All Combinatorial Hexachords

A hexachordally all-combinatorial 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) retrograde-inversion, 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.)

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: T_n
• Inverting H₁: I_n
• Reversing the order of H₁ (retrograde): R_n
• Reversing the order of H₁ and inverting it (retrograde-inversion): R(I_n)

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.

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, T_6.. 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-combinatorial 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, T_n, I_n, R_n, or R(I_n), 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).