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Rhythmic Harmony 2

Musicians can improvise together on a form. The form is usually a chord pattern with few details. The pattern quite often belongs to a form family such as AABA. Improvisation is unpredictable by nature. If you wish to play something surprising together in a meaningful way and you cannot know what is going to happen, you must know at least where it is going to happen. The simplest and most reasonable hypothesis is that the song's overall structure points out where improvisation will deviate most radically from the expected. If the hypothesis has any merit at all, the form should chart an easy-to-find and memorable course through the piece which indicates to what degree chaos is "encouraged" at each point in the development of a theme.

I borrow elements of fractal geometry and harmonic synthesis in order to try to shed some light on how the form might appear to the improvising musician. This is a pictorial study of phrasing and rhythm where the song's structure is taken as a fractal or a waveform. It proves nothing.

Construction Technique

I use the following fractal construction: the divisions of the waveform are sub-divided following the same pattern. The sub-divisions can be pursued to infinity. This is known as a kochlike construction, after the Koch snowflake. The resulting line displays a property called self-similarity.

          /   \     
    o - o      o - o
The Koch snowflake. Each segment is subdivided 4 times for every generation. You get a striking squiggle. This classic is an inspiration for my own constructions.
         o - o         
    o - o     o - o    
The construction I use here. I will interpret the vertical axis as an intensity of disorder and the horizontal axis as time. Having disimilar axes, I will limit the action of division to horizontal segments only. This "ABA" figure is obviously similar to the Koch snowflake.

Initial shapes

click to enlarge!


This is the most basic even structure.
The succeeding divisions are of the form 2^n where n=0,1,2,3...

             o - o     
        o - o          

click to enlarge!


This is the most common even structure of all.
The succeeding divisions are of the form 4^n where n=0,1,2,3...

             o - o     
    o - o - o     o - o

click to enlarge!


This is an odd structure.
The succeeding divisions are of the form 3^n where n=0,1,2,3...

             o - o     
        o - o     o - o

click to enlarge!


This is an odd structure which might be compatible with the blues.
Again, we have the form 3^n.

             o - o     
    o - o - o          

Beat patterns, rhythm and song structure

may have the same shape

Measures serve up beats in groups of 2, 3 or 4 - and very rarely anything else. Songs come in structures such as AB, ABA, AAB, or AABA - there are others but these few account for the vast majority in many styles. This simplistic bent is one of the most striking features of Western and Creole music.

Now this simplicity would be helpful if self-similarity were an essential feature of how we play music. This seems to be the case with the AABA structure.

We could play 4/4 time with a special accent on the count of 3. The 3rd measure would be quite different from 1, 2, and 4. Measures 9 through 12 would be very different from the remainder of a 16-bar part and there would be 4 such parts in a 64-bar AABA structure. Obviously, each one of these divisions is of the form AABA. These words fit common musical usage and the AABA shape shown above.
One can easily verify that a self-similar construction principle is very common in a lot of (if not all) AABA music. It does not work very well for a shape such as AAB (blues). Odd ternary forms do seem to pose a bit of a problem: When you divide 12-bar blues into 3 parts, you are left with 4-bar segments which obviously won't be very cooperative if you want further sub-divisions by 3. I will demonstrate later on how limitations on the use of 3's in usage, far from being obstacles, can be the basis for new shapes.
I cannot fail to mention that walzes are counted in 3's but simply don't comme in 3-, 9- or 27-bar parts. They invariably come in 4-, 8-, 16-, 32-, and 64-bar groupings, often in AABA form. Having said that, I cannot fail to also mention that valse musette chord changes almost always occur in the 4th measure and almost never in the 3rd or the 5th - for an 8-bar part. This tends to give them a 3-flavor.
The chivalrous shark
A simple American waltz form
AABA in 16 bars
G             D            (D)            G             
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

G      G7     C      Am     D             G             
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

Le petit bal du samedi soir - part 2
A simple valse musette form
AABA in 32 bars
G                    D7    (D7)                  G      
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

G                    D7    (D7)                  G      
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

C             Cm     G      A7                   D7     
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

G                    D7    (D7)                  G      
/ / /  / / /  / / /  / / /  / / /  / / /  / / /  / / /  

The expected 4-bar AABA sub-structure is bent into something more like the AAB form we would sort of like to squeeze in - if it weren't contrary to common usage!

Self-similarity also accounts for a few informal observations. For example:

The vertical axis means chaos

If all music boils dow to a handful of basic shapes declined along kochlike principles, then one might expect to find only a handful of melodies. This is not the case. On the other hand (significantly), one does encounter only a surprisingly limited range of chords and chord patterns. The orthodox interpretation of song structure gives us a clue concerning the nature of the y axis.

