All-Trichord Hexachords

This is a continuation from an earlier post that looked at All Interval Tetrachords. Here we look at some recipes for building their bigger siblings, the All Trichord Hexachords, on the fly and some ideas for using them.

The All-Trichord Hexachords (ATHs) contain every possible three-note chord up to inversion (i.e. a "trichord") in a mere six notes. An example is 1-b2-#3-#4-5-6. As with the previous post, we're going to be in Elliott Carter territory, since analysis of his work seems to have identified the heaviest use of these chords "in the wild". Here's a short piece by him that uses it:

A quick way to find an ATH is to start with the Lulu chord (1-b2-#4-5) and add any non-overlapping major third. Visually, I find it easiest to locate these by seeing the Lulu chord as two pairs of notes a semitone apart; the legitimate major thirds must envelop one of those pairs. However, I don't usually play the Lulu chord in closed form, but rather as a fourth and a fifth in different octaves, or as two tritones in different octaves. This demands an extra step of visualization but it's not too bad.

This specific interplay of the Lulu chord with a major seventh plays an important part in another Elliot Carter piece, Gra, which I used to try to play during my brief stint as an over-ambitious clarinettist:

In fact ATH is not one chord but two: the one we've just met, 6-Z17, and its complement 6-Z43. One copy each of 6-Z17 and 6-Z43 can cover all 12 notes without overlaps, which makes them quite handy for making 12-tone music. As with the AITs we looked at last time, the distinction between the chords isn't very strong and it's often practical to just consider them variations on the same sound.

Here's a not-very-practical way to find the notes in an ATH in ascending order without memorizing anything. Both consist of three chromatic notes, a semitone and a single note, all separated by one or more semitones. Whenever you play one of these groups, always skip that many notes before starting the next one.

Suppose you start with two consecutive notes, D#-E. You skip two notes, so the next will be G. Let's make this the three-note group, so play G-G#-A. Since we played three notes, we skip three and that takes us to C#, which is the final note. That gives us D#-E-G-G#-A-C#, which is 6-Z43. If we'd gone for the single-note group second instead of the two-note group, we'd have got 6-Z17 instead.

OK that might get you finding these chords but it's too much calculation to work in real time. I
learned one good trick for this from this video by guitarist Johnathan Leathwood:

The idea is to play any augmented triad plus any trichord (0, 1, 6), which is just a tritone with a semitone placed above or below one of its notes (e.g. C-Db-Gb). As long as the two don't overlap, you'll get 6-Z17. Making sure they don't overlap is a bit more tricky -- it helped me to prioritize making the two adjacent notes in the (0, 1, 6) fit into one of the major thirds in the augmented triad.

Unfortunately the same idea doesn't quite work for the other ATH, 6-Z43. A similar approach would be to combine a diminished triad with (0, 1, 6) and this does work, but only if you put the (0, 1, 6) in the right place. The trouble here is that unlike the augmented triad, the diminished triad isn't symmetrical. We have to position the (0, 1, 6) so that the semitone (0, 1) leads up to the root of the diminished chord. For example, we'd pair C diminished C-Eb-Gb with Bb-B-E.

Here are some more "jazzy" ways to think of them in case you find those useful -- I'm sure you can find lots more:

  • A minor major 7 chord (C-Eb-G-B) with two chromatic notes leading up to the fifth (F-F#), giving 6-Z17
  • A minor major 7 chord (C-Eb-G-B) with a #11 and a b9 (Db-F#), also giving 6-Z17, which seems to be how Stanley Jordan thinks of it although how much he uses it I don't know)
  • An augmented sixth chord (C-E-G#-A) with two chromatic notes leading up to the third (D-D#), giving 6-Z17
  • A dominant 7 b5 chord (C-E-Gb-Bb) with added #9 and natural 7 (D#-B), giving 6-Z43
  • A minor sixth chord (C-Eb-G-A) with a b9 and a b13 connecting the 5 and 6 (Db-Ab), giving 6-Z43.

None of these is as elegant as I'd like, but if you pick one and practice it you should be able to find these chords pretty easily. For me the first and last are the easiest to visualize.

I mentioned that the two ATHs are complements of each other, i.e. they can cover the whole chromatic scale if you set them up right. If you pick a way to find 6-Z17 and a way to find 6-Z43, figuring out how to do that is probably worthwhile, at least assuming you're interested in 12-tone music. In my case, if I start with CmM7 as the basis for 6-Z17 I can build 6-Z43 from C#m6, which is nice and easy to remember.

Using Leathwood's approach it's probably best to identify where the three-semitone "gap" is in your current chord and fit the complementary chord into that; there's only one way to do that that will work and it's not too hard to spot. For now I think I prefer the "jazzy" version just because those chords are much easier for me to find reliably.