All Combinatorial Hexachords

An interactive look at the theory behind hexachordally all-combinatorial sets

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Published March, 2016.
All-Combinatoriality: The capacity of a collection to create aggregates with forms of itself and its complement under both transposition and inversion. Such a collection is all-combinatorial in that it possesses all four types of combinatorality: prime-, inversional-, retrograde-, and retrograde-inversional-combinatoriality. Of the six all-combinatorial hexachords, three are “first-order” in that they can create aggregates at only one transposition level for each of the four traditional orderings of the series: prime, inversion, retrograde, and retrograde-inversion. Of the remaining three hexachords, one is “second-order” (creating aggregates at two levels), one is “third-order” (creating aggregates at three levels), and one is “sixth-order” (creating aggregates at six levels.).
Milton Babbitt, 1987, Page 193.

What are hexachordally all-combinatorial sets?

A hexachordally all-combinatorial set is a collection of six pitch classes that when combined with a transformation of itself under 1) prime (transposition, Tn) 2) retrograde, R(Tn) , 3) inversion, In or 4) retrograde-inversion, R(In), creates an aggregate, or a collection of all 12 pitch classes. The resulting six pitch classes are referred to as a complement. (Note: Definitions below.)

Further Clarification:

Let's assume that we have six different pitch classes: C, C#, D, Eb, E, and F (the first six notes of a C chromatic scale). Treat these six pitches as a group named H1. "H" stands for hexachord and "1" represents group 1. Now, let's say we want to alter this group of pitches by applying one of the traditional transformations to the group as a whole. Below is a list of the traditional transformations:

  • Transposing H1: Tn
  • Inverting H1: In
  • Reversing the order of H1 (retrograde): Rn
  • Reversing the order of H1 and inverting it (retrograde-inversion): R(In)

Here, "n" is a variable for an interval to be represented by a whole number from one through six. 1 is a half-step, 2 is a whole step, and so on all the way through 6, a tritone (six half-steps). We don't go beyond a tritone (6) because anything from 7 through 11 can be inverted to an interval from 1 through 6. Note that in mod6, 7 becomes 5, 8 becomes 4, and so on. If this confusing, consider experimenting with my page on musical inversions or watching my YouTube video on inverting intervals/a musical phrase:

Back to our example (C, C#, D, Eb, E, and F ). Let's say we want to transpose our H1 up by six half-steps, T6.. To do this we need to add six half-steps to each pitch class, or transpose each note by a tritone. The resulting pitches of such a transformation is as follows: F#, G, G#, A, Bb and B. Notice that not a single note from the original group is present in this new group! This is what's important: all-combinatoral hexchords are unique because the successful application of any of the above transformations (at a specific interval) will result in an entirely new set of pitch classes. In other words, an all-combinatorial hexachord must form aggregates at each of the traditional transformations.

However, notice that the application of any of the above transformations to any group of six notes will produce a new set. The catch is that one or more notes from the original six-note set will most likely be present in the second set. This means that combining two such sets will not result in an aggregate (a chromatic scale). Thus, the successful transformation of a hexachordally all-combinatorial set will result in a collection of six pitches that is distinct from the original set, and the combination of these two sets will constitute the makeup of a chromatic scale. Post-tonal theory typically refers to these collections of six pitch classes as hexachords (or sometimes hexatonic scales). The first group (the original set) is called H1, and the second group (H1's resulting transformation) is known as H2 , also known as H1's complement.

Hexachordally all-combinatorial sets are a rare phenomenon; there are only six sets which fulfill the above definition. Some hexachords are just combinatorial (not all-combinatorial), in that aggregates can be formed with a transformation of itself at only, Tn, In, Rn, or R(In), or even a subset of these, but not all four. An all-combinatorial hexachord must form aggregates at each of these transformations. The six hexachordally all-combinatorial sets are listed below in their prime form in integer notation. The "0" within their Interval Class Vector (ICV) is colored red (more information on ICVs in the next pane).

The six hexachordally combinatorial sets
Tone Row
Label H1 H2
(hexachordal complements to H1)
Interval Class Vector for
H1
Babbitt Label
(A): 0 1 2 3 4 5 : 6 7 8 9 T E <543210> First-order
Chromatic Scale
(B): 0 2 3 4 5 7 : 6 8 9 T E 1 <343230> First-order
(C): 0 2 4 5 7 9 : 6 8 T E 1 3 <143250> First-order
(D): 0 1 2 6 7 8 : 3 4 5 9 T E <420243> Second-order
(E): 0 1 4 5 8 9 : 2 3 6 7 T E <303630> Third-order
(F): 0 2 4 6 8 T : 1 3 5 7 9 E <060603> Sixth-order
Whole-tone scale

Hexachordal Interval Class Vectors and all-combinatorality

Click on a tab below for an in-depth description of each of these sets


The Second-Order All-Combinatorial Hexachord's
12-Tone Matrix

The twelve-tone matrix below is that of the second-order all-combinatorial hexachordal set used by Milton Babbitt in in his composition All Set. A second-order hexachordally all-combinatorial set has two complements for P0H1 at each of the four traditional orderings of a collection. The interactive matrix below is an illustration of each of the combinatorial complements for P0H1.

Click on a button below to see P0H1's complement under that specified transformation.

Understanding the colors:

P0H1

Below P0H1 is highlighted with the hexadecimal color #00BFFF
...

P0H1's Complement

The stated complement to P0H1 will be highlighted with #00BFFF’s complementary hexadecimal color, #FF4000.
...
Click on a button to the right and see P0H1's Complement.
I0 I4 I5 I11 I6 I10 I7 I3 I1 I2 I9 I8
P0 C E F B F# Bb G Eb Db D A Ab R0
P8 Ab C Db G D F# Eb B A Bb F E R8
P7 G B C F# Db F D Bb Ab A E Eb R7
P1 Db F F# C G B Ab E D Eb Bb A R1
P6 F# Bb B F C E Db A G Ab Eb D R6
P2 D F# G Db Ab C A F Eb E B Bb R2
P5 F A Bb E B Eb C Ab F# G D Db R5
P9 A Db D Ab Eb G E C Bb B F# F R9
P11 B Eb E Bb F A F# D C Db Ab C R11
P10 Bb D Eb A E Ab F Db B C G F# R10
P3 Eb G Ab D A Db Bb F# E F C B R3
P4 E Ab A Eb Bb D B G F F# Db C R4
RI0 RI4 RI5 RI11 RI6 RI10 RI7 RI3 RI1 RI2 RI9 RI8

P0H1's Complements:

  • Prime Combinatoriality:

  • Inversional-Combinatoriality:

  • Retrograde-Inversional-Combinatoriality:

  • Retrograde-Combinatoriality:

The Intervallic characteristics of the second-order all-combinatorial hexachords

The figure below shows the comparative intervallic content of P0H1 and I7H2 with their inversionally related hexachordal complements.


The second-order hexachord produces what Babbitt refers to as “a very special kind of twelve-tone double counterpoint” (Babbitt, 1987, Page 115). This twelve-tone double counterpoint can be observed by comparing the intervals produced between complementary second-order all-combinatorial hexachords. The first group, viewed vertically, (P0H1 and I7H1) results in the intervals 1, 3, and 5, while the second group’s intervals are 7, 9, and 11; neither set shares an interval. This only occurs with the second-order all-combinatorial set. Furthermore, notice that if the bottom line of the second grouping, (I7H2), were to be transposed up an octave that the resulting intervallic content would be identical to the intervals produced between P0H1 and I7H1; the resulting intervals are inversions of eachother. Therefore, depending on the register choices used, the use of such a hexachord and its complement could result in obtaining “the regestrially defined intervals of one hexachord expressing the structurally defined intervals of the other hexachord – a real double counterpoint” (Babbitt, 1987, page 116).

Terms and definitions

Collection:

An unordered group of pitch classes.

Aggregate:

A collection of all twelve-pitch classes.

Mod12:

Mod12 is short hand for modulo 12, which is Modular arithmetic. modulo 12 wraps around 12. Nearly everyone is already quite familiar with modulo 12 because it is how the 12-hour clock functions. As an example, take 8 and 5: we know that normally 8+5 = 13, however, in mod12 8+5=1.

Think of it like a clock, 8pm + 5 hours = 1am, not 13pm. In music 8 + 5 = 1 as well, which, in this case should be interpreted as pitch-class Ab + 5 half-steps, which is C#, 1.

Another example: 11+3=2, or B + three half-steps = D.

In mod12 12 is mapped onto zero, so C is both 0 and 12 in mod12. Notice that 6 (F#) is the middle point, and therefore never changes.

Complement:

A collection of pitch classes (with less than 12 pitch classes) that when combined with another collection of pitches forms an aggregate.

Interval Class Vector ([ICV]):

[ICV]'s represent the intervallic content of a collection. They're typically notated between greater than and less than brackets, "<" and ">". Between these carrots there will six numbers, between the digits zero and six.

  • < x 0 0 0 0 0 >: x = the number of half-steps present in the collection.
  • < 0 x 0 0 0 0 >: x = the number of whole-steps present in the collection.
  • < 0 0 x 0 0 0 >: x = the number of minor-thirds present in the collection.
  • < 0 0 0 x 0 0 >: x = the number of major-thirds present in the collection.
  • < 0 0 0 0 x 0 >: x = the number of perfect fourths present in the collection.
  • < 0 0 0 0 0 x >: x = the number of tritones present in the collection.
Thus, an ICV of <123450> has 1 half step, 2 major seconds, three minor thirds, four major thirds, 5 perfect fourths, and no tritones.

Combinatoriality:

The capacity of a collection to combine with some transformation of itself (or its complement) to form aggregates. (Babbitt, 1987, Page 194)

H, or, hexachord:

A collection of 6 pitch classes.

H1:

Hexachord 1. Pitch classes 1 through 6 of a 12-tone row.

H2:

Hexachord 2. Pitch classes 7 through 12 of a 12-tone row.

Sources

Babbitt, Milton. Collected Essays of Milton Babbitt. Princeton, NJ, USA: Princeton University Press, 2011. Accessed May 5, 2015. ProQuest ebrary.

Babbitt, Milton. Milton Babbitt: Words About Music (The Maddison Lectures). Maddison Wisconsin: The University of Wisconsin Press, 1987.

Castaneda, Ramsey. An Analysis of Milton Babbitt's All Set. Unpublished.

Stuessy, Clarence Joseph Jr. “The Confluence of Jazz and Classical Music from 1950 to 1970.” Doctor of Philosophy Dissertation, Eastman School of Music of The University of Rochester, New York, 1977.