Some applications of the axis system
I’ve recently been looking into the the axis system, particularly at ways of applying it to improvisation and composition.
The system was developed by the music theorist Erno Lendvai in his study of the music of Bela Bartok, the basic idea being that chords and tones related by the intervals of a minor third and tritone can function as tonal substitutes for one another, and do so in many of Bartók's compositions.
This is best explained by a chart:
Subdominant axis F Dominant axis G Tonic axis C
D E A
B Db F#
Ab Bb Eb
Lendvai argued that the chords along each of these axes are related in a functional way. Eg, in the subdominant axis D is the relative minor of F, B is the relative minor of D, Ab is the relative minor of B and F is the relative minor of Ab and so on. There are other correspondences eg, F7 and B7 both share the same tritone interval (A and Eb) as do Ab7 and D7 (C and Gb). This of course forms the theoretical basis for tritone substitution in jazz harmony. Lendvai’s contention was that because of the functional relationship between these keys they can effectively be used as substitutions for each other. So an Ab chord could be substituted for an F chord, a D chord or a B chord and the function of the chord would remain the same in the progression.
The most familiar application of this in jazz is the tritone substitution which I referred to earlier, where a chord can be substituted for a chord a tritone away from the original chord. E,g Db7 can be substituted for G7. Most commonly this occurs with dominant 7th chords, however other chord types can work as well.
Tad Dameron in his composition Ladybird used this progression derived from using tritone substitutions for a typical I - VI - II - V turnaround
Original turnaround progression:
Cmaj 7 / A7 / | Dmin7 / G 7 / |
Cmaj7 / Ebmaj7 / | Abmaj 7 / Dbmaj7 / ||
Here the chords substituted are major type chords, but the function of the sequence remains the same.
‘Coltrane changes’ and the axis system
However as well as the more familiar tritone relationship within the axis chords, there is also the minor third relationship, whereby a chord can be substituted for a chord a minor third away from the original chord. It’s here that a link emerges between the system of axis substitution and the’ Coltrane changes’ developed by the tenor player John Coltrane. Below I’ve outlined a typical use of the Coltrane changes over a II - V - I progression
Dmin7 / / /| G7 / / / | Cmaj 7 / / / |/ / / / |
Cmaj7 / Eb7 /|Abmaj7 / B7 / Emaj7 / G7 |Cmaj 7 / / / |
Coltrane used this progression in many of his compositions most notably the classic Giant Steps, but he used it in numerous other tunes which often involved him reharmonising standards utilising this progression. Examples of this are 26-2 (based on Confirmation), Countdown (based on Tune-up) and Satellite (How High the Moon).
The ‘Coltrane changes’ are normally analysed as a series of superimposed descending major 3rd intervals arriving at a particular tonic. So the above progression could be simplified as:
Cmaj / / / |Ab maj / / / |Emaj / / / | Cmaj / / / |
Each of the major chords is preceded by its relevant dominant chord creating a cyclic progression. Although the progression can certainly be seen in this way, from this explanation it is not immediately clear why a series of descending major 3rds should work as a substitution for what is in effect a cycle of 5ths. The cycle of 5ths works because of the natural relation between the dominant and the tonic chord established in the harmonic series, but why does a descending series of maj 3rds work? I feel the answer may lie in the axis system.
If we go back to the original Coltrane changes:
Cmaj7 / Eb7 / |Ab maj7 / B7 / |Emaj7 / G7 / | Cmaj7 / / / |
I’d like to suggest that this sequence makes more sense if seen as a set of substitute changes for this progression:
Cmaj7/ / / | Dmin7 / / / |G7 / / / |Cmaj7 / / / |
In this version of the original progression the crucial V - I resolution happens at the same point in both sequences, also I feel this reflects more accurately how the Coltrane progression sounds. If we were to assign functions to these bars the C maj7 chord would be a tonic function, the Dmin a subdominant function, the G7 a dominant function, finally resolving back to the tonic function in the last bar. The below diagram shows the ‘Coltrane changes’ again but this time I’ve included underneath all the other axis tones for each of these functions:
|Cmaj7 / Eb7 / |Ab maj7 / B7 / | Emaj7 / G7 / | Cmaj7 / / / |
|Tonic axis (C Eb Gb A) |Sub dom axis (F Ab B D) |Dom axis (G Bb Db E ) |Tonic axis
It’s easy to see here that the Eb7 in the first bar can be seen as a substitute chord from the tonic axis, the chords in the second bar from the subdominant axis , and the chords in the third bar from the dominant axis, finally resolving back to the tonic axis.
So, I would suggest that the reason the descending cycle of major 3rds works is because of the relationship between the axis chords, and the fact that the chords in the descending 3rds progression fulfil the same function as in the original progression. To show this more simply:
Tonic |Subdom |Dom |Tonic
Cmaj / / / |Ab maj / / / |Emaj / / / | Cmaj / / / |
Cmaj / / / |Dmin / / / | G7 / / / | C maj7 / / / |
It occurred to me that if the axis system is correct the ‘Coltrane changes’ were only one possible route through the system, and if the axis chords were essentially interchangeable it should be possible to create different progressions which functioned in the same way.
Here is a chart showing all the possible routes through the system with some basic analysis. I haven’t given the chords functions as these can change, all the examples are in C.
F - G - C | IV - V - I The basic subdom, dominant, tonic sequence.
D - G - C | II - V - I The ‘jazz’ version of the above sequence.
Ab - G - C | bVI - V - I Ab functions as a tritone sub for the D
B - G - C | VII - V - I
F - Db - C | IV - bII - I Db functions as tritone sub for G
D - Db - C | I - bII - I Standard tritone substitution
Ab - Db - C | bVII - bII - I Common tritone sub, |Abmin7 |Db7 |Cmaj7 |
B - Db - C | VII - bII - I
F - Bb - C | IV - bVII - I The so-called ‘backdoor cadence’
D - Bb - C | II - bVII - I Variation of the ‘backdoor cadence’
Ab - Bb - C |bVII - bVII - I
B - Bb - C | VII - bVII - I
F - E - C | IV - III - I
D - E - C | II - III - I
Ab - E - C |bVII - III - I Coltrane changes!
B - E - C | VII - III - I
In the following tune of mine called Upcycling I used the IV - bII – (I) progression.
Ebmaj7 / A7 / | Bmaj7 / F7 / | Emin7 / / / |A7 / / /|
Dmaj7 / Ab 7 / | Bbmaj7 E7 | Ebmin7 / / /| Ab7 / / /|
Dbmaj7 / / / | G7b9 / / / / | A maj7 / / / | Eb7b9 / / /|
Dmin7 / / / | G7 / / / | Cmin7 / / / |F7 / / / |
Bbmaj7 / / /| E7b5 / / / /|
The piece is in Bb major and the first two bars use an axis substitution for a II - V - I leading to Bb major (although they don’t actually resolve to the tonic). So I’ve used the Eb and the A from the subdominant axis of Bb and the B and F from the dominant axis, this creates a different progression from the Giant Steps sequence but one which functions in essentially the same way.
I used a different route through the axis (the VII - III – I progression) for another tune of mine called ‘One for Hughie’
The first 4 bars are:
Bmaj7 / Amin7 D7 |Emaj7 / Dmin7 G7 |Cmaj / F#min7b5 B7 | Emaj7 / Dmin7 G7|
Here the progression is in C major and again uses an axis substitution for a II - V - I progression in C (which this time does resolve). In the first bar I’ve used the B and D from the subdominant axis the E and G from the dominant and the C from the tonic axis. (I’ve also used the relevant II chord for each of the Dominant chords this time.)
Below is a chart showing all the possible symmetrical permutations of the basic ‘Giant Steps’ progression when the different axis chords are substituted in . By symmetrical I mean that the interval in both the subdominant bar and the dominant bar are the same. Here I’m following Coltrane’s lead, as this is how the Giant Steps progression works. The symmetrical nature of the progression tends to gives it a greater sense of stability and resolution (and also mirrors the symmetrical nature of the original II V I progression). It also conveniently limits the number of permutations, although theoretically there’s no reason why the unsymmetrical permutations shouldn’t work just as well.
1 Bmaj7 / D7 / | Emaj7 / G7 / | Cmaj7 / / / | / / / /|
2 Abmaj7 / B7 / | Emaj7 / G7/ |Cmaj 7 / / / |/ / / / | (The Giant Steps progression)
3 Fmaj 7 / B7 / | Dbmaj7 / G7 /| Cmaj7 / / / | / / / /|
4 Abmaj7 / F7 | Bbmaj7 / G7 / | C maj7 / / / | / / |
5 Dmaj7 / F7 / | G maj7 / Bb7 / | Cmaj7 / / /| / / / / |
6 Bmaj7 / D7 / | Gmaj 7 / Bb 7/ | C maj7 / / /| / / / / |
7 Abmaj7 / D7 / | Emaj7 / Bb7 / | Cmaj7 / / / | / / / /|
8 Bmaj7 / Ab7 / |Db maj7 / Bb7 / | Cmaj7 / / / | / / / /|
9 Fmaj 7 / Ab7 / | Bbmaj7 / Db7 / | Cmaj7 / / / /|/ / / /|
10 Dmaj7/ F7 / | Bbmaj7 / Db7 / | Cmaj7 / / / | / / / / |
11 Bmaj7 / F7 / | Gmaj7 / Db7 / Cmaj7 / / / / |/ / / / |
12 Dmaj / B7 / | Emaj7 / Db7 / Cmaj7 / / / / | / / / / |
13 Abmaj7 / B7 / | Dbmaj7 / E7 / | Cmaj7 / / / |/ / / /|
14 Fmaj7 / Ab7 / | Dbmaj7 / E7 /| C maj7 / / / | / / / /|
15 Dmaj7 / Ab 7 / | Bbmaj7 / E7 / | Cmaj7 / / / | / / / /|
16 Fmaj7 / D7 / | Gmaj7 / E7 / | Cmaj 7 / / / |/ / / / |
Tune-In and Drop-Out
As an exercise I thought I'd try and come up with my own versions of the Miles Davis piece Tune-Up using a couple of the substitute progressions above - I'm calling them Tune-In and Drop-Out. Tune-In uses progression number 1 from the chart above and Drop-Out uses progression number 3. I've followed the model of Countdown and started each progression with the original II chord before switching into the substitutions.
