Combinatoriality

In music using the twelve tone technique combinatoriality is a quality shared by some twelve-tone tone rows whereby the row and one of its transformations combine to form a pair of aggregates.^[1] Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create, "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."^[1] Combinatoriality is a side effect of derived rows where combining different segments or sets such that the pitch-class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic. Combinatorial properties are not dependent on the order of the notes within a set, but only on the content of the set, and combinatoriality may exist among tetrachordal and trichordal sets, as well as between pairs of hexachords.^[2] It may also be applied to dyads.^[3]

A complement is in this context half of a combinatorial pitch class set and most generally a complement is the "other half" of any pair including pitch class sets, textures, or pitch range. Most generally complementation is the separation of pitch-class collections into two complementary sets, one containing the pitch classes not in the other.^[1] More restrictively complementation is "the process of pairing entities on either side of a center of symmetry".^[4]

Combinatorial tone rows from Moses und Aron by Arnold Schoenberg pairing complementary hexachords from P-0/I-3^[5]

Definition

The term, "'combinatorial' appears to have been first applied to twelve-tone music by Milton Babbitt" in 1950,^[6] when he published a review of René Leibowitz's books Schoenberg et son école and Qu'est ce qu la musique de douze sons?^[7] Hexachordal inversional combinatoriality refers to two rows, a principal row and its inversion. The principal row's first half, or six notes, are the inversion's last six notes, though not necessarily in the same order. Thus, the first half of each row is the other's complement. The same conclusion applies to each row's second half as well. When combined, these rows still maintain a fully chromatic feeling and don't tend to reinforce certain pitches as tonal centers as might happen with freely combined rows. Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions. Retrograde hexachordal combinatoriality is considered trivial, since any set has retrograde hexachordal combinatoriality with itself. Combinatoriality may be used to create an aggregate of all twelve tones, though the term often refers simply to combinatorial rows stated together.

Semi-combinatorial sets are sets whose hexachords are capable of forming an aggregate with one of its basic transformations transposed. There are twelve hexachords that are semi-combinatorial by inversion only.

(0) 0 1 2 3 4 6 // e t 9 8 7 5
(1) 0 1 2 3 5 7 // e t 9 8 6 4
(2) 0 1 2 3 6 7 // e t 9 8 5 4
(3) 0 1 2 4 5 8 // e t 9 7 6 3
(4) 0 1 2 4 6 8 // e t 9 7 5 3
(5) 0 1 2 5 7 8 // e t 9 6 4 3
(6) 0 1 3 4 6 9 // e t 8 7 5 2
(7) 0 1 3 5 7 9 // e t 8 6 4 2
(8) 0 1 3 5 8 9 // 7 6 4 2 e t
(9) 0 1 4 5 6 8 // 3 2 e t 9 7
(t) 0 2 3 4 6 8 // 1 e t 9 7 5
(e) 0 2 3 5 7 9 // 1 e t 8 6 4

There is one hexachord which is combinatorial by transposition (T6):

(0) 0 1 3 4 5 8 // 6 7 9 t e 2

All-combinatorial sets are sets whose hexachords are capable of forming an aggregate with any of its basic transformations transposed. There are six source sets, or basic hexachordally all-combinatorial sets, each hexachord of which may be reordered within itself:

(A)  0 1 2 3 4 5 // 6 7 8 9 t e
(B)  0 2 3 4 5 7 // 6 8 9 t e 1
(C)  0 2 4 5 7 9 // 6 8 t e 1 3
(D)  0 1 2 6 7 8 // 3 4 5 9 t e
(E)  0 1 4 5 8 9 // 2 3 6 7 t e
(F)  0 2 4 6 8 t // 1 3 5 7 9 e

Note: t = 10, e = 11.

Hexachordal combinatoriality

Hexachordal combinatoriality is a concept in post-tonal theory that describes the combination of hexachords, often used in reference to the music of the Second Viennese school. In music that consistently utilizes all twelve chromatic tones (particularly twelve-tone and serial music), the aggregate (collection of all 12 pitch classes) may be divided into two hexachords (collections of 6 pitches). This breaks the aggregate into two smaller pieces, thus making it easier to sequence notes, progress between rows or aggregates, and combine notes and aggregates.

Occasionally hexachords may be combined with an inverted or transposed version of itself in a special case which will then result in the aggregate (complete set of 12 chromatic pitches).

A row (B♭=0: 0 6 8 5 7 e 4 3 9 t 1 2) used by Schoenberg may be divided into two hexachords:

B♭ E  F♯ E♭ F  A // D  C♯ G  G♯ B  C

When you invert the first hexachord and transpose it, the following hexachord, a reordering of the second hexachord, results:

G  C♯ B  D  C  G♯ = D  C♯ G  G♯ B  C

Thus, when you superimpose the original hexachord 1 (P0) over the transposed inversion of hexachord 1 (I9 in this case), the entire collection of 12 pitches results. If you continued the rest of the transposed, inverted row (I9) and superimposed original hexachord 2, you would again have the full complement of 12 chromatic pitches.

Source

1. ^ ^{a b c} Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p. 272. New York: Cambridge University Press. ISBN 978-0-521-86341-4 (hardback) ISBN 978-0-521-68200-8 (pbk).
2. ^ George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, London: University of California Press, 1977), 129–31. ISBN 0-520-03395-7
3. ^ Peter Westergaard, "Some Problems Raised by the Rhythmic Procedures in Milton Babbitt's Composition for Twelve Instruments", Perspectives of New Music 4, no. 1 (Autumn-Winter 1965): 109–18. Citation on 114.
4. ^ Kielian-Gilbert, Marianne (1982–83). "Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity", Perspectives of New Music 21: 210.
5. ^ Whittall, 103
6. ^ Whittall, 245n8
7. ^ Milton Babbitt, Untitled review, Journal of the American Musicological Society 3, no. 1 (Spring 1950): 57–60. The discussion of combinatoriality is on p. 60.