The all-but-six chords

The all-but-six (ABS) chords are Forte 6-1, 6-8, 6-14 and 6-32. They're all six-note chords that contain every interval except the tritone. Of these, I think 6-8 and 6-14 are particularly promising.

Let's look at all four and see why I think 6-8 and 6-14 are the interesting ones:

• 6-1 is just a stack of six notes separated by semitones (e.g. C-C#-D-D#-E-F). While clusters like this have a place, they don't seem to me to benefit from deep study.
• 6-32 is six notes of the diatonic major scale (e.g. C-D-E-F-G-A), which is also not very fruitful to study. You can think of it as a c major triad plus a D minor triad, for example.
• 6-8 is a run of four semitones with a whole tone at either end (e.g. C-D-D#-E-F-G).
• 6-14 is very similar in structure to the ATH chords we looked at in a previous post, with groups of one, two and three notes (e.g. C-Db-Eb-E-F-Ab). This one was a favourite of Elliott Carter, who's been presiding over a few recent posts here.

On a keyboard we can play these "straight". 6-14 can be played easily by combining an augmented triad with a three-note whole-tone fragment (e.g. C-E-G# plus Db-Eb-F). As you'll see by experimenting, there's essentially only one way to do this: surround one of the notes in the augmented triad with a whole tone, then add another one above or below it.

Visualization

On guitar, a six-note chord rarely sits well on the fretboard and voicing options are very limited. We can start by writing out the two chords as arpeggios over the whole fretboard. I've chosen root notes to reflect a structural point I'll make in a second:

Both of these are made by combining a minor triad with another three-note chord that consists of a whole tone followed by a semitone (this is an inversion of Forte 3-2). So C 6-8 is C-D-D#-E-F-G, which is C-Eb-G (C minor) plus D-E-F, whereas C 6-14 is C-Db-Eb-E-F-Ab, which is F-Ab-C (F minor) plus Db-Db-E.

In both diagrams above I've made the minor triad A minor to make this relationship clearer; you just need to add the right 3-2. If we take the lowest note of 3-2 to be the root, we add it at the 2 (i.e. B-C#-D) to make 6-8 and at the b6 (i.e. F-G-G#) to make 6-14.

Here is a whole-fingerboard diagram for 3-2. Pick any selection of three notes of different colours to make a chord fingering; superimposing it over a minor harmony will give you one of the two ABS sounds:

Chords from 6-8

Obviously, 6-8 is very dissonant, right? It's a big stack of semitones! And surely any harmony that came out of it would be undifferentiated dissonant mush, right? But looked at another way, 6-8 is mostly diatonic: it's the first five notes of the major scale plus the b3, which means it's perfectly possible to pull out major, minor and suspended chords from it that are very consonant. The following example is rooted on C; play a set of same-coloured notes, or some of them in this case as playing all six is awkward:

So one way to think about 6-8 is as a non-diatonic expansion of quartal voicings, which are ambiguous between major and minor. It contains the Sus 2 and Sus 4 chords that are typical of quartal sounds, but also both the major and minor triads. That superimposition of major and minor means we also have the "major-minor chord" 1-b3-3-5, which you can think of as a Hendrix chord (e.g. C7#9) whose b7 has gone astray. That makes a huge difference to the sound. Here's the arpeggio of that chord along with some suggested voicings:

This chord is found in the Coltrane Cycle too. If you play CMaj7 and superimpose either EMaj7 or AbMaj7 (going up or down a major third) you introduce the b3, so you can play all three chords as minor-major chords without leaving the "home" Augmented Hexatonic scale. I'm inclined to think of 6-8 as material for an "extended Coltrane cycle" in which 6-8 moves in major thirds but I haven't yet had a chance to experiment with that.

Chords from 6-14

In a similar way we can see 6-14 as mostly diatonic: it starts off as 1-b3-5-b6, which is very Natural Minor-sounding, but then there are two sevenths, the minor and major versions. If you add just the minor seventh, it gets even more Natural Minor-ish (you've just missing the 2 and 4). But if you add just the major seventh, you get what sounds like a chunk of Harmonic Minor.

