This post is a continuation of the last one in which we look at supplementing the rather limited whole-tone harmony with one or more added notes. The resulting chords take us away from "pure whole-tone harmony" in the same way that, for example, secondary dominants and other borrowed chords take us away from "pure diatonic harmony" while simultaneously enriching it.

Fortunately some of the combinatoric work has already been done. The notes we can add to a whole-tone scale form the other whole-tone scale, so the question of which distinct choices we have is the same as the one answered in the previous post. For the sake of brevity I'll call these added notes "foreigners".

We'll only go through a few of the possibilities here, in particular those that involve adding one or two notes to a whole-tone chord or three or four notes. These sounds stay recognizably close to the "home" whole-tone scale. Adding more foreigners, or indeed piling up many notes in a single chord, risks taking us so far from the "pure" sound that the connection with it no longer seems useful. (Which is not to say such chords can't be useful in their own right of course.)

Adding a Single Foreigner

Adding one note is surprisingly fruitful. It yields the Neapolitan scale, 1-b2-b3-4-5-6-7. All the chords already discussed can be extended by the addition of (any) one note that isn't in the whole-tone scale and we have Neapolitan harmony. When the added note is a semitone away from one or two other notes in the chord, the overall roughness of the sound is obviously increased. Still, to my ears most of these sonorities, even the very rough ones, are significantly sweeter than they would be without the added note, especially if they're not voiced too closely.

Some of the resulting sounds also belong to octatonic harmony; that's a topic for another series of posts.

Clusters with Foreigners

The 3W-cluster (e.g. C-D-E) can have three kinds of foreigner added to it: two that are within the cluster (in this case, C# or D#), two that abut it (B or F) and two that lie outside it (G or A). These are very different: the first create clouds of dissonance whereas the last are diatonic and very conventionally pretty -- the major add 9 and the minor add 11. The middle cases are in between; they are "diatonic clusters" that are tonally ambiguous but still very familiar-sounding thanks to their containment in the major scale.

The 4W-cluster (e.g. C-D-E-F#) allows three foreigners within (C#, D#, F), two abutting (B, G) and one outside (A). Again, the second and third classes are all diatonic and can be interpreted tonally.

The insertion of a "foreigner within" increases the roughness of the cluster and changes its character without containing any obvious tonal implications; these three chords are very interesting and sound wildly different depending on the voicing. One way to think of these is as 3S-clusters (that is, clusters of three notes separates by semitones) with two additional notes a whole-step away:

• C-D-E-F# plus C# is a 3S-cluster with two whole-tones above
• C-D-E-F# plus D# is a 3S-cluster with a whole-tone on either side
• C-D-E-F# plus F is a 3S-cluster with two whole-tones below

Note that all three also contain minor seconds and minor thirds and the first also contains a perfect fifth (F#-C#). All these intervals are "foreign" to pure whole-tone harmony.

The Augmented Triad with a Foreigner

Only two different chords can be made this way:

• C-E-G# plus C# -- can be thought of as C# minor maj7
• C-E-G# plus D# -- can be thought of as G# major add b13

I've given "tonal" interpretations of these as it might make them easier to find but these are relatively rare sounds in functional harmony. Although not unknown there, without any other context they certainly don't seem to pull the listener towards a tonal interpretation.

They're inversions of each other and as pure PC sets they have Forte numbers 4-19A and 4-19B. To me they really do seem to be essentially the same chord; at least for the moment I'll consider adding a foreigner to an augmented triad to be a single sound. Different voicings emphasise the characteristic semitone to different degrees.

The Augmented Add 9 with a Foreigner

Because the starting chord is asymmetrical, there are more possibilities and in fact these sounds are much more interesting than they deserve to be.

• C-D-E-G# plus C# -- a "foreigner within"
• C-D-E-G# plus D# -- a "foreigner within"
• C-D-E-G# plus F -- a "foreigner on the threshold"
• C-D-E-G# plus G -- a "foreigner on the threshold"
• C-D-E-G# plus A -- a "foreigner outside"
• C-D-E-G# plus B -- a "foreigner outside"

The first two have the character of cluster-like chords consisting of a mixture of semitones and tones, with another note a major third suspended above it. The middle two have the character of five-note clusters, again mixing semitones and tones; each does contain a minor third but to my ears it is a bit "swamped" by the other notes.

The last pair are the sweetest-sounding, perhaps because they contain perfect fifths. The first can be thought of as a perfect fifth (D-A) interlinked with an augmented triad (C-E-G#). In the case of the second the perfect fifth (E-B) is not so useful for visualization since the remaining notes (C-D-G#) are not particularly familiar; it could be "seen" as a major third (C-E) interlinked with a diminished triad (G#-B-D) or as a dominant chord (E-G#-B-D) plus the b13.