The B part is understood to be, in some fundamental way, different from the A part. The B part is "not A" but otherwise unspecified. This idea should be familiar to anyone acquainted with popular jazz and classical standards.

The A part represents a kind of rule. In the AABA structure this notion is obviously reinforced by repetition. The B part is understood as a strong deviation from the rule. A represents order and B represents chaos. In my pictures, A is always 0. The vertical axis therefore represents chaos or disorder.

Chaos isn't just the breaking of rules. It is essentially unpredictable, regardless of what rules one might choose to break. Chaos ignores rules. Chaos is, in the numerical world, truly random, unlike the pseudo-random stuff you get from computer programs and other mechanical contrivances. "True" chaos is a very important part of playing and (obviously) improvising music. [This explains in a nutshell why computer programs used to generate music from scores are so boring!]

Musical sounds set up the expectation that something will happen. It is pleasing when, in certain spots, the expected does not occur. The forms examined here reveal where these spots could occur in common song structures. The pictures show the degree of surprise, the intensity of chaos, not the actual shape of the music as shown for example in a melodic score. Most music is made with many independant voices. With several voices, chaotic spots will be where the chaos will have the greatest density.

What makes this little study exciting to me is that it suggests how improvisation is at all possible. It also explains how several people can improvise together. To improvise is to have an adventure. One does have the sense that this involves some risk-taking. The risks involved can be minimized if you know in advance where you can get away with almost anything. If improvisation were rocket science then it wouldn't be done by people of all ages and all backgrounds. It is not rocket science.

Rhythmic and tonal harmony

Now let us turn to an extrapolation of the Koch construction. I, for one, wouldn't play music if the above drawings, cool as they may seem, were all there is to it. They are too constraining and a few are obviously downright boring. There is another approach: instead of viewing the song structure as a Koch generator, one can view it as a wave. No matter what the wave-form, it can be used as the basis for synthesis using a series of related wavelengths. This is very similar (if not identical) to harmonic synthesis of chords using tones.

Octave synthesis with AABA

generators Each wave is twice the frequency of the one directly below it. This is similar to constructing the sound of octaves in the tonal world.
Spectrum One can obviously represent the synthesis (ignoring the waveform) as a spectrum. This is roughly equivalent to a tonal spectrum with harmonics 1, 2, 4, 8, 32, 64, etc. This spectrum might go from 0.002 to 0.25 Hz, with an arbitrarily decreasing amplitude. The scale is much different of course in the tonal world for timbre, typically measured between 20 and 20000 Hz. Be that as it may, this is a true spectrum measuring something as a function of frequency.

Here is the new waveform which obtains with the octave synthesis I have just outlined:

New Aabaland

AABA octave fractal

Much to my surprise, the shape is self-similar but much more interesting than those obtaining with a direct kochlike construction.

Returning to Koch, one can simply superimpose AAB and AABA divisions of the same period.

Here is an "analytical" view of AABA+AAB. The two movements are obviously contradictory and non-coincident. This is expected of shapes based on prime numbers.


click for more!

The picture is relatively legible and perhaps even recognizable. A binary pattern emerges: there is a "low city" and a "high city". The shape is not self-similar.

This is roughly equivalent to the chord produced in the tonal world by combining a set of octaves with dominant tones stepping through the cycle of fifths - similar to the chord made by the open strings of the violin. A very busy sound indeed! A fundamental difference between this and the sound of two simultaneous tones lies in the fact that the AABA and AAB shapes have dissimilar spectra. All tones of one instrument have similar spectra, by nature. This shape seems to fit almost nothing in common musical usage. There is, however, John P. Birchall's observation that playing 3 on 4 is an essential ingredient of jazz.

I cannot recommend John's website too strongly. Much of what I develop here I found there first. And there is much I needn't say because John has already said it better than I ever could.

To obtain a decent "3 on 4" picture was one of the main reasons for doing this study. I honestly don't know if "this is the one" but it does express the enormous tension known to be associated with the musical performance of "3 on 4".

Initial conclusions

The astounding accuracy with which the AABA self-similar decomposition (using either the crude "kochlike" or the "octave" method) matches established usage may explain why the form is so popular among improvisors. It has been most popular in Europe since the Renaissance. Indeed, lute pieces by Cutting, Dowland, Pickering, Robinson, and many other early composers follow this structure.

The application of the idea to ternary structures is more problematic. It seems to me that music just doesn't seem to be organized along self-similar ternary lines. A case study might be helpful.

The strange case of the blues

The structure is AAB. But the next sub-divisions follow a binary pattern. The 12-bar blues is commonly felt to be:


that is, a very binary shape. Let us look at an example!

