Aspects of Symmetry in Contemporary Jazz
The concept of symmetry holds a central position in the history of the Arts and Sciences. Throughout the ages, people have long been fascinated by the principle of reflection and have sought both to find symmetrical constructs in the natural world, and to incorporate elements of reflection and balance into their own creative work. In the modern world, graphic and numeric symmetry has become an increasingly common feature of our daily consciousness, a trend reflected in the art of our times.
Symmetry is the creation of order. Whether that order is created by us or perceived in our habitat, its recognition momentarily lends a comprehensible pattern to an otherwise chaotic universe. In many diverse cultures around the world, symmetry is representative of balance, of harmony, or of divine perfection.
By focusing in this short blog on the application of symmetrical thinking to the world of contemporary jazz, I wish to show how certain artists are employing concepts of symmetry, balance and logical supra-formal principles in their work. These explorations can be roughly divided into three main areas: melodic or intervallic symmetry, symmetry in harmonic structures, and the application of symmetry in composition.
Jazz demands constant creativity, and it flourishes most when only the basic structure of a idea is fixed, leaving the individual free to respond to what they hear around them and to have the space in which to develop their own ideas, lending the performance of a composition a certain unknown quality. Pure symmetry demands complete adherence to a form that once initiated is entirely determined and continues to follow a set sequence ad infinitum. How does the jazz musician strike a balance between employing models that are perfectly symmetrical and fulfilling the over-riding aesthetic demand that the work be full of life, emotion and narrative that has been a part of the history of jazz since its conception?
Melodic or Intervallic Symmetry
Some of the clearest examples of the application of symmetrical principles to the formation of a melodic line can be found in the work of Steve Coleman. Through his research into ancient cultures and into the science and mathematics of sound, rhythm, harmony and composition, Coleman has created a vast world of logical meaning that informs his real time decision-making at an improvisational level, and through the development of a high level of technique he has been able to express these decisions clearly and articulately.
Many of the examples quoted here are taken from a transcription and subsequent analysis of his solo on a tune called ‘The Creeper’ from the Marvin ‘Smitty’ Smith recording ‘Keeper of the Drums’ made in New York in 1987. There are also some examples from his arrangement of Charlie Parker’s ‘Confirmation’ on his own album ‘Def Trance Beats,’ recorded by the Five Elements in 1995.
When looking for examples of his work to transcribe which would shed some light on the symmetrical processes that Coleman employed in his improvisations, I sought to find a solo which was based on a traditional and simple harmonic structure, so that his super-impositions on the original chord sequence would be all the more apparent. His solo on ‘The Creeper’, a blues in F, proved to be fit these specifications perfectly.
It is also a great example of how an artist achieves a successful synthesis between innovation and tradition, shown in the way that he marries his own complex harmonic and rhythmic language with the traditional vocabulary and procedure usually associated with improvisations on this formal type.
This solo begins in the way countless solos throughout jazz history have begun before it, taking the previous soloist’s closing statement as its opening theme. This is a practice common in jazz since its conception and is probably a descendant of the call and response patterns found commonly in African music. An example of this is shown below, in the interchange between the consecutive solos of Lee Morgan and Benny Golson, on a recording of Art Blakey’s ‘Moanin.’
Example 1: Lee Morgan and Benny Golson consecutive solos on Moanin’
On the Creeper, we see another interchange (coincidentally between trumpet and saxophone again) as Coleman follows Wallace Roney’s solo. It is a great indicator of the depth of improvisational ability of the musician in moments such as these. There is no time for prior preparation, the theme arrives in real and decisions as to its application must be made near-instantaneously, both processing the information and in some way developing the material. Merely reproducing the phrase verbatim is not the object of the exercise; it is far more exciting musically to develop the idea and thereby personalise the original theme. For this reason, Coleman’s treatment of the phrase sheds light on his melodic theory and shows the degree to which he has worked on concepts of symmetrical development and practised their application in real time.
Example 2: Steve Coleman’s opening statement on ‘The Creeper’
This is an incredibly well constructed musical statement; there is so much compositional logic in the phrase. But it is all the more impressive because we know from the way in which he responds to his musical surroundings that this symmetry is being created spontaneously. Coleman’s improvised development of the idea is built in accordance with a strong symmetrical principle.
Coleman takes as his theme Roney’s descending semitone fragment, the G, F#, F triplet figure. He restates this figure three times; once to start his phrase, once in the centre of the phrase, and once more to close the phrase, where he resolves it to the third of the G7 chord. In between these three fragments he twice plays the five-note figure C, E, B, E, C. This is our first example of a symmetrical pattern, taking the B to be 0 this phrase translated into semitones is 1 4 0 4 1. This pattern is in itself a microcosm of the phrase as a whole, which is also divided into five parts and completely symmetrical, A B A B A, where A is Roney’s semitone fragment motif and B the inserted pattern. In the second playing of the B figure however he truncates the length of the phrases by altering the duration of each note, thereby resolving the phrase at the crucial movement to the four chord at the start of the fifth bar.
This is an example of an artist bending a pure mathematical construct of symmetry to fit a greater aesthetic principle, here defined by the harmonic progression of the blues. Using symmetry within the form of the structure in which he is improvising, his idea nonetheless makes musical sense. The perfection of the mathematics is subjugated to the harmonic scheme.
We see an example of the opposite of this at the end of his first chorus; where rather than breaking the perfect symmetry to fit the harmony, he continues the logical construct across the natural resolve point of the underlying chord.
Example 3: Carrying a logical construct across a natural resolution point
The resolution note suggested by the harmonic progression would be to move from the E in the penultimate bar to the root note D in the last. But Coleman instead continues the descending semitone sequence from F to E and through an Eb (flat 9) on the downbeat to finally play the consonant D on beat two of the last bar. The dissonance created by adhering to a logical sequence rather than bending the sequence to the asymmetric templates of the diatonic scale is a characteristic consequence of symmetrical or logical thinking it leading outside of the scale.
As well as incorporating symmetrical ideas into the construction of entire phrases, Coleman also uses a lot of small symmetrical cells within longer passages. In a master class that he gave at the School of Improvised Music in New York, Coleman described how he initially practised these symmetrical patterns repeatedly to ‘get them into my fingers’ in the same way that he had learnt bebop vocabulary and ornamental mordents whilst studying the music of Charlie Parker. It is his employment of these that lend his melodic lines their distinctive atonal character. We can see many good examples of these in this solo.
Example 4a: Symmetrical cell
This two-part cell uses bilateral symmetry about the mid-point. The first four-note group is composed of an interval of a rising tone followed by two descending semitones; the second four notes are an exact mirror inversion of this, two descending semitones followed by a rising tone. It is played as part of this much longer phrase.
Example 4b: phrase containing symmetrical cell
The example below is a symmetrical bi-polar augmentation of a two-note axis. Taking the initial two notes to be an axis of a tone, B to C#, the consequent Bb and D augment this axis by a semitone above and below. The initial note B is restated to form a five-note pattern. Here he plays the cell in isolation but develops it sequentially by repeating the entire pattern in the next beat, now transposed a semitone lower.
Example 4c: Symmetrical cells in sequence
As well as using these medium-sized symmetrical cells, Coleman also regularly creates a small symmetrical axis movement around a single tone. Frequently he will use this device at the end of a line, particularly when resolving the line at a crucial moment in the harmonic sequence. This is not a new idea in jazz, and can be found in the vocabulary of many contemporary jazz musicians. It probably traces its lineage to a technique commonly found in bebop voice-leading when the resolve note is approached via its surrounding scalar tones.
Example 5a: Approaching a resolve note via the surrounding scalar tones
(Charlie Parker, Donna Lee)
Parker uses a symmetrical cell used at the end of this line, but it is a symmetry rooted in the asymmetric diatonic scale. With the increased chromaticisation of the music jazz musicians moved away from the prevalence of the seven or eight note modes and explored the wider possibilities of the entire chromatic scale as an available tone-set. Approaching a note from its surrounding scalar tones, which would usually mean a movement of a semitone and a tone above and the below the axis note, gave way to the entirely symmetrical approach of a semitone on either side of the target note. Steve Coleman frequently employs this device, and the increased amount of harmonic tension that is created by its use of total intervallic symmetry approach has become a characteristic of his sound, and of many others within the world of contemporary music.
