We have been thinking about dividing the 12 notes of our ordinary tuning system (12-EDO) into two distinct regions, each of which is a whole-tone scale. Each region can be thought of as a tuning in itself, 6-EDO, the equal division of an octave into six notes. In this post we consider how the Augmented Hexatonic can help us bridge the gap between these two regions.

The whole tone scale is often thought of as the result of superimposing two augmented triads a whole tone apart; for example, C+ (C-E-G#) plus D+ (D-F#-A#) gives C Whole Tone (C-D-E-F#-G#-A#). The scale I call the Augmented Hexatonic (1 b2 3 4 #5 6) can also be thought of as a pair of augmented triads, but this time separated by a semitone. For example, C+ (C-E-G#) plus Db+ (Db-F-A) gives C Augmented Hexatonic (C-Dd-E-F-F#-G#-A). These two arrangements are the only possibilities for a pair of augmented triads; all other intervals yield one of these two or a perfect superimposition, as of C+ (C-E-G#) with E+ (E-G#-C).

Note that the Augmented Hexatonic has two modes whereas the Whole Tone only has one. In this series we are thinking largely atonally but this issue may come up as we go along. Where a root note matters, I will call the example above (C-Dd-E-F-F#-G#-A) the "short mode" of Augmented Hexatonic and the other mode (Dd-E-F-F#-G#-A-C) the "long mode", referencing the size of the first interval in each case.

We will need some notation, so I write:

• C#^W for the Whole-Tone scale on C# (C#-D#-F-G-A-B)
• C#^a for the short mode of the Augmented Hexatonic on C# (C#-D-F-Gb-A-Bb)
• C#^A for the long mode (C#-E-F-G#-A-C)

I'll refer to the long mode by default if the distinction isn't important.

The idea of inserting this new collection into the whole-tone harmony is that C^A contains three notes from C^W and three from C#^W. Furthermore, these form augmented triads, which are very characteristic of whole-tone harmony in general. So these scales are the perfect choice to act as "bridges" between the whole-tone regions.

(Incidentally, a year or more ago I wrote some blog posts about expanding C^A into a seven-note collection, which can be done in exactly two ways; this one contains some theory and this one contains some guitar chords and other observations. This "system" can be seen as a very usable extension of what we're describing here, although it takes us away from the sound of whole-tone harmony that I'm trying to capture.)

Relationship of the Augmented Hexatonics to C^W

Each whole-tone region has two augmented triads to contribute to the formation of an Augmented Hexatonic; in C^W these are C+ and D+. For each we can add an augmented triad either a semitone above or below, making four possibilities:

• C+ and D#+ = C^A: C-D#-E-G-Ab-B
• C+ and C#+ = C#^A: C#-E-F-G#-A-C
• D+ and C#+ = D^A: D-F-Gb-A-Bb-C#
• D+ and B+ = Eb^A: Eb-F#-G-A#-B-D

We're choose one augmented triad from one whole-tone region and adding another from the other region, which will necessarily be either a semitone below it or a semitone above. Depending on your instrument it may be useful to practice the following steps:

1. Visualize the two intertwined augmented triads in each of C^W and C#^W
2. Visualize C#^W as the "holes" in C^W (while playing the latter) and vice versa
3. Visualize one of the two intertwined augmented triads from C#^W while playing C^W and vice versa
4. Play one triad from C^W and one from C#^W while visualizing C^W (ditto for C#^W of course).
5. Visualize the two intertwined augmented triads in C^A, then in the other three Augmented Hexatonics.
6. Visualize the two intertwined augmented triads in C^A while playing C^W, and so on with the other combinations.

Of course, simply memorizing the four Augmented Hexatonics is also necessary, but visualizations like these really help to orient you in these shapes and "see" their overlaps, which are crucial for using them in whole-tone harmony.

Augmented Hexatonic Chords

The most distinctive chords contained in C^A (C-D#-E-G-G#-B) are the major sevenths: CΔ (C-E-G-B), EΔ (E-G#-B-D#) and G#Δ (G#-C-D#-G). Since these are strongly suggestive of tonal harmony -- in particular, jazz harmony -- they must be treated with care. But they are of great structural value, since they're four-note chords that each contain two notes from one whole-tone region and two from the other. They can therefore be treated as chords of tension; they "want" to resolve to pure whole-tone chords. This is in complete contradiction to the usual situation in jazz where whole-tone harmony has a dominant function and major sevenths are chords of resolution.

We'll pick out some noteworthy 3- and 4-note chords; I will give one example spelled out as an element of C^A but note that any chord will appear as three copies at major third intervals from each other.

Ignoring the augmented triads, diatonic chords (e.g. triads) or fragments thereof (e.f. C-E-B, the root-shell voicing of CΔ), the noteworthy three-note chords can all be considered a semitone plus another note -- I give one example of each:

• Minor and Major Thirds: C-D#-E
• Minor and Major Sixths: D#-B-C
• Fifth plus Augmented Fifth: C-G-G#
• Major Third plus Fourth: B-D#-E

These combinations are distinctive of octatonic harmony and are not found at all in pure whole-tone harmony. The connection is that octatonic harmony (so-called "alpha harmony") comes from the Half-Whole diminished scale, which is formed by alternating semitones with tones. The Augmented Hexatonic scale comes from alternating semitones with minor thirds, and minor thirds are themselves characteristic of the octatonic universe.

The second of these can be thought of as the generator of C^A: play C-D#-E and then repeat the pattern starting on E (E-G-G#) and so on. In the same way, D#-B-C generates B^a. This Slonimsky-like approach -- decorating the notes of an augmented triad with three-note cells -- generates melodic patterns; see Chapter 9 of my Slonimsky book (in those days I called the Augmented Hexatonic the "1dom + b6maj Scale"; no, I don't really know what I was thinking either).

Chords 3 and 4 on the list above are diatonic subsets; C-G-G# respelled as C-G-Ab belongs to C natural minor and B-D#-E belongs to B major. So these might be more useful in combinations than alone.

Turning to the four-note chords, I'll again leave out those with obvious tonal interpretations as well as those that seem to me to be strongly characteristic of octatonic harmony. We'll deal with that separately; here I'm in search of the "Augmented Hexatonic sound", which I hear particularly in these two:

• Major Thirds a Semitone Apart: C-D#-E-B
• Major add b13: C-E-G-Ab

The first is especially interesting since each takes two notes from one whole-tone region and two from the other. It's an almost magical chord in this context that acts as an ideal pivot between them.

There are significant overlaps between the four-note chords in this scale and those in the Half-Whole Diminished; I've omitted those but they're genuinely useful in the whole tone context. I'm doing my best to keep that relationship separate from the present one, though, in order to bring out what's distinctive about the Augmented Hexatonic.

[EDIT: The original version of this post set a new personal best for typos per square inch. Hopefully I've fixed most of them.]