The Power of the Circle of Fourths – A Hero’s Continuum

As musicians we have all had our interpretive encounters with the famous Circle of Fourths (or fifths, depending on how you look at it). Western classical music anchored the importance of this harmonic cycle by embedding in it the very key signatures we use to delineate tonal areas.

We have entire forms and episodes of music built on the foundational momentum gifted to us by dominant cycles: developmental sections in sonatas and concertos, rhythm changes B-sections in jazz, and, lest we forget, the ubiquitous II – V – I progression that has been spicing up cadences in just about all genres for decades.

There’s no denying the staying power and structural fundamentalism conveyed through the circle of fourths: it is, in no uncertain terms, the Rosetta stone of western music.

The 21st century, however, has seen the emergence of a new (or old, depending on the context) harmonic approach that has insidiously supplanted the circle of fourths: “neo-Riemannian Transformation” (nRT), as reinvigorated by David Lewin (1987). Whether disguised under the myriad pop renditions dubbed “The Axis of Awesome” (VIm – IV – I – V) or flaunted within the vertically-padded, ostinato-driven, electronically-sampled and minimalist-triadic approach favoured by many modern film composers, nRT’s philosophy of basic triads connected via tonal interactions derived from symmetrical nodal networks abandons the idea of a tonal centre and, with it, the structural implications accompanying the circle of fourths.

Incidentally, the circle of fourths may be gifted its own network within the nRT (T5 or DOM) but, while this approach to harmony has yielded vast orchards of fruit when it comes to aesthetic exploration, atonal and chromatic composition, film scoring and popular music, a basic tenet of music has been lost, in my opinion. The reduction of harmony to mathematical – and essentially functionless – tonal interactions has diminished (pardon the pun!) the narrative power potential of which music is capable i.e. “The Hero’s Journey”.

Pioneered by Joseph Campbell (1949), this narrative philosophy has influenced plays, books and films for decades. Simplified, it refers to the various stages taken by the hero of a classic adventure tale, and how said stages result in a roughly 3-act structure: DEPARTURE – INITIATION - RETURN. For the purpose of adapting this journey to the harmonic potential of the circle of fourths, I’ll redefine them as: HOME – CONFLICT – RESOLUTION. When put into harmonic terms these translate to: TONIC – PRE-DOMINANT/SUBDOMINANT – DOMINANT.

Any western-trained musician will recognize these as the three primary harmonic functions within the major scale. They represent the starting and ending point of the home key (TONIC), the foreign excursive area immediately resolved to by the tonic in the circle of fourths (PRE-DOMINANT/SUBDOMINANT), and the penultimate cadential area that resolves back to the tonic (DOMINANT) i.e. [I – IV – V].

In my second year studying jazz theory at university, I became obsessed with solving a mystery surrounding a basic harmonic concept in jazz music: The manners in which a dominant chord may resolve. I expect anyone reading this could list at least two or three possible ways, but it might surprise you to know that there are, in fact, SIX in total! We all know about the “perfect cadence” ([V – I]), jazz musicians are aware of “tritone substitutes” ([bII – I]: the butter to jazz’s bread!), and perhaps some may have come across the “subdominant minor cadence” (IVm – [bVII – I]). The full range of possible dominant resolutions is as follows:

While the “perfect cadence” ([G7 – C]) and “tritone substitution” ([G7 – Gb]) made sense back then, if for no other reason than they were drummed into me ad infinitum, the other resolutions mystified and intrigued me --- there had to be some hidden force connecting them! MODAL BORROWING and CHROMATIC COMMON-TONE MODULATION were offered as place-filler solutions but, much as with neo-Riemannian Transformation, these substitutions and modulations merely exploited individual tonal relationships existing across chordal interactions i.e., form without function. There had to exist some functional driving force that lent aesthetic and narrative value to these harmonic relationships. It wasn’t until years later that I came across Darryl Lee White’s theory of the “Octatonic Metaphor” (2006), and my own personal hypothesis was confirmed.

If we picture the circle of fourths as a clock, we may notice a pattern with the first four potential resolutions (target areas) from the example above:

These four areas are symmetrically connected at 3, 6, 9 and 12 on the clock. Note that the relative minors happen to coincide with related positions of the same set e.g., C and Am. The same connection exists for the other nine areas i.e. [F, Ab, B, D] and [Bb, Db, E, G]. White identified these areas as being unified by the octatonic scales – half-whole and whole-half diminished – and pointed to the inherent functional interchangeability connecting them. The metaphor goes further to group these three sets into the very categories associated with the “hero’s journey”: TONIC (T), SUBDOMINANT (S) and DOMINANT (D):

This approach accommodates both directions of the clock i.e., “perfect cadence” ([D – T]: clockwise) and “plagal cadence” ([S – T]: anti-clockwise). The actual potency of the metaphor, however, lies in the user’s (me and you) freedom to instantly transfer between both function (T, S or D) as well as octatonic set members (e.g. [C, Eb, F#, A]). Thus, we end up with a continuum of many potential possibilities and variations for our “hero’s journey”. It is for this very reason that I began to use the phrase “CROSS-MODULATION” (notice the cross shape in the clock above?) instead of modal borrowing or chromatic common-tone modulation.

