Continuing from my previous post, this is an exploration of the tritone sub hexatonic (e.g., C-Db-E-F#-G-Bb) and its complement (e.g. D-Eb-F-G#-A-B). They're the same pitch class set (6-30) but feel and sound very different on the instrument.

What are we talking about today?

Here are full-fingerboard diagrams of the Tritone Sub Hexatonic (top) and its complement (bottom), with characteristic triad arpeggios highlighted in red and green:

The top structure is the union of a pair of major triads a tritone apart -- here, C major (C-E-G) and F# major (F#-A#-C#) combine to make C-Db-E-F#-G-A#. The bottom structure is its complement (i.e. all six notes that we haven't already covered), which can be thought of as a pair of minor triads a tritone apart -- in this case D minor (D-F-A) and G# minor (G#-B-D#) combining to make D-Eb-F-G#-A-B.

Besides being transpositionally symmetrical (each is identical to itself when transposed up a tritone, which is obvious by the way they're constructed), these are also inversions of each other. They both have the Forte number 6-30; this is not very surprising since one is made out of major triads and the other out of minor triads, and the major and minor triad share the same set class. That of course doesn't mean there's no musical difference between them!

For the sake of having names for them, I'll use TSH to abbreviate "tritone sub hexatonic" and call the one made from major triads "Major TSH" and its complement "Minor TSH". However, it's worth noting that Major TSH already has a name, the Petrushka Chord, due to its extensive use in Stravinsky's ballet of the same name.

Chord tones and tonal applications (or lack thereof)

In terms of chord tones, Major TSH can be written 1-b2-3-#4-5-b7. In the previous post I pointed out that this is, like the Whole Tone scale, neither fish nor flesh when it comes to jazz harmony; there's a natural fifth and #11, pointing towards Lydian dominant, and a b9, pointing towards altered.

Minor TSH can be written 1-b2-b3-#4-5-6, making this a mixture of two Dorian-related scales, minus the seventh: Dorian b9, which is a mode of Melodic Minor, and Dorian #11, which is a mode of Harmonic Minor. This combines the spicy notes from both without the minor seventh cluttering things up. If you want to play it in a jazz context, I don't see why it couldn't be made to work on the ii in a ii-V-I.

The two scales form a complementary pair when Minor TSH is played a whole tone above Major TSH. This would be like, for example, playing D minor and Ab minor over C major. The D minor is diatonic but does contain the 4 (11) which jazz college types will tell you is an "avoid note"; the Ab minor contains the #9, b13 and major 7, which is a weird combination. I'd be inclined to interpret this as a sort of more bluesy harmonic-major-ish sound, which is weird if you're also playing Major TSH in the root position as that's a dominant sound. So you'd be implying C7 resolving to C Maj 7. That's not impossible, but it's pretty far out.

Or you can think of it the other way -- you're playing Minor TSH and looking at its complement, Major TSH a whole tone below. Staying in C, that's like playing Bb major and E major over C minor. Again, one of these is diatonic and very "tame", giving the upper structure b7, 9 and 11. That's not likely to sound jazzy but it can provide a very inside sound. On the other hand, E major over C minor is lunacy -- you get the b11, b13 and major 7 on a minor triad, which is quite the combination. More Bartok than Bird, that's for sure.

Twelve-tone applications

Of course, the reason for looking at hexatonics and their complements isn't to play bebop, it's to play music using all twelve tones in constructive and interesting ways. That's often about finding ways to recombine the same intervallic material -- in this case, a specific collection of plain old major and minor triads -- in different ways.

Thnking of them as

• C Major TSH = C maj + F# maj
• D Minor TSH = D min + Ab min

we're encouraged to try the other parings (there are only two because of transpositional symmetry):

• C maj + D min = C D E F G A (C major scale without the 7)
• C maj + Ab min = C D# E G Ab B (C Augmented Hexatonic)

The first one isn't terribly interesting, although being able to pivot to and from a diatonic sound is actually quite useful if you're not making "pure" atonal music, so it's still worth being aware of. The second one sends us somewhere else, into the augmented hexatonic or "Coltrane cycle" language that I've written about a fair bit on this blog already. To tie in with the way I usually think about that, I'd see this as playing the minor triad a major third below.