Example 5b/c: Approaching a resolve note from a semitone above and below (Coleman)
Here we see the same device employed by another saxophonist, David Liebman.
Example 5d: Approaching a resolve note from a semitone above and below (David Liebman)
Examples 2 through 4 all show a decidedly conscious use of symmetry when constructing improvised lines. However, through continued investigation into a particular area artists absorb the principles that they are studying on an unconscious level, and analysis of their work may yield findings that were not necessarily conscious decisions whilst improvising, or indeed composing.
There is another symmetry in this solo that occurs on a slightly subtler level, and would have required either enormous mental facility to have implemented in real time, or to have been the unconscious product of an instinctual balance and innate symmetrical understanding.
In bar 9 of the penultimate chorus, Coleman plays this line using the Eb lydian scale to create a line with a high degree of tension that he resolves to the tonic on the fifth:
Example 5a: Eb lydian
In almost exactly the same place a full chorus later, in bar 9 of the final chorus, he plays a line using the C# lydian scale to create another tension phrase that he resolves to the tonic on the root.
Example 5b: C# lydian
Traditionally, this position in the form of the blues would have been where the action took place, after the initial theme statement and its re-statement, therefore it is not surprising that he plays a line with a high degree of harmonic tension at the same place in both choruses, nor that they both resolve to the tonic on the final note. However, what is particularly interesting in symmetrical terms is that he uses the same mode to create this tension but derives it from the two notes that surround the tonal centre D in a semitone axis. The argument as to whether this was a conscious or unconscious decision can only be answered by the composer himself.
Symmetry in Harmonic Structures
In the flourishing world of jazz in the late 1950’s and early 1960’s, many musicians began experimenting with increasingly complex harmonic structures. There was a shift away from the interpretation of popular song forms and an increase in the composition of original structures to provide the basis for the improvisation. By writing their own tunes the musician could focus more precisely on specific areas that they were investigating at the time. These new structures inspired new ways of improvising and spawned new kinds of melodic lines that although still based on the same principles and borrowing heavily from bebop vocabulary, began to sound quite different from the lines that were rooted in the diatonic chord progressions dominant in the music of the 30’s and 40’s.
During this period, there was an increased focus of interest in the use of symmetrical structures in direct application to harmonic progressions. Jazz musicians such as John Coltrane and Joe Henderson wrote tunes which employed root movements through equal divisions of the octave, or which moved in fixed sequential steps. These structures demanded a high degree of technical precision and harmonic understanding to articulate clearly, and had a characteristic new sound.
Without a clear tonal centre they have a more cyclical feel in comparison to the diatonic standard, which is linear in comparison and features clearly delineated moments of harmonic tension and equally clearly defined moments of release when the tonic is stated. In a symmetrical system however, the equal importance of more than one key centre creates a sense of tonal ambiguity. As Erno Lendvai points out in his book ‘Symmetries of Music,’ symmetry creates atonality and conversely tonality demands asymmetry. It is important to note however that although this new kind of harmony did not have one clear tonal centre, it was not exactly atonal, as the improvised lines which were developed during this period clearly outlined the tonic keys as they passed by in succession, using very simple pentatonic shapes or arpeggio type figures played in quick It might be more accurate to describe this kind of harmony as being pan-tonal.
The idea of using a symmetrical division of the octave to formulate the root movements of a harmonic progression had been incorporated into jazz at a far earlier stage, through its use by composers of Broadway standards subsequently adopted by the jazz musician. Richard Rodgers’ song ‘Have You Met Miss Jones?’ uses a symmetrical major third movement in its bridge, and it is said that Coltrane received the inspiration for his subsequent extensive exploration of the augmented axis when whilst in a record date with Duke Ellington, the composer sat at the piano and showed him other examples of songs which used similar harmonic progressions whilst in the studio. This is an interesting curio in the history of jazz, one of the most powerful surges forward in the harmonic development of the music instigated by an innovator from an earlier generation. Ellington’s own composition ‘In a Sentimental Mood’ uses a major third movement when it moves from F major at the end of the A section to begin the bridge in Db major.
