Dictionary

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]. Schoenberg 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.

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" [2].

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

Combinatoriality

The term was first described by Milton Babbitt. 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 would 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 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 utilizes all 12 chromatic tones (particularly serial music), the tone row may be split into two hexachords (collections of 6 pitches). Occasionally hexachords may be combined with an inverted or transposed version of itself in a special case which will then result in the complete set of 12 chromatic pitches.

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

```Bb E F# Eb 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.

Complementation may be imagined visually as two one color outfits worn by two different women. One is wearing a red top and black pants and the other is wearing a black top and red pants. If they stand next to each other, your "row" would consist of looking horizontally across the image (both women's shirts or both women's pants). The full complement of a red garment and a black garment is also present, however, vertically in each woman's complete outfit. Each woman's outfit is a visual representation of the concept of combinatoriality.

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. ^ Kielian-Gilbert, Marianne (1982–83). "Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity." Perspectives of New Music 21: 210.
3. ^ Whittall, 103