Emin7 / B7 / | Dbma7 / E7 / | Gbmaj7 / A7 / | Dmaj7 / / / |
Dmin7 / A7 / | Bmaj 7 / D7 / | Emaj7 / G7 / | Cmaj7 / / / |
Cmin7 G7 / | Amaj7 / C7 / | Dmaj7 / F7 / | Bbmaj7 / / / |
Emin7 / / / / | F7 / / / | Bbmaj7 / / / | A7 / / / ||
Emin7 / F7 / | Gmaj7 / Db7 / | Ebmaj7 / A7 / | Dmaj7 / / / |
Dmin7 / Eb7 / | Fmaj7 / B7 / | Dbmaj7 / G7 / | Cmaj7 / / / |
Cmin7 / Db7 / | Ebmaj7 / A7 / | Bmaj7 / F7 | Bbmaj7 / / / |
Emin7 / / / | F7 / / / | Bbmaj7 / / / | A7 / / / ||
As Coltrane does I'm using the maj7 – dominant 7th chord functions, however this is somewhat arbitrary, and other chord functions can work as well. I’ve experimented with writing a sequence using a ‘minor’ Giant Steps progression.:
Abmin7 / B7 (b9) / | Emin7 / G7 (b9) / | Cmin7 / / / |/ / / / |
Alternatively, all the chords could be major (as with the 'Ladybird' tritone substitution mentioned earlier) or all minor. The important factor really is the root movement, as that is what gives the progression its sense of momentum and resolution.
‘Ipanema Bridge’ and the axis system
The famous bridge from the tune Girl from Ipanema utilises a slightly different form of axis substitution. The original sequence is:
Gb maj7 / / / |/ / / /|B7 / / / | / / / / | F# min7 / / / /| / / / / | D7 / / / / | / / / / |
Gmin7 / / / / | Eb7 / / / | / / / /| etc
If we take the first chord to be a relative major substitution for Ebmin7 we can see this progression as being fundamentally a string of II – V7 progressions where the
V7 chord has been substituted for the 7th chord a min 3rd away.
Ebmin7 / / / /|/ / / / |Ab7 (B7) / / / |/ / / / | F#min7 / / / /| / / / / |B7 (D7) / / / |/ / / / |
Gmin7 / / / / | / / / / | C7 (Eb7) / / / / |/ / / / |
If we apply axis system theory to this and try swapping in another chord from the dominant axis we can produce another version of the bridge (it works -try it at your next jam session and get some funny looks!)
Ebmin7 / / / / | / / / / | F7 / / / | / / / / | F#min7 / / / / | / / / / | Ab7 / / / |/ / / / |
Gmin 7 / / / | A7 / / / /| / / / /|
This progression could also be seen as a tritone substitution for the original axis substitution!
In theory we could also substitute a D7, Ab7 and F#7 respectively into the dominant bars, but these clash with the melody.
I used the ‘Ipanema Bridge’ progression as the basis of a composition of mine called Naming Clouds, here is the A section:
Bbma7 / / / |/ / / / | F#min7 / / / | D7#11 / / / | Gmaj7 / / / | / / / / | Ebmin7 / / / | B7#11 / / / |
Emaj7 / / / |/ / / / | Cmin7 / / / |Ab7#11 / / / /|Dmin7 / / / / |G7 / / / /| Cmin7 / / / | F7 / / / / |.
I find that axis substitution often makes sense of progressions which otherwise seem obscure. I remember I was once teaching the Stevie Wonder tune Sir Duke to a group of students at a summer school. The chord progression in the chorus is :
C / / / / | F#min / / / / | Dmin / / / / | G11 / / / / |
One of the students asked where the F# min chord comes from. Initially this completely stumped me, as it doesn't seem to have any obvious connection to the rest of the progression. Eventually however I realised that it's simply an axis substitution for the more obvious VI chord Amin which virtually everybody apart from Stevie Wonder would have used!
I realise that none of this is particularly new. I know that David Baker for one has written extensively about the 'Coltrane changes' and their possible variations and applications. What I find significant is the link between the axis system and the 'Coltrane changes' and how exploring this provides both a sound theoretical explanation for how the substitutions work, and also opens up a lot of possibilities with regard to expanding the harmonic language.
Any thoughts, comments or ideas appreciated.