Coltrane’s ‘Giant Steps’ is a distillation of the augmented axis root movement. The octave is divided into three equal parts and the rapid movement from one tonal centre to the next creates the tune’s dense harmonic rhythm, which can be divided into two parts.
Example 6a: Breakdown of ‘Giant Steps’ changes
A section: │B v│G v│Eb│ii v│G v│Eb v│B│ii v│
B section: │Eb│ii v│G│ii v│B│ii v│Eb│ii v│
If we extract two 7 bar sections of the tune we can see the symmetry of the two sections more clearly. In each section he begins and ends in the same tonal centre, but uses a different sequential patterning of the augmented axis in each section. The A section has a kind of fold-back pattern, moving rapidly from one end of the axis to the other, B to Eb, but then returning to the G and repeating the sequence in order to return to the B at the end. In the second section he follows a more obvious sequential pattern, moving through the axis in straight succession.
Example 6b: Breakdown of ‘Giant Steps’ changes (7 bar sections)
A section: │B v│G v│Eb│ii v│G v│Eb v│B│
B section: │Eb│ii v│G│ii v│B│ii v│Eb│
Although the octave is divided equally into three parts, the three tonal centres are not equally weighted. In a tune composed of exactly the same essential building blocks, major chords and their subsidiary building blocks, dominant and secondary dominant chords, one way of attempting to define the importance of each tonal centre is by gauging the length of time spent in each centre or on its secondary chords. This tells us that the tonal centre G is held for 18 beats, that of B for 20 beats and that of Eb is held for 26 beats in total.
It is easy to see from this that the G tonality has the least dominance, and is used only as a passing tonality. But to go on and infer that the Eb is the most dominant tonality overall is actually incongruous with the feel of the tune. Because of the rhythmic spacing of the chord progression, particularly the effect created by beginning and ending the first section of the tune in B, the B tonality seems to have a particular dominance, despite being played for a total of less beats than the Eb tonality. This shows us the importance of rhythm in disrupting a symmetrical sequence, and this tune is a good example of how an asymmetric rhythmical structure can affect a pure symmetrical construct and turn an otherwise formulaic logical idea into a tune with a strong contour and shape.
Joe Henderson’s ‘Inner Urge’ is another tune clearly divided into two sections. In this tune however, symmetry is only employed in the harmonic construction of the second section, which contributes in part to the strong contrast the two sections have with one another. This is also accentuated by the harmonic rhythm of the two sections, with each chord being held for four bars in the A section and for only one bar in the B part.
In this tune Henderson almost entirely abandons the use of traditional dominant-tonic functional chords. The second section of the tune is almost entirely composed of major seventh chords, with the exception of the last chord whose quality is altered from major to dominant to provide a stronger movement back to the top of the form. The root movement of this progression follows an entirely symmetrical sequence.
Example 7: B section of Henderson’s Inner Urge
The root movement here follows a set sequential movement, moving down a minor third and then up a semitone. Seen from an invisible axis at the mid-point, this is an example of a bi-lateral symmetrical sequence.
Down 3, up 1, down 3, up 1, down 3, up 1, down 3
Although this period still has a large influence on current trends within contemporary jazz, it is not strictly true to class it as being from the contemporary period. Almost fifty years have passed since ‘Giant Steps’ was recorded, and contemporary harmony has continued to increase in complexity. Tunes like ‘Inner Urge’ and ‘Giant Steps’ employ symmetry in a direct linear sense, with the progressive root movements of the chords defining the symmetrical pattern. This approach has been incorporated into the harmonic technique of contemporary jazz composers like Jim McNeely, Maria Schneider and Bob Brookmeyer, who use symmetrical divisions of the octave to create substitute chord progressions based on the augmented or diminished axis, and Coltrane’s major third substitution has been absorbed into the mainstream jazz vocabulary.
For many artists working today however, there has been a movement towards tonal ambiguity, moving in and out of given key centres to create movement and contrast. The concept of linear symmetry as represented by a tune like ‘Giant Steps’ seems to have become slightly obsolete, to be replaced by greater chromatic freedom and an increased focus on complex rhythmic rather than harmonic form.