Practically speaking, the functions are broadly distinguished by their presented harmonic forms/voicings:

• Tonic: major6, major7, 6/9, minor6, minor/major7, minor6/9

• Subdominant: minor7, minor7b5, diminished7 (descending)

• Dominant: dominant7, diminished7 (ascending), half-diminished7 (as a rootless 1st inversion dominant7)

Some of these functions may overlap depending on the stylistic context e.g., in the Blues a tonic chord may be dominant. Generally speaking, these functional idiosyncrasies will determine where you are in the journey i.e., TONIC, SUBDOMINANT or DOMINANT:

There is also noticeable functional ambiguity regarding, for example, a minor6, minor7b5 and half-diminished chord. The organic fluidity present herein is highlighted by the fact that certain chord voicings are, in fact, identical to others but are distinguished by inversion (bass note) and/or enharmonic spelling:

In practice, the user determines the function based on the voicing style and spelling, and then decides on the target destination based on said function. This destination may be as conventionally expected: e.g. [G – C] (perfect cadence) or [G – D] (plagal cadence); but may also be unexpected based on the various available options in the octatonic set: e.g. [G – A] or [G – F]. Likewise, one may wish to simply change the TONIC set (HOME) by cross-modulating to another harmony within the same set: e.g. [G – Bb]. This then also allows a change of function within the same area: e.g. [G6 (T) – Bb7 (D)].

Let’s use as an example the most basic harmonic progression and treat it using the octatonic metaphorical lens:

May become…

In the above example the Cmaj7 (TONIC) is replaced by A7. This A7 still exists within the TONIC set but has changed function to DOMINANT and essentially becomes a V7 of the harmony that follows (Dm7) - also known as a secondary dominant. The Dm7 is a SUBDOMINANT structure from the SUBDOMINANT octatonic set, but it may just as easily have changed its structure to TONIC (Dmaj7; see “Don’t you worry ‘bout a thing” by Stevie Wonder) or DOMINANT (D7alt), whilst still retaining its overall functionality as a SUBDOMINANT constituent (from the set of D, F, Ab & B) in relation to the TONIC set (C, Eb, F# & A). A more advanced example may include multiple set and function changes:

As for the remaining two dominant resolutions ([G7 – Ab] & [G7 – E]), a slight amendment to the octatonic metaphor was in order i.e., “HEXATONIC CROSS-MODULATION”. This approach instantly substitutes the TONIC set chord with one of either two common-tone alternatives derived from the hexatonic and/or whole-tone scale:

This hexatonic cross-modulation allows for an unexpected departure away from the tonic area, as you’ll notice that each of the substitutions – E & Ab - are in the DOMINANT and SUBDOMINANT sets, respectively. Like a deceptive, interrupted or incomplete cadence, this offers the option of landing on a tonic-sounding chord (TONIC function) but in a non-tonic set which happens share common notes with the original tonic chord:

Conversely, cross-modulating a DOMINANT function from the TONIC set with one of these hexatonic substitutes bestows additional driving power to an already interesting progression:

If you’re thinking that G7 resolving to E is strange (from the original dominant resolution possibilities), you’d be justified. This resolution, while not unheard of, is somewhat redundant. In the above example, substituting E7 with its other hexatonic substitute would result in Ab7 resolving to F i.e., SUBDOMINANT to SUBDOMINANT. While the voice leading proves to be smooth between these chords (go and try it!), this resolution is peculiar at best and superfluous at worst. This may be because, according to White’s metaphorical model, subdominant chords have a natural falling motion (i.e., plagal cadence) whereas dominant chords have a natural rising motion. Also, these two chords share several notes, meaning that the “plagal effect” that would result in this instance can’t be “amplified” (as with the DOMINANT effect in the above example) but would become “static” instead:  

Nevertheless, even stasis has its appropriate place in a narrative context…

Ultimately, that ‘dominant dial face’ reveals itself as a veritable clockwork of limitless creative potential, offering myriad variations of progressional direction, chordal function and form, and narrative-driven harmonic movement. I encourage you to experiment with this and it is my sincere hope that you’ll discover new realms and possibilities for composition, reharmonization and improvisation. Above all else, remember that music exists temporally, and is a journey of emotion and story… a hero’s journey!