theory, a trichord () is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row. In musical set theory there are twelve trichords given ...

s, or from which all twelve possible trichords may be derived. The

prime form In algebraic geometry, the Schottky–Klein prime form ''E''(''x'',''y'') of a compact Riemann surface ''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The prime form ''E'' is not quite ...

of this set class is and its

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 ...

is 6-Z17. 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 6-Z43 and they share the

interval vector In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interva ...

of <3,2,2,3,3,2>. It appears in pieces by Robert Morris and

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- ...

. Carter uses all-interval twelve-tone sets consisting of all-trichord hexachords in his '' Symphonia: sum fluxae pretium spei''.Schiff (1998), p. 41.

References

*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). . *Boland, Marguerite (2006). "'Linking' and 'Morphing': Harmonic Flow in Elliott Carter's ''Con Leggerezza Pensosa''". ''

Tempo In musical terminology, tempo (Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (often ...

'' 60, no. 237 (July): 33–43. . *Boros, James (1990). "Some Properties of the All-Trichord Hexachord", ''

In Theory Only ''In Theory Only'' () was a peer-reviewed academic journal specializing in music theory and analysis. It began publication in 1975, under the auspices of graduate students in music theory at the University of Michigan School of Music, Theatre & ...

'' 11/6: 19–41. *Capuzzo, Guy (2004). "The Complement Union Property in the Music of Elliott Carter". ''

Journal of Music Theory The ''Journal of Music Theory'' is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl (Yale University) in 1957. According to its website, " e ''Journal of Music Theory'' fosters c ...

'' 48, no. 1 (Spring): 1–24. . *Capuzzo, Guy (2007). "Registral Constraints on All-Interval Rows in Elliott Carter's Changes". '' Intégral'' 21:79–108. . * Morris, Robert D. (1990). "Pitch-Class Complementation and Its Generalizations". ''

'' 34, no. 2 (Autumn): 175–245. . *Morris, Robert (1995). "Compositional Spaces and Other Territories". ''

Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was first ...

'' 33, nos. 1 & 2 (Winter–Summer):328–358. . *Ravenscroft, Brenda (2003). "Setting the Pace: The Role of Speeds in Elliott Carter's ''A Mirror on Which to Dwell''". ''

Music Analysis Musical analysis is the study of musical structure in either compositions or performances. According to music theorist Ian Bent, music analysis "is the means of answering directly the question 'How does it work?'". The method employed to answe ...

'' 22, no. 3 (October): 253–282. . . *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. .. *Sallmen, Mark (2007).
Listening to the Music Itself: Breaking Through the Shell of Elliott Carter's 'In Genesis'
, ''

Music Theory Online ''Music Theory Online'' is a quarterly peer-reviewed open access academic journal covering music theory and musical analysis, analysis. It was established in 1993 and is published by the Society for Music Theory. The initial issues were designat ...

'' 13/3 (Alegant 2010, p. 307n4). (Accessed 31 March 2014) Chords Hexachords Twelve-tone technique Hemitonic scales Musical set theory {{music-theory-stub

See also

References

Further reading