FRACTALS, CHAINS AND THE GOLDEN RATIO:

MUSICAL APPLICATIONS



mp3 examples

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about fractals

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Of course, some fractals, such as the Mandelbrot above,  involve more complex mathematics which are far beyond the purpose of this explanation, but I think this “simpler” example illustrates the point beautifully.

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As far as the development of the fractal music system I ended up using is concerned, the whole thing came about almost by chance. I was already having lots of fun rendering graphic fractals as sounds, but I began to wonder about the possibilities of applying this geometry to form and structure, instead of pitch, dynamics, timbre and rhythm. Preferably a system that I could incorporate into my personal style, as opposed to something I worked on in parallel.

Anyway, I just happened to be studying the piano version of Stravinsky's Rite of Spring. I was quite taken by the way in which Stravinsky managed to maintain interest and ascertain forward momentum while using harmonically static material such as ostinatos (loops) by alternating them with other ostinatos, and nearly always changing their length. This can be clearly observed in the beginning of the Games Of The Rival Tribes:

I was also particularly captivated by the way he created forward momentum by making motifs come out from inside another, so to speak. As opposed to minimalist-styled transformations, where textures just fade in and out of each other, Stravinsky alternates progressively larger instances of a new motif with progressively smaller instances of an ongoing one. This is best illustrated by what happens from fig.84, in the ‘Mystic Circle of the Adolescents’. We first hear what we’ll call Motif A for 3 bars:

 

A Motif B pops in for a single isolated bar surrounded by rests and a special motif which we’ll call X (we’ll come back to Motif X later on), until Motif A comes back slightly altered, and only for 2 bars this time:

Motif B comes back and stays for longer, this time for 4 bars:

On the fourth bar, we hear a two-bar hint of Motif C overlapping with Motif B:

 

Motif B returns one last time for only 2 bars:

 

...before Motif C settles in for 10 bars:

Also of interest is the transition between the ‘Mystic Circle of the Adolescents’ and the ‘Glorification of the Chosen One’, which is where Motif X we mentioned above becomes the “seed of change”. We first hear a hint of Motif X 28 bars into the Introduction to the second part, the section before Mystic Circle, as if something very large is slowly approaching (a feature that was very effectively exploited in the movie ‘Fantasia’), at figure 85. This is followed by 28 more bars of the Introduction:

We hear it again 49 bars into the Mystic Circle, one bar before fig. 101. That’s 77 bars after we first heard i! It’s also a little bit more pronounced this time. But now we only have to wait for 5 bars until it comes back again, this time as a major climatic component that lead us into the very intense Glorification:

Note for movie buffs out there: this is very similar to the way Dennis Hopper edited the transitions of the classic Americana flick ‘Easy Rider’, which he probably assimilated from the French Masters Goddard and Truffaut.

Meanwhile, I had just come across an analysis of John Cage’s percussion sextet First Construction (In Metal). I was fascinated by his multi-scalar, self-similar (fractal-ish?) textures. Micro-macrocosmic relationships strongly feature in most of his percussion music for almost 20 years. This is how that piece in particular works:

He starts with a 16 bar series of phrases - everything about that piece revolves around the number 16 – and the length of each is laid out in the following pattern:

 

From this pattern we can build the first section by reiterating it 4 times, since the first phrase length number is 4:

By relating the number of phrase lengths in bars to the size of each section we can get the shape of the whole piece:

I decided then to attemp a combination of both formal devices. So, I came up with a bunch of ostinatos, and mapped them out according to a self-similar pattern, in the process which I describe below. The first difference one may note between this approach and Cage’s is that while Cage only specifies the proportionality of phrases occurring in the structure, what I did was to assign each proportion to specific material, and emulated the Stravinsky-styled transitions I mentioned above by making a motif “emerge” from within another.

Throughout the years I have experimented with many different shapes and forms, and eventually settled on a relatively simple and straightforward one:

 Q (unit = 1 bar):

 

meaning: 3 bars of A, 1 bar of B, 2 bars of A, creating a 12 bar-long structure out of two distinct elements, Let’s call this Structure Q (we can call it anything, but if we perform 4 iterations of the process, we’ll need all letters from A to P, as we’ll have 16 different motifs). 

There are two main problem with using Cage’s Proportional Durations for these purposes however. Firstly is the potentially excessive length of the final structure. After two iterations the structure would be 144 bars long (12 sections of 12 bars each), and after three iterations, 1728 bars long (12 x 12 x 12). Secondly, after so much repetition the pattern would eventually become predictable to an attentive listener. There are compositional ways of disguising this, of course, but that would defeat the whole purpose of this exercise entirely. 

So, instead of creating 11 more structures using these same proportions, we can make just one more, which we’ll call Structure R:

 R (unit = 1 bar):

 

Now we can do to Q and R what we did for A and B, and C and D, and create a Structure X, where each proportional unit is 2 bars long:

 X (unit = 2 bars):

 

If we peek into Structure X, we can see all the bars it’s made of:

 

If you look at the proportions of A, B, C and D, the whole thing is pretty chaotic-looking already, so the pattern would not become predictable. And this is only after the first iteration. If we were to create a structure Y in the same manner as X (introducing structures S and T, each composed of E, F, G and H):

 Y (unit = 2 bars):

 

...this is how the whole thing would look like, after mapping them into the proportions of an even larger structure Z, where each proportional unit is 4 bars long:

 Z (unit = 4 bars):

 

And this is what Z is made of:

Now if we look the pattern of occurrence of each of the 8 original elements, we'll find that there is only one, and it’s not very apparent (I only found it by accident!): if we split Z down the middle, the second half mirrors the first, but only as far as element changes are concerned. For all intents and purposes, we got an apparently random sequence out of a simple, straightforward process. And only 48 bars are necessary, as opposed to 1728! That's the power of fractal geometry at work: complexity of result created out of simplicity of process. 

And, yes, there's nothing to stop you from bringing in another similarly generated structure to be chained with this one, and so on, theoretically forever, but in practical terms one must stop at some point. After running numerous formal experiments for over a decade, I found that in order to get that really nice feeling of balance between randomness and predictability that one gets from contemplating fractals on a computer screen – what I like to call The Escher Effect - a minimum of 3 iterations is necessary. Coincidentally, Harlan Brothers also specifies - and here I am just paraphrasing - that in order for a structure to earn the title of fractal, a minimum of 3 levels of scalar affinity must be present, as with less than that, mathematical power-law relationships will not exist (meaning we cannot see if the transformation is constant at larger scales).

I decided to name it Fractal Würfelspiel, after Mozart’s and Haydn’s Würfelspiel (“game of dice”) pieces. These consist of individual bars which can be arranged in any order by the throw of dice, and the result will sound musically coherent. Of course, the system itself does not guarantee that, therefore the original material has to be carefully composed.

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Another geometric process that I thoroughly enjoyed investigating was to perform significant changes of direction in the musical structure at specific points, by mapping out the Golden Ratios of the entire structure recursively. The Golden Ratio, also known as Golden Section, Golden Mean, or even Divine Proportion, is an irrational mathematical constant, a ratio according to which the sum of two quantities to the larger one equals the ratio of the larger one to the smaller. It is represented by the Greek letter Phi:

This proportion is found to occur naturally in quite frequently. It is present in the proportions of the human body, the human face, certain branching structures of plants, shells, etc, and it relates to a perceived sensation of aesthetic pleasure. It just simply feels right! It has been used consciously or intuitively since ancient times in architecture, arts and, naturally, music. Works by Chopin, Debussy, Bartók among many others display an awareness of this proportion.

Recursive mapping of these sections in a piece involved dividing its full length by an approximation of the value of Phi (as it is an irrational number), in this case 1.61803398874989. Let’s say that we start out with a 10 min long piece. The golden section of the entire piece, which we’ll call Phi1, will be at 6’11” approximately. Now we find Phi2, the Phi of each side of the main section, which will happen at 3’49 and 8’32. Now we have four segments of the piece, and we can find the Phi3 of each one of them: 2’22”, 5’17”, 7’38” and 9’27”: 

The most significant change in a piece will happen at Phi1. Phi2 will still be quite significant, but not as much as Phi1; Phi3 will be less significant than Phi2, etc. In order to make the result more musically coherent, I usually round it up to the nearest bar line, unless, of course, I’m working on a free-time electronic or improvisatory piece. I actually made a nice little Excel utility in order to speed-up the process. Later, I found out that Bartók used exactly the same process, most notably in his Music for Strings, Percussion and Celesta, although he probably didn’t need Excel...

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One final and very important influence here was Lutosławski’s Chain Form. Chains consist of multiple overlapping layers of musical material of diverse lengths and starting points. Say that we start with a texture A. Sometime later a texture B will begin, and once it has established itself, A would end and maybe a texture C would emerge from within B, and so on. Another great way to achieve forward momentum using non-tonal material (or even tonal, for that matter):

I particularly enjoy using each fractal structure as a separate texture in the chain. This allows for much more flexibility and expressivity as far as the musical flow is concerned, and it does help to bring out the Escher Effect.

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