What's the Big Deal about All-Interval Tetrachords and the All-Triad Hexachord?
It's no secret that American composer Elliott Carter (1908-2012) spent multiple decades of creative work absolutely fascinated by just a few types of sonorities, including what we sexy music theory nerds commonly refer to as the all-interval tetrachords and the all-triad hexachord. Henceforth I will abbreviate the former as AITs and the latter as ATH because, hey... I'm a music theorist and we just love abbreviations that make us sound science-y.
For being such a prolific composer (especially in his last two decades), how could our Modernist Granddad Elliott possibly create so much music focused on just a handful of chords within the entire chromatic universe? What's the big deal about all-interval tetrachords and the all-triad hexachord? I fell down this particular rabbit hole myself in my dissertation, but here I'm going to give you the quick and dirty version without all the footnotes.
There are 29 different four-note pitch set classes you can create from the standard twelve-note chromatic collection. These include everything from our traditional fully diminished seventh chords  to a cluster of four adjacent chromatic pitches . For a neat prime form calculator you can play on and experiment with, click here. Now of all these 29 tetrachords, there are only two that have a special property we are about to discuss:  and . Check it out.
Collections  and  are extraordinary among the twenty-nine four-note set-classes in that each includes one and only one instance of the six interval classes (in further set theory terms, each has an ecumenical interval-class vector of <111111>.)
Follow with me: within the  chord in the above example, you can extract a minor second/major seventh (G/A-flat - dyad class ), a major second/minor seventh (B/C# - dyad class ), an enharmonically spelled minor third/major sixth (A-flat/B - dyad class ), a major third/minor sixth (G/B - dyad class ), an enharmonically spelled perfect fourth/perfect fifth (A-flat/C-sharp - dyad class ), and a tritone (G/C-sharp - dyad class ). Each interval within an octave is present and accounted for - and each happens exactly once!
Let's try the same thing by dissecting the  we see above: a minor second/major seventh (G/A-flat - dyad class ), a major second/minor seventh (A-flat/B-flat - dyad class ), a minor third/major sixth (G/B-flat - dyad class ), a major third/minor sixth (B-flat/D - dyad class ), a perfect fourth/perfect fifth (G/D - dyad class ), and a tritone (A-flat/D - dyad class ). Same story as the  - each of the intervals within the octave is here and appears only once.
Bonus Geekery: Why do I have arrows in this diagram? If you break down  and  into pairs of intervals, they share a mutual partitioning of  (minor third/major sixth) and  (tritone). That's pretty rad, right? I think so. So did my boy EC, but that's a dissertation for another day...
Because you can extract all of the intervals within the octave from each of these two special chords (of the available 29 four-note-chord types), we call  and  the all-interval tetrachords. Makes sense, right? These are the only two tetrachords you can play this fun game with. Furthermore, you can obviously transpose and invert both  and  and the intervallic content would remain the same.
From a creative standpoint, all-interval tetrachords are extremely powerful due to their ability to provide almost limitless intervallic flexibility while maintaining harmonic uniformity; put differently, an industrious composer could construct a piece from a single tetrachord (or pair of tetrachords) but still have access to the expressive capabilities afforded by the entire spectrum of intervals.
With AITs, you can write melodies with an incredible amount of intervallic variety but still maintain super tight harmonic consistency. It is precisely this malleability that may account for why Carter was fascinated with  and  for well over a half century.
In a manner of thinking, the all-triad hexachord  is the six-note analogue of the all-interval tetrachords. Just as AITs are singular among tetrachords for containing all six interval classes, the all-triad hexachord (ATH) is the only one of the 50 hexachord classes from which one may extract each of the twelve trichord classes. Although a composer could hypothetically realize the six-note set as pentachords plus singletons, the more likely scenarios (and those recurrently drawn on by Carter in his compositions) feature textures that separate tetrachords/dyads and trichords/ trichords. The diagram below displays the aforementioned latter partitionings.
The ATH is the only hexachord of the fifty that exist containing at least one subset of , , , , , , , , , , , and . Hence our name all-triad hexachord.
Concerning trichordal partitioning, certain three-note chords will always leave the same complementary trichord when extracted from an ATH. As an example, we could remove an  augmented triad subset from an all-triad hexachord superset and always be left with an  trichord remainder (I just love these little combinatorial magic tricks). This property does not always work in the opposite direction though (i.e. it is noncommutative), since we could easily round out an ATH by holding some realization of an  in one hand and particular realizations of an , , , , or  in the other. Due to the prevalence of  as an ATH trichord subset, the five +[xyz] partitions are listed as the top row of my diagram above; not surprisingly, Carter repeatedly takes advantage of  as a means to connect various transpositions of the all-triad hexachord in his music.
Carter spent decades exploring both AITs and the ATH in his compositions, pondering their inherent musical properties, finding ways of linking them to create formal continuities, and connecting their presence to larger referential sonorities (such as all-interval twelve-note chords). By now, AITs and the ATH are closely associated with Carter's compositional style and harmonic language — much like the whole-tone scale is with Debussy and late French Impressionism. Unlike whole-tone sonorities, however, the AITs and ATH still seem fertile ground for other composers to explore despite Carter's decades of investigation.
For greater detail about AITs, the ATH, and analyses of Carter's usage of them in musical contexts, see Chapter 2 of my dissertation (specifically pages 31-49).