Jazz Mapping

Dimitri Vassilakis 

Jazz Mapping - basic principles to explore  (c) 2016

(for the research team @ Georgia Tech & Georgia State Universities)

On this writing some basic harmonic conceptions and principles will be presented - from the standpoint of a jazz improviser as a real time performer - with the purpose to find solid departing points for the research.

The approach is from the point of view of tonal jazz and at first we will be examining single line horn solos over a given harmonic structure.

Few examples will be given to open a further discussion on the mapping, the descriptors the values and the methodology. 

Some examination will be made on the different styles and approaches for tonal performance, looser forms and freer playing, the meanings of phrases and the intent behind some of the examples.

The tritone, as a multi-vehicle for harmonic movement, will be examined, under a new light, comparing its use in functional, modal and blues harmony:

  • As the 11th harmonic F# comes high in the harmonic series.
  • C-F# is dissonant and needs to resolve when part of a D7 Chord 
  • C-F# is not dissonant and doesnt need to move if part of Cmaj7#11
  • C-F# is used as color when part of a C blues scale
  • Modal uses of tritone are extensive both for creating tension and release
  1. Harmonic series and tritone points 
  2. Chords hyper structures and symmetry 
  3. Harmonic tension and movement
  4. Other descriptors vertical, horizontal, rhythmic movements, thematic development. 
  5. Mapping of simple short phrases to compare descriptors and possible values 
  6. Comments on «meaning and intent» behind phrases in a real time jazz performance   
  7. Texture, sound, expression, nuances.
  8. How can we code «interesting» 
  9. Harmonic series and tritone points 

        a) Harmonic series and tritone points

        One of the basic rules behind tonal music/harmony/harmonic movement – or else behind organized meaning in music - is the principle of tension and release, arrival and departure.

This is especially evident in mainstream jazz improvising and we will be looking at it from that point of view.

A simple tune, phrase, melody or song has is a departure and an arrival point. 

The most important harmonic movement is the so-called V-I cadence. 
All other harmonic, momentary and so-called secondary cadences will be considered in this setting mostly complimentary, as to contribute to the variety and development of the larger scale harmonic movement. 
They also carry different values of color, tension and release and help in forming harmonic structures. 

To simplify even more and to strip down harmonic meaning to the basic core – similar to 
schenkerian analyses  - we understand harmony in reality as a bipolar of 2 harmonic areas: 

V = Tension, movement, momentarily a departure point to 
I = release a momentary arrival point. 

Under that light we can analyze easily all harmonic movement in song form and jazz standards. 

But why do we consider V as tension and I as release in tonal functional western harmony and especially in the jazz genre?

The answer maybe lies in the harmonic series.

Since the beginning of time we humans listen to sounds that mostly come from sources that produce the harmonic series, so even subconsciously we are aware of these byproducts of a fundamental sound. 

Below is an illustration of the harmonic series in musical notation. The numbers above the harmonic indicate the number of % difference from equal temperament (rounded to the nearest percent). Blue notes are flat and red notes are sharp.

harmonic series

Consonant intervals - as we all agree to perceive in western tonal music - are the ones that are produced first from the fundamental tone so we get 8ves 5ths, 4ths and 3rds. 

As we move further up, on the series of byproducts, we realize that these upper structure notes are very weak in the mix and we barely hear them. 

Looking further at the upper structures we see that the 7th/2nd are far to appear and even further higher on the series we finally find the augmented 4th.

That interval really made to serve the harmonic framework of western tonal music and especially jazz as in functional harmony.

And the reason is that this interval is found between the 3rd and the 7th of the V chord - so called dominant chord. 

The function of this interval - when it has its position there - is very important, as it needs to resolve into a consonant interval like a 3rd or even a 5th
Thus the V chord becomes the vehicle that enables harmonic movement and we can move to another chord in a natural and somehow compulsory way. 

(Par example on a G7 chord moving to C then tritone B-F resolves to C-E, or on a Db7 to Gb major then tritone F-Cb moves to Gb-Bb).

Thus we are able compose and harmonize songs with a harmonic framework starting from a minor or major arrival/starting point and move, with the aid of different tonal chords, to points of more tension. These can be a V chord and other secondary dominants, as departure points or secondary tonics as arrival points and we can continue in that manner until the final cadence.