Adding a Foreigner to the Suspensions

By the "suspensions" in whole-tone harmony I mean these chords:

• The "sus 2 b5": C-D-F#
• The "sus 2 #5": C-D-G#

I will use the same (admittedly silly) terminology as above -- first for "sus 2 b5":

• C-D-F# plus C#, Forte 4-5A -- a "foreigner within" (fifth = F#-C#; surround the top note by semitones)
• C-D-F# plus D#, Forte 4-12A -- a "foreigner within" (no fifth)
• C-D-F# plus F, Forte 4-z15B -- a "foreigner on the threshold" (fifth = F-C; add the major third a major sixth above)
• C-D-F# plus G, Forte 4-16B -- a "foreigner adjacent" (fifth = G-D; add the tritone a semitone below)
• C-D-F# plus A -- a "foreigner outside" -- but this is just D7
• C-D-F# plus B, Forte 4-z29B -- a "foreigner adjacent" (fifth = B-F#; add the major second a semitone above)

All but one of these contain perfect fifths that make them sweeter than they ought to be, and more interesting. We might think of these (excluding the D7) as four ways to ornament a perfect fifth. These deserve more attention.

The second is the odd one out. It can be seen as a diminished triad with an added 9. The fourth option is not of interest since it just rediscovers a normal dominant seventh chord.

Only the first three are non-diatonic (the others are subsets of the G major scale). The last of the lot, Forte 4-z29B, is an all-interval tetrachord. Apparently this is an important feature of Carter's Night Fantasies, which immediately makes it interesting to me:

The "sus 2 #5" options are similar but a bit different:

• C-D-G# plus C#, Forte 4-5B -- a "foreigner on the threshold" (fifth = C#-G#; surround the lower note by semitones)
• C-D-G# plus D#, Forte 4-12B -- a "foreigner within" (fifth = G#-D#; can be thought of as G# add #11: G#-B#-D#-D)
• C-D-G# plus F, Forte 4-27 -- a "foreigner within" (fifth = F-C; but this is just Fm6: F-Ab-C-D)
• C-D-G# plus G, Forte 4-16A -- a "foreigner on the threshold" (fifths = C-G-D with one semitone above, could also be thought of as G#-B#-D-F##, which is G# maj 7b5)
• C-D-G# plus A, Forte 4-z15A -- a "foreigner adjacent" (fifth = D-A)
• C-D-G# plus B, Forte 4-z29B -- a "foreigner adjacent" (C augmented with a major 7)

The dominant 7b5 with a Foreigner

This one can be dealt with rapidly, since 7b5 is already a familiar chord:

• C-E-F#-A# plus C# -- C7b5b9 (Forte 5-31)
• C-E-F#-A# plus D# -- C7b5#9 (Forte 5-32)
• C-E-F#-A# plus F -- C7b5 add 11 (Forte 5-29)
• C-E-F#-A# plus G -- C7#11 (Forte 5-28B)
• C-E-F#-A# plus A -- C7b5 add 13 (Forte 5-28A)
• C-E-F#-A# plus B -- Forte 5-15

The first two of these are familiar as jazz altered dominants. Because of this, and since much of whole tone harmony already has a strong "dominant character", these seem not to be particularly useful. Certainly the two inversionally-related ones, Forte 5-28A and Forte 5-28B, share a common sound, but again 7#11 is a core jazz sound.

The most promising for a non-tonal language therefore seem to be Forte 5-29 and Forte 5-15. These are worthy of further investigation but care will still be needed to voice them in a way that doesn't just imply a dominant function.

More Foreigners

The available pairs of foreigners are C#-D# (major second), C#-F (major third) and C#-G (tritone). Of course, this means we have up to three times as many possibilities to consider, and grinding through all of them seems unhelpful.

As a general rule, any whole-tone harmony with a major second added will retain its whole-tone character with added ambiguity; are we sure which whole-tone region we're in? Adding a major third has a similar effect but is likely to create less roughness and risks re-inventing familiar sounds from tonal music. Adding a tritone risks exacerbating the "dominant character" of whole-tone harmony even when the tritone comes from the other whole-tone scale.

Structurally, these may be useful when working with whole-tone harmony by creating an external "axis" that exerts a more or less strong gravitational pull, as we see sometimes in Bartok. Treating such a dyad almost as the tonic chord gives an interesting effect where the main harmonic materials do not contain the tonic at all, and chords of "resolution" are likely to contain both tonic and non-tonic elements. Improvising in this way adds some potential for ebb and flow (even a sort of "tension and resolution") to the pure whole-tone language without trivializing it.

Alternatively, adding two foreigners to a 3-, 4- or 5-note chord begins to create ambiguity about which of the two whole-tone regions we're currently in and provides a smooth stepping-stone from one to the other. This is how I usually think of these sounds; as liminal regions between the two "home keys" that are the pure whole-tone collections. Some delicious sounds can be found here, but there are also "dangers"; paths that can lead you away from whole-tone harmony entirely. Of course, sometimes you want to follow those will-o'-the-whisps and sometimes you don't.

It's odd that both of these views make sense to me despite them being contradictory. One sees these chords as points of rest, the other as transitional. Perhaps that's the heart of this kind of harmonic language: every sound is simultaneously both and neither.