An interactive look at the theory behind hexachordally allcombinatorial sets
MusicTheoryPractice.comAllCombinatoriality: The capacity of a collection to create aggregates with forms of itself and its complement under both transposition and inversion. Such a collection is allcombinatorial in that it possesses all four types of combinatoriality: prime, inversional, retrograde, and retrogradeinversionalcombinatoriality. Of the six allcombinatorial hexachords, three are “firstorder” 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 retrogradeinversion. Of the remaining three hexachords, one is “secondorder” (creating aggregates at two levels), one is “thirdorder” (creating aggregates at three levels), and one is “sixthorder” (creating aggregates at six levels.).
A hexachordally allcombinatorial set is a collection of six pitch classes that when combined with a transformation of itself under 1) prime (transposition, T_{n}) 2) retrograde, R(T_{n}) , 3) inversion, I_{n} or 4) retrogradeinversion, R(I_{n}), 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.)
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 H_{1}. "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:
Here, "_{n}" is a variable for an interval to be represented by a whole number from one through six. 1 is a halfstep, 2 is a whole step, and so on all the way through 6, a tritone (six halfsteps). 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.
Back to our example (C, C#, D, Eb, E, and F ). Let's say we want to transpose our H_{1} up by six halfsteps, T_{6.}. To do this we need to add six halfsteps 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: allcombinatorial 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 allcombinatorial 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 sixnote 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 allcombinatorial 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. Posttonal theory typically refers to these collections of six pitch classes as hexachords (or sometimes hexatonic scales). The first group (the original set) is called H_{1}, and the second group (H_{1}'s resulting transformation) is known as H_{2 }, also known as H_{1}'s complement.
Hexachordally allcombinatorial sets are a rare phenomenon; there are only six sets which fulfill the above definition. Some hexachords are just combinatorial (not allcombinatorial), in that aggregates can be formed with a transformation of itself at only, T_{n}, I_{n}, R_{n}, or R(I_{n}), or even a subset of these, but not all four. An allcombinatorial hexachord must form aggregates at each of these transformations. The six hexachordally allcombinatorial 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  H_{1}  H_{2} (hexachordal complements to H_{1}) 
Interval Class Vector for H_{1} 
Babbitt Label  
(A):  0  1  2  3  4  5  :  6  7  8  9  T  E  <543210>  Firstorder Chromatic Scale 
(B):  0  2  3  4  5  7  :  6  8  9  T  E  1  <343230>  Firstorder 
(C):  0  2  4  5  7  9  :  6  8  T  E  1  3  <143250>  Firstorder 
(D):  0  1  2  6  7  8  :  3  4  5  9  T  E  <420243>  Secondorder 
(E):  0  1  4  5  8  9  :  2  3  6  7  T  E  <303630>  Thirdorder 
(F):  0  2  4  6  8  T  :  1  3  5  7  9  E  <060603>  Sixthorder Wholetone scale 
The twelvetone matrix below is that of the secondorder allcombinatorial hexachordal set used by Milton Babbitt in
his composition All Set. A secondorder hexachordally allcombinatorial set has two complements for P_{0}H_{1} at each of the four traditional orderings of a collection.
The interactive matrix below is an illustration of each of the combinatorial complements for P_{0}H_{1}.
Click on a button to the right and see P_{0}H_{1}'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 
The figure below shows the comparative intervallic content of P_{0}H_{1} and I_{7}H_{2} with their inversionally related hexachordal complements.
The secondorder hexachord produces what Babbitt refers to as “a very special kind of
twelvetone double counterpoint” (Babbitt, 1987, Page 115). This twelvetone double counterpoint
can be observed by comparing the intervals produced between complementary secondorder allcombinatorial hexachords. The first group, viewed vertically, (P_{0}H_{1}
and I_{7}H_{1}) 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 secondorder allcombinatorial set.
Furthermore, notice that if the bottom line of the second grouping, (I_{7}H_{2}), were to be transposed up an
octave that the resulting intervallic content would be identical to the intervals produced between
P_{0}H_{1} and I_{7}H_{1}; 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).
An unordered group of pitch classes.
A collection of all twelvepitch classes.
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 12hour 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 pitchclass Ab + 5 halfsteps, which is C#, 1.
Another example: 11+3=2, or B + three halfsteps = 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.
A collection of pitch classes (with less than 12 pitch classes) that when combined with another collection of pitches forms an aggregate.
[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.
The capacity of a collection to combine with some transformation of itself (or its complement) to form aggregates. (Babbitt, 1987, Page 194)
A collection of 6 pitch classes.
Hexachord 1. Pitch classes 1 through 6 of a 12tone row.
Hexachord 2. Pitch classes 7 through 12 of a 12tone row.
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.