If we do have a tritone as part of a tonic type chord then as this tritone is not present between the 3rd and the 7th of the chord it doesn’t alter its character.  
These tritones (say on Cmaj7#11 = C-E-G-B-F#) add more color and harmonic richness to the tonic chord.
On the other hand we can make a V chord more intense/rich/ if we add more tritones. So a V7b9 chord has another tritone between the 5th and the flatten 9th, if we add the #9 then there is another tritone between the 6th and the #9 and if we add the augmented 4th we have another one between tonic and aug4th. 

In that manner we achieve the necessary harmonic movement and we can depart from one point to another until we hit another arrival or another/different departure point.

b) Chord hyper structures and symmetry

Here we will discuss another function/role of the tritone somehow opposite from the approach on a). 

One of the existing problems in tonal music regarding chord extensions and upper structures is to preserve the initial meaning of the chord and its place in the harmonic framework. 
If we built a full chord in C major adding thirds we will arrive soon on the 11th that is an F and an avoid note! as harmony textbooks declare. 

This is because we have made an agreement that C major is a tonic - release - arrival structure and this type of chord cannot be also a dominant chord. 
But this is clearly the case if we keep the F (11th) as now we got a tonic type chord up to the 9th C-E-G-B-D together with an overlapping dominant chord G-B-D-F… 

The most important incompatibility though comes from the breaking of symmetry or the initial “agreement” or code that we started on and that was: 

Major 3rd - minor 3rd or M3m3

With B-D-F on top we get 2 minor 3rds and that loss of symmetry breaks the agreement and the nature of the chordal structure that sounds also not at all agreeable. 

So when we add the 11th on a major chord then we sharpen it and we thus get 
Cmaj7#11 = C-E-G-B-D-F# that sounds good…

So here the tritone C-F# is quite welcome.

If we continue to built the full chord until we get to the tonic C, we get:

and again we are faced with the same problem of having two minor 3rds in a row. 
These we need to correct to avoid the tonic/dominant duality and also to avoid breaking the symmetry = our agreement. 
So we must arrive at a structure that has C# on top of the C major chord!!
Although we didn’t expect this it sounds good, as symmetry rule is not broken, by keeping the M3m3 pattern. 
If we continue on this manner we will get a really big structure or polychord in order to arrive finally at the tonic C obeying to the M3m3 rule:


Harmonically is like having a polychord: Cmaj7/Dmaj7/Emaj7/F#maj7/Abmaj7 

Of course in a jazz tune we don’t often use such huge polychords, but when we do want to play the perfect 4th on a linear solo - on top of a usual major chord - then we avoid keeping it as a long melody tone, but we resolve it nicely to the 3rd or to the 5th

See example from Clifford Brown’ s tune: “Joyspring”.


Or from Charlie Parker’s solo on “Now’s The Time” 

parker short

Now if we examine the case of the Dorian mode - that is extensively used in jazz, modal and bebop styles - we see that the agreement here is the opposite: 


If we built the full chord in say D minor tonality we get:

So in that case we do arrive safely to the tonic, not breaking the initial agreement and keeping the symmetry. 

In result we find that the Dorian mode is an extremely useful mode as any note of the mode played on top of the chord - even as long melody tone - will sound agreeable with no necessity to move it or to correct it. 

These upper extensions will produce a very agreeable sound and a different harmonic color, value and intensity, but still a point of arrival.  

In comparison we can examine the equally popular Aeolian mode or natural minor. 

This mode is not functioning that well under the symmetry prism.
We see that in this case the scale has a Bb as its 6th degree.

The result is that when we built the full chord = 
D-F-A-C-E-G-Bb-D we are faced with two minor 3rds between E-G-Bb, so while this chord structure starts initially as a tonic – release - arrival D minor, it moves on to sound more like a dominant – tension C9. 

When we improvise we tend not to keep that Bb as a long melody tone on top of the D minor chord, but we resolve Bb note usually to note A.

For these symmetric structures/chords there are some very interesting patterns that Professor Thanos Economou has designed to reveal the level of symmetry in the structures.

Below we see a first set of figures describing the major system.
The first is the cycle of 5ths, second is the M3m3 pattern discussed above, and third figure is the combination of both.