A minor triad with both sevenths stacked on top is 5-14, which I suppose I'm inclined to call "double seventh minor" -- we could see 6-14 as this with an added b6, which makes it quite easy to find but might not help us hear or apply it:

We can see it as Cm7 (C-Eb-G-Bb) combined with AbmM7 (Ab-Cb-Eb-G), overlapping on the Eb and the G; you could play the Ab as a bare triad instead, since the major 7 (G) is already present in the Cm7 chord.

For a different perspective, adding the major third gives us a seven-note scale that's a mode of Gayakapriya, a scale that comes up quite often around here:

Gayakapriya is related to the Coltrane Cycle -- hold that thought for a minute.

Here's one last oddball way to think of 6-14 that yields interesting results. We're still looking at 6-14 in C, but we're thinking of Ab major 6, which is the arpeggio outlined in green, along with he augmented arpeggio a semitone below it in red. They make an odd couple but, like many such, they rub along in interesting ways:

All of this is kind of interesting as far as it goes but it's not very decisive. They tell us what 6-14 is like but they don't give a clear picture of what it actually is. The next section provides a broader context that I think helps with that.

The Expanded All-But-Six Universe

It's a little bit striking that the two versions of 3-2 that we need are a tritone apart. Without going too far into a digression, we could combine these to make another six-note symmetric scale 1-2-b3-b5-b6-6, sometimes called the Petruschka chord because of Stravinsky's use of it. It's a subset of the Whole-Half diminished scale and is its own complement, which are very nice properties.

So we can think of superimposing of 6-8 and 6-14 so that their minor triads line up; the notes that aren't the minor triad form the Petrushka chord. I'd like to think of the "big picture" here being a Cm triad with the Petrushka chord suspended over it. That's a 9-note scale and the All-But-Six chords are just two of the many structures it contains -- in fact it contains almost all possible 6-note scales. Its complement is F#m triad, the minor triad that's a tritone away from C. There seems to be plenty of grist here for the compositional mill, and since these ideas are mostly quite easy to build from familiar blocks there's scope for creating very flavoursome improvisational vocabulary as well.

The Even Bigger Picture

The total chromatic of 12 notes can be partitioned into four sets of three. There are lots of ways to do that -- I have a post brewing about the ways to do it only with common triads, for example. In our case we're interested in the following partition: {C-Eb-G} (C minor triad), {D-E-F} and {Ab-Bb-B} (from Petrushka) and {F#-A-C#} (F# minor triad). We've been having fun combining these four trichords in different ways to get different effects, and the whole set of results is:

• {C-Eb-G} + {D-E-F} = 6-8
• {C-Eb-G} + {Ab-Bb-B} = 6-14
• {D-E-F} + {Ab-Bb-B} = Petrushka
• {C-Eb-G} + {F#-A-C#} = Petrushka
• {F#-A-C#} + {D-E-F} = 6-14
• {F#-A-C#} + {Ab-Bb-B} = 6-8

so the scales we've already looked at includes all the possibilities.

I'm interested in partitions of the 12 notes that are like this. By "like this" I mean the four trichords come in pairs of identical PC sets separated by a tritone. It seems clear enough to me that the list of these is nothing but the list of six-note scales that are symmetrical at the tritone, which is a very short list. They are (with example partitions starting on D, for consistency):

• The Whole Tone scale: 1-2-3-#4-#5-#6 {D-E-F# and Ab-Bb-C}
• Petrushka: 1-2-b3-#4-#5-6 {D-E-F and Ab-Bb-B}
• Inverted Petrushka: 1-b2-b3-b5-5-6 {D-Eb-F and Ab-A-B}
• Tritone Chromatic Scale: 1-b2-2-b5-5-b6 {D-Eb-E and Ab-A-Bb}

The last one's name is as per the book but it's not traditional. Here is how I'm thinking of these visually:

Whole Tone	Petrushka	Inverted Petrushka	Tritone Chromatic

Note that none of these besides Petrushka combines with a C minor triad -- they all overlap with it. That's expected; once you fix one pair of trichords we'll have limited choices for the other pair. So in each case you need to go hunting for your 20 options all over again.

It's at least somewhat interesting that Petrushka is one of only three scales that behave this way (or four, if you count the inversion separately). If you like these kinds of sounds there's a fairly large number of possibilities here that could be worth exploring. And I think this is one overcomplicated but suggestive answer to the question: "What is 6-14 really?"