Combinatoriality

Combinatoriality is a fairly simple concept, but, before explaining it, let's review two related terms:

Aggregate: A collection of all 12 pitch classes.

Complement: The collection of pitch classes needed to form an aggregate with an existing set. For example, a hexachord that contains all chromatic pitches from C to F would have a complement that contains all chromatic pitches from F# to B. Similarly, a set of 9 pitch classes would have a complement that contains the 3 pitch classes missing from the 9 pitch class set.

A 12-tone row, as we know, contains one of every pitch class, and is therefore an example of an aggregate. Some 12-tone rows are constructed in such a way that the first six notes of the prime form form an aggregate with the first six notes of a different form of the same row. Or, put another way, the first hexachord of two different row forms complement one another (which, by necessity, means the second hexachords of those two row forms also complement one another).

Rows that have this property are said to exhibit hexachordal combinatoriality.

Here are some examples:

Berio, Sequenza 1:
P-0:   A  G# G  F# F  E  | C# D# D  C  A# B
P-6:   D# D  C# C  B  A# | G  A  G# F# E  F

Babbitt, Semi-Simple Variations
P-0:   A# F# B  G# G  A  | D# C# D  F  C  E
P-6:    E  C  F  D  C# D# | A  G  G# B  F# A#

In each of the above rows, the first hexachord of P-0 forms a complement with the first hexachord of P-6 (and the same is therefore true of their respective second hexachords as well).

The row from Berio's Sequenza 1 is an example of the easiest type of hexachordally-combinatorial row to construct: You arrange the first 6 pitches of a chromatic scale in any order, then do the same with the remaining portion of that chromatic scale, and you will automatically have created a row that is combinatorial with a tritone transposition of the same row (because the first 6 notes of a chromatic scale are a tritone away from the last 6 notes of the chromatic scale).

I'll take a few questions now…

What's the Point?  

• It allows composers to use two (or more; a tone row may be combinatorial with more than one different form of itself) row forms simultaneously without having the same pitch appear at the same time in both row forms, assuming the rate at which pitches are deployed is similar.


• It also allows composers to create a new, related row (called a secondary set) by combining, in the above examples, the first hexachord of P-0 with the first hexachord of P-6.

Uh huh... Are there different types of combinatoriality?

There are four types of combinatoriality, corresponding with the four forms of a row:: Prime, Inversional, Retrograde-Inversional, and Regrograde. The two rows above are both Prime Combinatorial. A row that is combinatorial with all four types is called "all-combinatorial."

Retrograde Combinatoriality? Isn't any 12-tone row combinatorial with its retrograde? What's so special about that?

Yes, every 12-tone row is combinatorial with its own retrograde, so there is nothing special about that. Because of this, Babbitt, who came up with this term, originally excluded it from his list of combinatoriality types.

Babbitt, eh? Isn't he the title character in a book by Sinclair Lewis?

Exactly! However, in this case, the reference is to Milton Babbitt, the composer/theorist, not the fictional character/fictional character.