This next set comes from the minor system and the m3M3 pattern as it forms gradually. 


                Symmetry versus tonality is an interesting area for research and discussion. Other very common symmetric structures are the diminished and whole tone scales and chords. When we use      
                these harmonic structures in jazz we very often tend to play symmetric patterns, producing syntax that is moving away from functional harmony.

               c) Harmonic tension and movement 

                When we look at it from the point of view of a linear horn solo then harmonic tension and movement is achieved by the momentary resolving of tendency tones, called voice leading.
                These are commonly the 3rd and the 7th of the chord and in jazz are also called guide tones. 

We very often see Charlie Parker, and many other bebop or hard bop player who followed playing licks/phrases resolving by moving the 3rd or the 7th.
In order for a computer system to sound like there is a direction and intent in producing a jazz solo, patterns are not always functional in a harmonic environment, as they have no harmonic direction and meaning. 

Patterns can successfully sound as licks or phrases, but if we bind many of them together - with no harmonic direction - then the result will be patchy.

Phrases in jazz/bebop - and even in more open forms like modal - serve as bridges between the chords to describe the harmony the theme and use syntax like language. 

Harmonic rules can make a difference but the real development will come when the meanings of phrases and the syntax are fed to the system.

d) Other descriptors vertical, horizontal, rhythmic movements, thematic development 

Apart from harmonic issues other important descriptors that we need to add values to are the rhythmic movement, vertical or horizontal playing and thematic development. 

A simple phrase can develop with repetition, augmentation, and diminution, can be transposed to other keys, when the harmony modulates, can work in counterpoint with another thematic idea etc.
Other uses of the thematic tool are to develop the initial theme/fragment by altering one or few notes, the attack, articulation, sound etc. It will be a very interesting tool for a computer system to utilize, as it will produce improvisations that will sound more “human”.

Rhythmically the displacement, the syncopation, how dense or how sparse the phrases are, the accent, the flow and most importantly the different degrees of “swing feel” will have to be encoded. Techniques like “laid back” or “on top of the beat” are commonly used to characterize jazz players for their particular style of play.  

e) Mapping of simple short phrases to compare descriptors and possible values

If we start by examining just a couple of notes like this 1st example then it is evident that they can carry much different harmonic value/intensity/interest depending on the underlying harmony.


Another example below for a 
Gm7-C7-Fmaj7, II – V – I  movement.

Ex.1 has little harmonic tension as it uses the basic chordal tones in an arpeggiated manner. 

Ex.2 carries more harmonic tension as it uses the #5 of the dominant chord that resolves to the 9th of the tonic. 

Also the rhythm is more interesting as it has triplets and 8th notes, plus a better flow. 

Ex.3  we got same harmonic changes but we have put one chord per bar to leave more space for the phrases - we have few more added elements. 

Apart from the rhythmic movement and added syncopation we get more harmonic tension by a more intense dominant chord with the added b9, #9, b13, plus the #11 on the tonic F major chord. 

Ex.4 we move even further in harmonic and rhythmic intensity by using the famous Coltrane «A Love Supreme» reference theme, that produces tonal notes and extentions on the upper structures plus with a hint of motivic development.









These and more elements are incorporated into a solo for the next longer example by hard bop tenor saxophonist Hank Mobley. (Ex.5)
This comes from his solo on “This I Dig Of You” and I site here his 1st chorus on the 2nd A of the form. 

He offers a very good example, as he speaks the bebop language in a fluent and simple manner and we can extract some clear meanings and uses of the syntax.



He starts diatonically describing chords with the basic important notes and with an easy swing relaxed time feel.

Here we see on bars 7-8-9 the same II – V – I movement on F major as above and how nicely he builds his ascending phrase, adding more intensity as he goes higher also by adding the b9 and the #9 on the dominant chord C7.

Notice also how the voice leading works almost all the time, connecting chords and bars and contributing to the flow of the harmonic rhythm.

See how the 7th resolves to the 3rd on:
bar 8 to bar 9 from Bb to A, 
bar 10 to 11 from A to G, 
bar 11 to 12 from D to C#, 
bar 12 to 13 from G to F, 
bar 14 to 15 from F to E…

He achieves a nice rhythmic and melodic flow combining diatonic playing, nice swing phrasing, voice leading and rhythmic variety. He is one of the “in” players. 

The following is a famous Charlie Parker phrase that opens his solo on his 12 bar blues tune “Now’s The Time”, that describes very clearly a major chord. 



This is used very frequently by jazz players to make a Parker reference and “says” - describes the chord very clearly, 
This phrase usually sets up a good time swing feel and if we examine the end of it we get contrast and rhythmic interest. This is done by the use of dotted quavers creating a feeling of rhythmic division in 3 - thus adding to the intensity and excitement - to give even more harmonic meaning, although Parker here just plays the tonic note.

Next example comes from Sonny Rollins a master in the hard bop language and a very adventurous improviser, who was also especially interested in thematic development and fragmental improvisation. 

Here we see how he opens his solo on his well-known tune “St. Thomas”. We again get a famous opening that sometimes is used by players as a Sonny reference.



Here he starts the motivic development by a very simple opening interval between A and D on D major chord. This he develops throughout his solo by slightly changing notes to neighboring tones that correspond to the harmonic movement but also with his very strong rhythmic ability.
He does that using the so-called rhythmic displacement effect, when he places that phrase or similar intervals in different beats of the bar.

Another important aspect in jazz improvisation is the “blues” sound. 
The blues scale in D is: D F G Ab A C

This is something unique that jazz has brought to enrich the harmonic and melodic language of western music. 
The blues scale is a very common and useful harmonic and melodic tool in jazz that also sheds another light on the tritone case. 

Any jazz player to be considered seriously has to have the blues feel in his playing, not only by playing on the blues form or using the blues scale, but by knowing how to use that sound in different harmonic settings and project successfully its rich meanings and references. 
That is a very important aspect of jazz improvisation, as important as swing feel and voice leading. 

It will be a very crucial tool to add to any computer model for improvisation to serve the melodic and harmonic development in the way that the greats were able to express.
Here again from Charlie Parker “Now’s The Time”, see how he plays the D blues scale on tonic chord D major in bar 4 of a blues sequence.


Here the tritone is not used for a V-I cadence but just to add tension and bluesy character to the solo especially evident as it’s played over the major chord.

The next one comes from Oliver Nelson and his opening choruses on “Blues And The Abstract Truth”.



This is a minor blues sequence and here we see the use of open intervals 4ths – 5ths, use of longer notes, sparse playing, repetitive figures and thematic/motivic references. 

Most importantly here we witness the modal style of playing, in the sense that the phrases in the opening bars don’t need to resolve but they hang out there on top of the underlying harmony well placed in form. 

Modality is the opposite of functional harmony where phrases resolve with voice leading rules. Modal playing has opened another huge and rich area for jazz improvisation to explore and when we map jazz solos these different approaches have to be noted. 

The next two examples come from great contemporary players that have also been considered as “out” players as they have explored more extreme and complex areas in harmony and chromatisism. 

Michael Brecker often uses superimposed triads over a simple harmonic framework as in the following example over a simple Cm7 chord. These triads can create more tension and also add movement to a static underlying harmony. 



The next one comes from Dave Liebman from his book “Chromatic Approach” and shows substitute chords that add chromaticism and tension to the II-V-I sequence.



These few examples serve just as a starting point for discussing the mapping process and to reveal some of the syntax tools of the language.

f) Comments on «meaning and intent» behind phrases in a real time jazz performance 

Resolving phrases by realizing the voice leading, using the blues sound, utilizing motivic/thematic development, extracting the meanings from the phrases and analyzing sound nuances, will give an air of musical “intention”, “meaning” and “excitement” to a computer improviser model. 

In a sense although we are looking for Parker – ism, Sonny – isms, Trane – isms, Brecker – isms, etc, arriving into any – isms, for the new developing player when he thrives to realize his personal voice, we need to see all these under the proper perspective into a harmonic and melodic framework. 
Pattern - isms can describe the players and the styles and can be extremely useful for players or computer systems, but when they acquire harmonic, melodic, voice leading etc. “meaning” and direction then we can start touching upon creativity into jazz improvisation.

g) Texture, sound, expression, nuances

Another different subject is to examine sound. 
All in the jazz world agree that one of the most important aspects that make a great player is that he is immediately recognizable from his “sound”.