I've Got My Mojo Working

Frank Frost

I've Got My Mojo Workin' but it just don't work on you                       

            C                   F7                 C              I've Got My
            /  /  /      /      /   /     /    /   / / / /  / / /      /     

            Mojo Workin' but it just don't work on you                       

            F7                                     C             I love you  
            /  /  /      /      /   /     /    /   / / / /  / / /  /         

            So  bad I just don't know what to do              I've got a...  

            G7                                 C   F C  C G                  
            /  /  /  /     /    /     /    /   / / / /  / / /      /         

Mojo Structure

Mojo guitar details

It is easy to be convinced that this blues, despite having an AAB structure, is completely binary in spirit. It is in 4/4 time with equal 8ths. Even the 16ths are equal! This appears to be a glorification of the gods of binarity. There are however a few hints to show that there is some threeness here:

When you improvise an instrumental chorus, it is possible to attempt ternary shapes. This is a bit like constructing an Eiffel tower with a triangular base using rectangular Lego bricks. It might be feasable but it is not easy. It takes imagination. It can be successful...well, almost. The challenge is very entertaining - so much so that one tends to do it over and over again!

Conceptuel leap: while we are busy playing chorus after chorus, all we really need to do is to inject more and more wild disorder as we go along and we draw out the AAB pattern (or any other we might choose, for that matter) almost without effort - not in the details of a single chorus but in the overall progression through the song. Obviously, self-similarity can be also constructed with multiplication as well as division!

The fact that ternary sub-cells don't fit the structure doesn't mean that such an organization can't be reflected in playing. It is very probable that shapes are based on alternating generators as well as simultaneous ones. One might not be tempted to express a 4-bar cell as 3. On a smaller scale, a 4-beat bar may well yield 3 beats for every 2. Incompatible divisions may be anchored to each other from scale to scale without mixing. Alternating generators suggested by the "Mojo" structure might look like this:


This picture gives a fair account of some aspects of the blues structure as described above. The AAB, AB, and AABA forms are all kept in separate boxes, so to speak. Here are other pictures made with AB and AABA (octave version) structures but this time, they are allowed to mingle. As before, the self-similar divisions are restricted to individual 4-bar cells.

click to enlarge! AAB + nothing but AB
click to enlarge! AAB + nothing but AABA (octave version)
click to enlarge! AAB + a combination of AB and AABA (octave version)

If we can take the notion of "more chaos at the end" to be an essential characteristic of the AAB structure, this could explain why some narrow-minded Fundamentalists see the blues as a satanic ritual. Rock concerts tend to finish on a distinctly chaotic note. This could also be why Jimi Hendrix often smashed his amps and burned his guitar at the end of his concerts. There can be no soft landing with an AAB structure!

Is this real?

My best guess at this point would be that the improvising musician might handle combinations of fractal generators of frequency 2, 3, and 4 with greater ease than one might think. The following commonplaces should be dismissed:

If you have the assumption that 2 is easier than 3 drilled into you from a very early age, that doesn't make it a fact, does it? If a belief is false, you can always change it!

It could turn out to be very easy for musicians to set up trains of motion having several consecutive generations of any simple generator, not noticing any real difference in comfort between, let us say, AB, AAB, and AABA. The same would be true of any relatively simple audible structure, such as the clave rhythm for example, possibly with very little training.

I would like to finish with a very simple and immediate application of this theory: the blues is swinging and swing is bluesey. Many bands play a combination of swing standards (mostly AABA) and blues (mostly AAB).

Sometimes I feel like a goldfish who took a few flying lessons and thinks he's an expert. The best possible application of patterns of chaos in your music is whatever suits you. You may not be aware of what this might be but it doesn't really matter. Such things do just fine in the sub-conscious.

What about the A part? This entire page has been devoted to some possible patterns for breaking rules. This is only a part of the story. Improvisation also involves rule-making. Paradoxically, there may not be a lot to say specifically about invention, other than: "be yourself!". This is easy to do because you really have no choice. The real key to improvisation is, therefore: "listen and pay more attention to others than to yourself!". This includes not only band members but also the entire audience. This is easy to do because your mind is more agile than your fingers and your fingers will continue to wiggle, no matter what you focus your mind on. You are most inventive when responding to others. Furthermore, rule-making very closely follows rule-breaking, in terms of form: obviously, some of the new ideas which inevitably spring up when you deal with chaos (and with other people) can be repackaged in a later chorus (once the smoke has cleared) as "new rules". This is not to say, of course, that making and breaking are the same. Rule-making has an entirely different feel and plays a very different role in the narrative development of a piece of music.

This is a subjective thing. The whole darn thing is completely subjective.

This page resolves some issues raised in an earlier article with the same title "Rhythmic harmony" found under the home page:

click on the clouds!

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