Giant Steps, Central Park West, and modulatory cycles


Along with his intellectual and creative gifts, John Coltrane obviously had a knack for finding just the right titles for his compositions. Giant Steps and Central Park West both employ modulatory ‘cycles’ — but they do so quite differently. His Giant Steps are the bold, breath-snatching modulations by which he audaciously displays the song’s underlying conceptual structure. By contrast, Central Park West is a sensuous, reflective walk in the park, in which structure quietly serves poetic expression. What the song titles don’t tell us, is the how and why — for these, we must look to the music itself.

Both Giant Steps and Central Park West owe their structures to equal subdivisions of the octave. Since an octave spans twelve semitones, it can be divided into two, three, four or six equal parts: divided by two, it produces two tritones; by three, it produces three major thirds; by four, it produces four minor thirds; by six, it produces six whole tones. Such equal subdivisions of the octave have been around for some time — Franz Schubert – in 1825 – based a series of modulations on them, in his Symphony in C major. However, for most of its history, western music has not used such symmetrical subdivisions, simply because most of the elements in our diatonic system of music divide the octave asymmetrically.

Giant Steps partitions the octave into three major thirds, so that it moves through the keys of B major, G major and Eb major. The tonics of these keys form a descending cycle of major thirds, that, when completed, form an equally-subdivided octave: B-G-Eb-B. Coltrane draws this cycle out, over two phrases: the song’s first phrase progresses through B, G and Eb; the second phrase returns to G, and then cycles down through Eb to B:

Central Park West divides the octave into four, producing an ascending cycle of minor thirds: B – D – F – Ab – B. Coltrane makes these the keys of his tune, and moves through the cycle one and a half times during its course. He slightly alters the cycle’s order, so that it becomes B – D – AbF – B:

None of these keys are closely related: each stop in Giant Steps' cycle of major thirds has four fewer sharps than the last, and there are three fewer sharps in each stage of Central Park West's cycle. If brusquely juxtaposed, these distantly-related keys produce jarring, angular modulations. Skillfully place them within a well thought-out design, though, and they can create profound and surprising beauty. Coltrane does both, applying the former approach in Giant Steps, and the latter in Central Park West. By examining his technique, we can learn how to go beyond the idea of modulatory cycles, to actually using them with artistic intent.

Giant Steps

The ‘giant steps’ of Giant Steps are the actual transitions from key to key, in which the distances between these disjointed tonalities are highlighted, rather than eased. Although there are ten modulations in the piece, all are derived from one of two basic designs — one for descending through the cycle of major thirds, and another for ascending through it:

Let’s begin with the ascending modulations, which retain features of more traditional harmony, and are therefore easier to grasp. The first of these connects the tonic of Eb major (bar 3) with the tonic of G major (bar 5):

The two keys are related via Eb major’s tonic chord, which is simultaneously the VI chord in the key of G minor. Using mixture1Mixture: the use of chords belonging to a major key’s parallel minor, and vice versa., the progression moves from G minor to G major, making the change at the ii chord:

The same procedure is used to modulate from B major to Eb major (bars 7-9), and G major to B major (bars 11-13), these being simple transpositions of the prototype.

The descending cycle presents a different picture. Here the tune wastes no time, completing the first step of this cycle, from the key of B major down a major third to the key of G major, in three chords:

The modulation is stripped down to almost nothing: apart from giving G major its V, Coltrane has done little to smooth the connection between the two keys. (In fact, the D7 prevents the use of a relatively effective smoothing technique, called common-tone modulation.) The transition between B major’s five sharps and G major’s one is sudden, drastic, and unmitigated.

Degrees of relation could be established between the chords involved — for example, the chord of G major is related to the key of B minor, where it is the VI chord — but Coltrane eschews any such attempt: he removes all possibility of key relation by immediately modulating to an even more distant tonality:

G is but a transient stop in the modulatory cycle, which immediately moves on to the key of Eb major, via that key’s dominant. This second modulation is an exact transposition of the first, and is just as abrupt and unadorned. The same pattern is repeated in two subsequent descending modulations, from G to Eb (once more, in bars 5-6), and Eb to B (bars 6-7).

At the end of the piece (bars 15 – 1), Coltrane employs a variant of this modulatory design. While performing a descending modulation from Eb to B, he borrows from the ascending cycle’s design, by giving B major both its ii and V:

There is now an opportunity for key relation, in that C#-7 suddenly casts Eb (=D#) as V of G# minor. Since G# minor is the relative minor of B major, there is potential for a closer relation between Eb major and B major:

This potential could be clearly realized were the melody to borrow a feature of common-tone modulation, and sustain one note so that it is common to all chords:

Here, the exposed, common-tone D# (the top-most note in the voicings) unifies all parts of the modulation, allows us to hear the tonal implications of pivot chords, and smooths out the transition between the keys.

