All-Interval Tetrachords

I was told something today that surprised me and sent me scurrying off to try it: Any non-overlapping combination of a minor third interval and a tritone interval contains all the possible intervals. Such chords are known as all-interval tetrachords (AITs). I've heard of them for years but never saw how to use them; as usual, it was just a matter of someone showing me the "right" way (for me) to look at them.

Specifically, if you start with a minor third, you can play tritones on top of it as long as they don't overlap and you'll always get an AIT. For example, on C-Eb we can play any of the following tritones: Db-G, D-Ab, E-Bb or F-B. These tritones form an octatonic diminished scale, making them even easier to find, since the minor third you started with belongs to the diminished chord that's the complement of the octatonic scale. This is another one of those things that's easier to see than it is to say.

The four options all produce different AITs, which pitch class set theorists call 4-Z15a, 4-Z15b, 4-Z29a and 4-Z29b. The 4-15s arise when the minor third and the tritone "interlock", whereas the 4-29s come from keeping them separate. They share such similarity of sound that people sometimes just refer to the AIT without differentiating between them -- theorists call such a collection of PC sets a "set class". Although it seems a bit abstruse, I think this is a useful approach; in practice, at least in my experience, fine distinctions generally take care of themselves. Grouping things together is just as helpful as dissecting them into parts.

This observation makes it suddenly very easy to play AITs improvisationally. A fun way to start with this is to evoke the dreamier parts of Elliot Carter's Night Fantasies, which to my ears seems to be made almost entirely from interlocking versions of these chords:

This is an example of the "Complement Union Property", in which (roughly) members of two non-overlapping set classes always combine to produce a member of another set class regardless of where you play them. This property seems to have been first noticed by music theorist Robert Morris in the early 1990s but for this post I found Guy Capuzzo's essay "The Complement Union Property in the Music of Elliott Carter" useful because it contains real musical examples. It's pay-per-view but you can get a PDF on SciHub.

Other ways to play an AIT include:

  • A minor triad with an added b9 (4-Z29a)
  • A major triad with an added #11 (4-z29A)
  • A major seventh with a whole tone interval positioned inside it, a minor third away from the nearest outer note (e.g. C-B with either Eb-F or F#-G# inside), giving 4-Z15a
  • A major seventh with a major third positioned inside it, a whole tone away from the nearest outer note (e.g. C-B with either D-F# or F-A inside), giving 4-Z29a
  • A minor seventh with a semitone interval positioned inside it, a minor third away from the nearest outer note (e.g. C-Bb with either Eb-Fb or F#-G inside), giving 4-Z15b
  • A minor seventh with a fourth positioned inside it, a semitone away from the nearest outer note (e.g. C-Bb with either Db-Gb or E-A inside), giving 4-Z29b

If you want the Elliott Carter sound, try voice leading between AITs so that each pair of consecutive chords has two notes in common. This is easiest to do by playing tritone + minor third, then moving only one of those while holding the other and repeating ad lib. But these chords also have plenty of promise as interesting colours for jazz and other music that keeps ties to the tonal system. After all, they're diatonic -- minor add b9 is Phrygian while major add #11 is Lydian. This also perhaps implies some ways into and out of this harmonic zone for those who like to mix up diatonic and 12-tone material.

Nevertheless, I'll leave you with one more Carter piece that uses AITs heavily, the Second String Quartet. If you find his music abrasive and alienating, maybe give it an hour and listen to the two pieces linked here after playing around with AITs a bit and see if it makes more sense: