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.

Debussy's La Fille aux Cheveux de Lin; Some Points of Harmonic Interest

→ Some points of harmonic interest in Debussy’s “La Fille aux Cheveux de Lin” (Preludes, book 1, #8)

Click here to link to a YouTube scrolling score of this prelude, which you may wish to do in order to follow the analysis.

1. The opening melody notes outline a Ebm7 chord, but is it functioning as a vi7 chord in Gb? If we hear the opening as I chord in Gb with an oscillating 5-6, the analysis of bars 1-3 would be:     I ^5-6-5-6(-5?) - IV - I
   
   • Is this the “correct” analysis? It is hard to say with certainty, which would suggest that the opening is harmonically ambiguous.
   • That said, my own feeling is that it is a I with oscillating 5-6, based in part on the frequency with which Debussy uses added 6ths in the harmony of the rest of the prelude.


2. The plagal cadence in mm. 2-3 may not sound particularly unusual, but it is! Plagal cadences are a rarity in tonal music (plagal extensions are relatively common, however; that is what you call the process of extending the I chord at the end of a composition by means of I - IV - I, which would typically be preceded by a V, so the cadence type in that case would be authentic, with plagal extension).


3. The first authentic cadence occurs in bar 6, but it tonicizes VI (major). Nothing particularly unusual about this, but VI is probably not the first region you'd expect to find an authentic cadence in a work...


4. Rock 'n Roll Chords!: The chords that follow (bars 8-9) are I^b7 - II^b7 - I^b7 - II^b7... in other words, major/minor chords (i.e., "non-functional" dominant 7th-types), the last of which turns out to be functional (V4/2 of V).
   • Also, each is in 4/2 position, but the 7th in the bass never resolves, another indicator that these are non-functional. Non-resolving chord 7ths are a common feature in this piece.
   • For that matter, the 7th in the bass never resolves in the last of these four chords either, and this chord is a functional chord (V4/2 of V)!


5. The last chord in bar 9 is a V¹¹/9/₇ chord. This too turns out to be a common feature of this composition; see bar15 (V¹¹/9/₇ of IV), bar 18 (V¹¹/9/₇ of VI; probably this is a modulation to VI), and the last chord in bar 25 (this one is without the 9).


6. Parallelism. Bars 24 to the first beat of 27 consist of chords in (mostly) parallel motion. The first (bar 24) is a I^{add 6}, and the second a vi⁷ (with unresolved 7th). This pattern of an added 6^th chord followed by a 7^th chord continues as follows in 24-26.1:
   
       I^{add 6} - vi⁷ - I6/4^{add 6} - vi⁷ - I^{add 6} - vi4/3 - I^{add 6} - vi⁷ - IV^M7 - V¹¹/₇ - I^{add 6}.
   
   Note that the pattern is broken for the last two chords of bar 25. The pattern is then resumed until bar 27 beat 2.


7. 27-32 contains a very interesting phenomenon: ii⁹₇ - V in bar 27 (beats 2-3) sets us up for a I chord in bar 28. But do we get a I chord?
   
   Apparently not! Debussy gives us a IV chord instead, apparently. Now, you may be thinking to yourself, "what do you mean, "apparently"? It is undoubtedly a IV chord!"
   
   Maybe so, but here is what I find interesting: When I first hear that "IV" chord, my brain thinks of it as the first part of a ^6-_4-⁵₃ decoration of a I chord, probably because that is what the apparent IV chord usually turns out to be in a ii - V - IV(?) - I progression.
   
   So... does that IV chord actually resolve to a I?
   
   Well... not in any obvious fashion, that's for sure! We finally get to the I at bar 32, having passed through a vi chord along the way (bar 31). Here's how I see it: If you take a somewhat bigger picture view of 28-32, it is fundamentally one big I chord. You could think of it as a big fermata, under which the opening theme comes back in its entirety, with the last three notes made longer. I suggested at the outset that the implied harmony of the opening theme was I ^5-6-5-6(-5?) - IV - I (which is itself one big I chord), so, in that context, you might be able to understand why my brain processes that relatively lengthy IV chord in bar 28 as the beginning of a decoration of the I chord, resolving finally to the expected I chord in bar 32. And if you're hung up on the vi chord in 31, don't be; vi and I are both tonic-function chords, so you can regard that as part of the embellishment of the I chord as well.
   
   Maybe?


8. There are more points of harmonic interest, but that covers most of them, and I'm running out of steam, so I'll leave it at that. A point of scalar interest is that pentatonic scales are used prominently in this prelude.
   
   Feel free to disagree with any of this!