Coltrane works against this calming effect, though, giving his melody a prominent leap at the very moment when it should maintain its pitch. Even more, the notes of his leap (BbF#) threaten to create a cross relation with the third of the immediately preceding Eb chord.

Why does Coltrane refrain from smoothing out these jarring modulations? Why does he sidestep those techniques and melodic changes that would produce more fluid transitions? Here we arrive at the crux of the matter: technique should always serve artistic intent. By exposing the rough edges and clashing tonalities, Coltrane lays his conception bare: the song is its harmonic structure, with its three-fold division of the octave. Coltrane wants its structure to show, and purposefully eliminates any effect that could interfere with that revelation. Giant Steps is his experiment, and his intention. The unmitigated, brusque, tonal juxtapositions serve that intention, and to have done otherwise would have been to undermine his goal.

Artistic intent is entirely different in Central Park West, and there Coltrane applies different techniques.

Central Park West

In tone, Central Park West couldn’t be more different than Giant Steps. Where the latter is audacious, exhilarating and wild, the former is gentle, sensitive and meditative. Structurally, however, they are very close:  both employ potentially harsh modulatory cycles, that divide the octave into segments of equal size; both consist almost exclusively of concise modulations, that are themselves mere transpositions of prototypical designs; and both have melodies that are developed almost entirely by tranposition. Given these structural similarities, what does Coltrane do to give Central Park West so different a spirit?

The six modulations of Central Park West are based on three prototypes:

As in Giant Steps’ cycle of major thirds, these modulatory intervals produce somewhat distant key relations: modulation up a minor third produces a key with three fewer sharps; modulation down a minor third adds three sharps; modulation by tritone produces a staggering difference of six sharps, to say nothing of the dissonance of the tritone relationship itself.

And yet, from this same structural material, Coltrane creates an astonishingly different aesthetic. For each type of modulation, he finds a wonderful, inventive way to transmute the aggressiveness of Giant Steps into a musical gesture of breathtaking tenderness.

To enable these changes, Coltrane alters the order of his structural material; in the process, he builds an elegant, symmetrical tonal design. He changes the cycle’s pattern of ascending minor thirds:

. . . to one of alternating thirds and tritones:

This order fixes B at its centre, and — transforming the cycle’s uniformity from obstacle to asset — creates an abstract symmetry of tonal relations:

Where the first modulation moves up a minor third from B, the second modulation creates a symmetry, by moving to the key that is down a minor third from B. The third modulation takes us to F, which, being a tritone from B, forms its own symmetry with that central key. Finally, F’s modulation to B completes the cycle. Through simple arithmetic (3+3 = 6), Coltrane ingeniously creates tonal order in a system that is normally antithetical to it. By combining two minor third steps of the cycle to produce one step of a tritone, he arranges all keys into a tonally symmetrical relationship to B.

Of course, the song doesn’t end there: the symmetry established, the remaining bars oscillate between B and D, to finally end the piece with a codetta-like passage that lends even more stability to the key of B.

Let’s look at how Coltrane exploits this re-ordering to create modulations that are so different in effect from those of Giant Steps. We’ll begin where the song begins — with modulation by minor thirds.

Modulation by minor thirds

Our cultural intimacy with the major – minor environment of diatonic music prepares us for a special relationship with modulation by minor third. We are habituated to the use of parallel modes — most commonly those of major scales and their parallel minors — and these modes are related by minor thirds. The parallel minor of the key of C major is, of course, C minor, and C minor shares the same key signature as Eb major. Since the keys of C major and Eb major are a minor third apart, so too are the parallel keys of C major and C minor — or, for that matter, the parallel majors and minors of any key.

Accordingly, modulation up a minor third produces the same relationship as that of a major key to its parallel minor: modulating up from B major to D major (the first modulation in Central Park West) is the same as moving from B major to B minor.

Modulation down a minor third simply reverses that order: modulating from D major down to B major (the sixth modulation in Central Park West, in bars 6-7) is the same as moving from B minor to B major.

As the principle of mixture permits the judicious use of chords from a parallel scale, modulations by minor third offer generous opportunity to discover pivot chords by which to effect the transition from key to key.

The modulations by minor third in Central Park West brilliantly illustrate this: Coltrane repeatedly finds just the right parallel-mode chord to transform a potentially dissonant relationship into a gesture of beauty.