While great players have a great and unique sound, they don’t use all their sound tools any more.

This is a very important aspect that has changed a lot in jazz history. 
As players develop more harmonic information they tend to rely on sound nuances less and they play with uniform sound, while traditional jazz players had a lot of expressive colors in their playing. 
We can compare here Ben Webster or Lester Young or Coleman Hawkins with modern players like Dave Sanborn or Michael Brecker etc., who play great, but nevertheless are using a limited sound vocabulary. 
This will become very evident when we examine ballad playing, where vibrato, subtone and other more human like vocal effects are evident on the old school cats.
We should examine samples from players to detect crucial aspects that are responsible for certain texture and timbre effects they produce.

Also harmonic tension can be produced by just sound itself. 
Difference in attack, articulation, bending, vibrato, timbre can make just a strategically placed note very important in a jazz solo and can create excitement.
Players like Miles Davis where able to express a lot with very few notes.

We can store favorite phrases from great players that characterize their playing and together with the topics and approaches touched upon above, these phrases will gain the necessary direction and harmonic meaning.

h) How can we code «interesting»

What is regarded as beautiful in a jazz solo is a matter of a big discussion with players and listeners and especially with the greats that are still alive. 

This discussion will get us deep into the insights of this great art form and will shed light on many gems of the language.
Usually when someone plays well - placed phrases, nicely executed in time, with good sound/tone/timbre/texture etc, with relevance to the harmony and some bluesy flavor, then we usually say he plays good. 
And usually his phrases describe the underlying harmony very well. 

There are many aspects on how to utilize and navigate at this huge area as - while we can list a bank of useful or hot “licks” or phrases - these will work nicely depending on what is been played before or after, but they can become very useful as teaching tools and as a means to build a good solo with proper references to the language. 

Same goes for finding phrases or fragments or patterns or longer structures that produce: “humor”, “wit”, “high energy”, “powerful”, “emotions”, “relevance”, “reference” etc.

We can also - in the first stages - feed the computer model with a large amount of ways to “say”/describe the chords and the harmonic structure. We can do with as many functional or modal phrases we can find from the repertoire and vocabulary, which is easily accessible with many transcribed solos available. The important thing is to isolate the phrases and map their meanings adding the descriptor values.

In a real performance when a jazz player improvises he can draw from his bank of phrases or his bag of tricks, but he is also trained to “spill out” the chords in different manners in real time. 
That way we can identify his special way of playing like when we talk about mannerisms in speech. 

The use of pre-learned patterns is also very important and self evident when we listen to many great players but this is only used partially. 

We could also plan how patternistic, modal, bebopish, thematic, linear, melodic, fragmental, bluesy etc we want the new solo to be and in that manner we could create many interesting and contrasting new improvisations.

In short here are some 
descriptors that can acquire values: 

harmonic tension/movement, timbre, articulation, attack, linear/vertical, voice leading, blues, phrase references, thematic development, rhythmic intensity, swing feel, density, pattern playing, functional-modal.  


To conclude I could suggest that we can focus on 3 main areas. 

One task will be the mapping itself that needs a lot of detailed work for every phrase, sound and approach.

Second task will be the more immediate use of some of the mapping principles, to feed the existing computer models, so they will become more creative, developing thus a kind of intent and direction when playing or reproducing a solo for performance and/or educational purposes. 

Third area is to start working with visual or other performing arts and connect the mappings with analogous valued descriptors.

We can present these descriptors for discussion and evaluation to jazz greats, while a method of evaluation has been discussed in relation to also presenting these mapping values to audience. 

Talking to living legends like Sonny Rollins, Wayne Shorter, Dave Liebman, McCoy Tuner, plus younger players like Jeff Tain Watts, Christian McBride, Dave Kikoski, Benito Gonzales, Roy Hargrove etc. the list is huge…(with many of them I am fortunate to record and perform with) will further evaluate, deepen and accredit the discussion.

If we can succeed in emulating, up to a point, the processes behind a jazz player, as a real time improviser, then we will be getting closer to making a computer system, that reproduces an improvisation, react in a more creative way.

Dimitri Vassilakis 
May 2017 revision

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