One of the most interesting examples of symmetry in more contemporary harmony comes once again from Steve Coleman. Rather than applying symmetry in a linear sense however, he applies the symmetry to the vertical aspect of the music, to the chords themselves. He calls this concept ‘negative harmony.’ Using negative harmony, chords are generated by inverting the original chord in a mirror image, maintaining the original intervals.
Example 8a: Negative inversion of a minor 7th chord
Here the original chord A minor 7 has been symmetrically inverted to form the new chord B minor 7. From the root up, the A minor chord is voiced minor third, major third, minor third. To derive a negative chord from this we voice the chord with the same intervallic structure but with the original root now the top voice.
Example 8b: Negative inversion of a major 7th chord
It is interesting to note that with major and minor seventh chords, the chord quality does not change, only the root. With dominant chords however, a negative inversion does change the chord’s quality, to half-diminished.
Example 8c: Negative inversion of a dominant 7th chord
By using these chords both as substitution options, as well as compositionally, Coleman has been able to generate new harmonic information from the traditional chord system. He talks of the possibilities of substituting minor and half-diminished chords for dominants, describing the negative chords as having a ‘darker’ sound as opposed to the ‘brightness’ of the dominant, which he ascribes to the use by those particular chords of the undertones of the tonic, rather than the overtones contained within the dominant. This he derives from Ernst Levy’s Theory of Harmony, which he quotes often in his own writings.
The application of symmetry in composition
Some of the clearest examples of symmetry used in modern jazz compositional techniques are found in the work of the pianist Vijay Iyer. Through his study of mathematics and his investigations into the world of Indian classical music he has developed some very strong compositional idea, some of which are based on entirely symmetrical forms. This example taken from a tune called ‘headlong’ on the most recent release from his Fieldwork trio, shows how the musical material is generated using a symmetrical process.
Example 9a: Opening section of Vijay Iyer’s ‘headlong’
from the album ‘Simulated Progress.’
There are three rhythmic voices in this opening passage. The first voice is made up of the alto saxophone and the right hand of the piano, the second is the piano’s left hand, and the third voice is the drum part, although this voice is closely related to the second and is not made up of entirely unique material. If we examine the first and second voices more closely, we can see that in rhythmic terms they are each composed of the same symmetrical cell, A B A B A.
Example 9b: Extraction of first rhythmic voice
Here A corresponds to a note held for a duration of six semi-quavers, B to a semi-quaver followed by four semi-quaver rests, which could be thought of as a group of five semi-quavers.
Example 9b: Extraction of second rhythmic voice
In this voice A corresponds to the pattern of 2 1 2 1, or of quaver, semi-quaver, quaver, semi-quaver. B is the 2 3 pattern, or a quaver followed by a dotted quaver. The symmetrical cell A B A B A is clearly discernible in both parts.
Although it has not been possible to do anything but skate the surface of such a potentially vast area of exploration, I hope that these findings will illuminate some of the ways in which symmetry has been incorporated into modern jazz, both improvisationally and compositionally. But what does symmetry actually mean in musical terms? Although symmetry is represented in many cultures are being representative of balance and equanimity, we can see that in pure musical terms symmetry often has the opposite effect. When used melodically, symmetrical forms will disrupt tonality and create a characteristically dissonant or atonal quality to a line. They are often used as superimpositions or axis devices to create a tension that is resolved to a tone contained within an asymmetric diatonic structure. In harmonic terms, symmetry creates cyclical forms that do not allow for extended cadential points, or darker chord possibilities. When used compositionally, the forms that develop from this approach are often angular and create odd metres and a highly complex rhythmic density.
Contemporary jazz musicians strive to create a high level of contrast, to reconcile opposites. As the music increases in complexity, more advanced techniques are required to generate material. Employing a symmetrical device does not in itself create balance in the music, in fact it imbalances the music. It is left to the musician to employ other techniques that will counter-act the effects of symmetry, lyrical playing or use of space. Yet by the use of symmetry and its strong impact on the music the stakes of the game are raised, the dramatic curves are amplified, the distance between peaks and troughs is increased. Although this increases the challenge for the musician, the rewards are increased too, and striking a balance between the many disparate elements that are now found in this music all the more satisfying when it is struck.