Modulation up a minor third

The song begins with an elegant, spacious modulation, moving up a minor third from B major to D major:

Coltrane knows that modulating up a minor third is the same as changing the first key’s mode from major to minor. Because a pivot chord is not otherwise available, he uses mixture to move to the parallel minor — B minor —, where there is an abundance of suitable chords. B minor’s iv — Em — is simultaneously D major’s ii, and is therefore ideal for the role of pivot chord. Since the progression I – iv (minor) commonly occurs in major keys (e.g., ‘minor plagal’), the Em will also flow naturally from the opening BM7 chord, and thus make good musical sense in both keys.

The shift from major to parallel minor is often experienced as a change of tone, colour, or mood. Because it is based on the same relationship, modulation up a minor third also conveys such transformations, but the tonal displacement adds even more dimension, and — as in this wonderful passage — an element of wonder. In a sublime gesture, Coltrane broadens the experience by calling in a technique normally associated with common-tone modulation: he sustains his melody’s F# throughout the entire progression. It is, in turn, a brash fifth, a poignant ninth, a floating thirteenth, and straight-forward third, each contributing to the cascade of nuance and colour. Common tones, sustained through modulations in this way, both stabilize transitions and add dimension.

In contrast with his hands-off approach in Giant Steps, Coltrane has drawn heavily upon technique to make this modulation gentle, reflective, and beautiful. He repeats it, note-for-note, as the melody begins its answering phrase, in bars 5 & 6.

Modulation down a minor third

Although it modulates down a minor third, not up, this third key change — from Ab major down to F major — has much in common with the B to D modulation just observed:

Coltrane repeats the technique of sustaining his melody note (through most of the modulation, rather than all of it), and looks to the parallel minor — this time of the goal key, as this modulation is down a minor third — for his pivot chord.

The key of Ab major has little in common with F major, but everything in common with F minor, its relative minor. Since modulating from Ab major down to F major is equivalent to moving from F minor to F major, Coltrane plucks his pivot chord from the key of F minor, and relies on mixture to make the transition to the parallel major.

In the B to D modulation, Coltrane used a pivot chord that could serve as the ii chord of his goal key. In this modulation by descending minor third, he uses the first key’s tonic chord as pivot, for Ab major is simultaneously the tonic of Ab and the III chord of F minor. (He could have used other chords — Bbm or Db major, for example — but his desire for symmetry, the idiomatic use of the ii – V – I formula, and the beauty of making an early mode change, happily led to this choice.)

There are two especially magical touches to this modulation: the first is the ‘lightness’, or release, of its resolution, produced as a result of the move from a key with four flats to a tonality with only one — an effect generated by modulation down a minor third. The other is the series of suspensions (or, really, one eleventh and one suspension) created by the interplay of Coltrane’s melody and chord progression:

. . . that is, the intervallic play between Ab-C G-C, and G-F C-E.

Here, as in the first modulation, Coltrane uses harmonic technique to create a softer, more introspective effect than that produced in Giant Steps.

The piece’s sixth modulation, from D major to B major (bars 6-7), is identical in all respects, save one: its melody is more active.

Modulation by tritone

Keys a tritone apart have almost nothing in common, and are extremely difficult to link by any usual mode of relation. Even mixture fails to produce any common chords.

There are two tritone modulations in Central Park West: the second modulation, that descends from D to Ab (bars 2-3), and the fourth, by which F returns to the central key of B (bars 4-5). Coltrane uses a different modulatory design for each. In moving from D to Ab, his answer to the riddle of key relation is original, ingenious — and may, in fact, have contributed to his choice of tonal design. Here’s that modulation — shown, in context, with the opening passage:

There is no apparent link between the two keys: Ab’s ii (Bbm7) bears no relation to the key of D — major or minor — , no other chord acts as pivot, and there is no common tone unifying the progression. And yet, by the time we hear the Eb7, we feel a sense of ‘returning’, of moving toward something inevitable and fitting — as though this strange modulation made perfect sense. Well — it does make perfect sense, because we are returning: the end of this modulation — the ii - V - I in Ab — relates far more to the key of B major, which opened the tune, than to D. Listen to how naturally the tune flows from B to Ab, with the modulation to D removed:

I’ve removed everything from bar 1, beat 3, to bar 2, beat 3, leaving the chord progression as it would sound had there been no modulation to D major (the melody, of course, cannot be similarly abridged). The result is a beautiful, perfectly natural pattern, in which Ab’s ii - V - I smoothly follows B’s ii - V - I, in a harmonic sequence that descends by minor third. To better perceive that descending minor third relation, here’s the same progression, enharmonically altered:

We can easily see that the harmonic sequence performs a modulation by descending minor third. It is exactly the same kind of modulation — with exactly the same design — as the Ab to F modulation we studied earlier (click here to review it). Even more, the entire sequence can be viewed as one diatonic progression in Ab, which begins in the minor mode and then moves to the parallel major, in the progression   ii of III    V of III | III parallel major: ii     V | I .