So I think there's a little collection of things to practice here to make this material really sing, and to integrate it with other things I already like to play. I don't think there's a strict sequence to these and they'd probably benefit from being worked on in parallel:

• Major TSH as a dominant sound: on a major triad, play the major triad a tritone away
• Minor TSH as a minor sound: as above but with minor triads
• Major TSH and Minor TSH a whole tone apart (minor above major) as a cover of all 12 notes
• Major triad with minor triad a whole tone above: diatonic (but think of this as "part of major TSH with the corresponding part of minor TSH")
• Major triad with minor triad a major third below: augmented hexatonic (but think of this as "part of major TSH with the tritone-away part of minor TSH")

The idea of covering the 12 notes with four triads is something I've mentioned here before but not dug into much; another way to do it is exploited in Berg's Violin Concerto to create lyrical, Romantic-sounding music that is atonal but often maintains a localised illusion of tonality:

Pentatonic departures

Incidentally, that diatonic scale without the seventh is its own complement and can be thought of as just the major pentatonic with an added fourth; so another way to travel to the TSH region is starting from pentatonic language.

To start with (I guess) the most familiar example, consider the blues application of minor pentatonic on a dominant chord. Adding the b13 bumps this up to the diatonic subset we've been looking at. If the root is A, we have A minor pentatonic (A-C-D-E-G) plus F (A-C-D-E-F-G), which is a mode of the one rooted in C we were working with above. We can think of this as C major plus D minor, and playing those arpeggios keeps us in that minor pentatonic language. It doesn't sound much like the blues any more, though -- in part because of the b13, and in part because we're not used to thinking of triads providing upper structures in that context.

Now, We can shift to Major TSH by switching the D minor triad out for F# major, which in the context of a tonic A gives us (major) 3, b9 and (natural) 13. The 3 and 13 are bluesy; the b9 is a note of tension that gives a Phyrgian flavour to things. Or instead we could shift to Minor TSH, retaining the D minor but swapping C major for an Ab major. Again in context that gives us b5, (major) 7 and (natural) 9, which you can play in a bluesy way if you wish (the b5 is of course the note that makes the "blues scale").

The point is that by listening and playing with care for the genre and context of the music can allow you to step into the 12-tone world without just abandoning everything familiar for apparent chaos. Yes, some of these notes are more sour and less "inside" than others but this context provides a kind of logic for them that's different from the usual "outside note that needs resolving by a half-step" kind of approach. That approach may even seem timid by comparison and also a bit arbitrary, since the "blue notes" aren't part of anything larger; they simply visit minor pentatonic briefly, then depart again.

As another example, consider now playing on A maj 7 and using the commmon trick of playing the minor pentatonic scale a semitone below. So on A maj 7 (A-C#-E-G#) we play G# minor pentatonic (G#-B-C#-D#-F#), which on its own implies the Lydian mode. We'll do the same trick again. First, we replace G# minor pentatonic with the combination of B major triad and C# minor triad, which just adds the fifth back into the scale that was in the chord already.

Now we drift over to Major TSH, replacing the C# minor triad with F# major (F#-A#-C#). This introduces the b9 and 13 (as well as reinstating the 3). The story is similar: the 13 is a very inside sound on A maj 7 but the b9 is rather dissonant and needs care (I do like this sound when it works, though).

Alternatively we can shift to Minor TSH, playing C# minor with G minor. The G minor triad changes the mood dramatically, since G is the b7 of A and the overall sound turns from tonic to dominant. The other notes are the natural 9 and 11, making this a peculiarly plain sound that produces dissonant results, functionally similar to playing the Mixolydian scale on a major 7 chord.

I think these suggest one way to start using these ideas in real musical situations. Of course, care and listening are all-important: these are 12-tone structures that provide more than enough banana skins should you wish to fall flat at your next jam session. They're more like ships bound for distant lands that aren't really convenient for day trips.