It is because of the unity of this sequence — the intimacy between its halves — that we experience a sense of ‘return’ as we modulate from D to Ab, for we recognize in the ii - V - I of Ab the completion of a pattern begun in the ii - V - I of B. Coltrane creates the initial pattern, interrupts it with a tangential move to D, and then restores and completes it via tritone modulation from D to Ab. Rather than hear the tritone modulation’s strangeness, we are captured by the recognition of the initial pattern’s return. It’s magical — ingenious — and quite far removed from the brusque modulatory approach of Giant Steps.

The last tritone modulation (bars 4-5), by which the completed cycle — now at F — returns to its starting tonality of B, is even more enigmatic:

Here, there is no immediately preceding passage or key to which the modulation relates — except the end of the cycle itself, and its return to B. It’s quite possible that, for Coltrane, this was sufficient basis for the modulation. That he repeats his melody’s beginning, here — despite the tritone leap required to do so — adds weight to that perspective. On the other hand, a composer who is stimulated by the idea of cyclical modulation probably wouldn’t shrink from the challenge of tritone relationship, and so Coltrane may have had other connecting relationships in mind. Here are two possibilities:

FM7 as bII of E

Here, FM7 serves as both I in F and bII in E. In turn, EM7 is simultaneously I in E and IV in B. Coltrane’s use of C#m7, rather than E, can be viewed as a simple chord substitution, that preserves the song’s ii – V – I formula.

FM7 as chromatic passing chord

Here, F major’s tonic is still the pivot chord, but this time as a chromatically raised IV in the key of B major, leading directly to the V of that key.

With this return to B, the four-fold division of the octave is completed, and Coltrane’s symmetrical tonal design fulfilled.

Let’s now take a step back, to place the details of these two pieces within the framework of our larger quest.

Putting it all together . . .

We see that, in Giant Steps and Central Park West, Coltrane has worked the same structural material in two very different ways, applying divergent techniques in his treatment of the modulatory cycle, and in his approach to tonality. Let’s summarize those differences:

Treatment of the modulatory cycle

Giant Steps

Coltrane eschews any technique that would soften the harshness of his modulatory cycle: he avoids pivot chords and common tones, choosing instead to highlight the cycle’s awkward key relations, by leaving them exposed.

Central Park West

Coltrane goes to great lengths to mitigate — and, indeed, transform — the cycle’s natural abrasiveness. In modulations by minor third, he uses mixture to produce effective and elegant pivot chords, and sustains his melody notes to create the smoothest possible transition. Where these aren’t available, he creates the illusion of seamlessness, by making the modulation itself into a parenthetical insertion that separates the two halves of a very closely related sequence.

Treatment of tonality

Giant Steps

The composer deliberately avoids any techniques that would more firmly establish one tonal centre. Though the piece cyclically returns to B, that key is given little more prominence than any other. The lack of a strong central tonality means that we hear all the keys equally — which in turn exposes the composition’s underlying structure, in which the octave is divided into three equal major thirds. The suppression of tonality is an important means by which Coltrane realizes his artistic intention.

Central Park West

Coltrane goes out of his way to create a symmetrical tonal design that places the key of B major at its centre. Once the structural cycle is completed, he restricts his modulations and progressions to those that strengthen B as tonic. Both actions minimize the modulatory cycle’s tonality-disrupting tendencies, and successfully establish a central key. This creates the unified base from which we can more calmly experience the cycle’s tonal trajectory, and delight in its unfolding. The establishment of a central tonality builds the perspective through which Central Park West can be properly experienced.

The sum of these differences is the realization of two distinct artistic intentions. Through Giant Steps and Central Park West, Coltrane shows us that the same material can be made to produce totally different results — if used differently:

It’s not what you use, but how you use it.

What was a source of jarring aggression in Giant Steps becomes the starting point for utter charm and beauty in Central Park West. What Coltrane makes axiomatically tender in one, he renders brusque and angular in the other — and it is all in the how: how the composer eliminates all smoothing techniques, so as to expose the underlying structure in Giant Steps, and how he finds wonderful, inventive ways of transforming the discordance of distantly-related keys into breathtaking charm and delight in Central Park West.

While these insights deal with composition, they apply equally to improvisation, where the realization of artistic intent is just as important. To solo well, one must understand the structures upon which one creates.

This little tour necessarily touched on but a few parameters — there is far more, in Coltrane’s creations, from which we can learn and grow. It’s my hope that our brief exploration will inspire further visits to his work.

Michael Leibson
May, 2009

Michael Leibson is a composer, music consultant, and music educator, who specializes in jazz and classical harmony. To learn more about him, click here; for information about studying with him, click here; and for copyright information, click here.