fractal music
Nearly all
of my experimental
works (and some of the mainstream ones) composed since 1995 were
developed by incorporating (to varying degrees) two main
principles/thought processes:
1. Non-linearity: or fractal, controls the surprise, the unexpected, and is achieved by the Fractal Würfelspiel;
2. Linearity: maintains the flow and carries a listener through a work. This is delivered by recursive (fractal, self-affine) applications of the Golden Section.
These concepts are not treated as opposing factions, but rather like polarities, without absolute positions. When thoughtfully combined, these polarities become a powerful fountain of "life-like" musical material. These are not some dusty old mathematical formulas, but natural processes in action all around and within ourselves.
If traditional music composition is analogous to 'playwriting' and aleatoric/algorithmic composition is analogous to 'genetic engineering', the best analogy to this working method would be 'gardening'. One sows the seeds and watches them grow, trimming and prunning here and there, chosing the right flower-bed for transplanting, and designing the overall layout of the garden, but it is the plants who have to grow themselves.
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The Fractal Würfelspiel
This is one of my tools/processes of choice, featuring heavily in pretty much all of my works between 1995 and 2003. Diverse sources of inspiration contributed towards its inception, the most relevant ones being:
1. The Würfelspiel compositions by Haydn and Mozart, where a piece of music was composed of various independent bars to be re-arranged at random. The stylistic qualities of the composer would always remain present, irrespective of the resulting sequence.
2. John Cage's First Construction (in metal), in which the technique of Proportional Durations is developed. This technique involves the application of the same proportions to both phrase lengths and section lengths, thus creating a micro-macrocosmic relationship within a piece:
3. Fractals. These are shapes that exhibit similar (or identical) features at different orders of magnitude. The most popular ones are the Mandelbrot Set:
the Koch Snowflake:
and the Sierpinski triangle:
The following interpretation of the T'ai Chi symbol is most relevant, as it contains in itself the fractal structure, the concept of interlocking polarities, and the ideal of balance (even between peace and chaos):
4. Of equal, if not greater relevance, are the natural growth of structures such as clouds, forests, mountains, reefs, broccoli and ferns.
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Following is a brief description of a Fractal Würfelspiel at work. In theory, it can be achieved by any means, even pen-to-paper and a blueprint.
Step 1: The creation of initial material, the cells, which could be pretty much anything: an ostinato, a drum loop, a melodic/polyphonic line, a sound sample or (even better) a mixture of all of the above. How much diversity is there in between these elements controls how dramatically broad the resulting work will be. All cells should be of equal length. For the purpose of this explanation, let's say that they will last for 11 bars.
Step 2: Now, let's get these elements and mix them up. Let's arrange them into a specific sequence, which we'll call a 'matrix' from now on. We ca use any sequence we feel like. We could revert to th classics ABA, AABB, ABACABA, etc. But instead of looking over our shoulders all the time, let's combobulate something different. Arranged according to one of my favourite matrix forms, the 3 elements would look like this:
...meaning: 3 bars of A, 1 bar of B, 2 bars of A, 2 bars of B, and so on. I', particularly fond of matrixes like this in particular because new elements seem to sprout from inside one another. Roughly speaking: we first hear lots of 'A' followed by a little bit of 'B', a little less of 'A' with a bit more of 'B' and a tiny bit of 'C', and so on. We get the sensation of 'A' going away and 'B' growing from within 'A'. The pattern then repeats with 'C' growing out of 'B' and end with 'A' growing out of 'C' to complete the cycle. It ends exactly where the whole sequence would (and could) begin again.
I'm not claiming to have invented this kind of sequence. Stravinsky uses devices such as this in many passages from the 'Rite' in particular. Dennis Hopper and László Kovács also used a similar principle when editing the transitions for 'Easy Rider' (far out, man!).
Step 3: New material (or variation on the same material) is needed in order to create 2 more segments of equal length to the first. We could call the new elements D, E, F, and G, H, I. These are arranged by the same matrix, which will give us:
Group DEF
Group GHI
Step 4: Now this really gets interesting: let's multiply the length of
the bar by the number of initial elements, in this case: 3.
So if
we started with a 3/4 bar, now we'll be
using 9/4. The original matrix had 33 bars in total, but by changing
the time signature, it's now 11 bars long. This is the same number of
bars that each group had to begin with.
Step 5: Now that the ABC, DEF and GHI groups have only 11 bars each, let's give them new names. Let's call them 'X', 'Y' and 'Z' respectively. So what do we have now? We have three distinct elements, each of them being 11 bars long. Sound familiar? Yes, that's exactly what we had to begin with, in Step 1! Now, just for fun, let's arrange these elements using exactly the same matrix as before:
If we peek into this new sequence by reverting to the original 3/4 time signature, apparent chaos will begin to show itself:
Step 6: Lather and repeat as needed! New material can be incorporated or developed, on an ever-expanding process. An interesting and rich effect can be achieved by having one or more diverse lines generated by the same process starting at different times. The result can have an effect very similar to Lutoslawski's 'Chain Form' (interlocked self-contained strands of musical material of different lengths and/or starting points binding into a cohesive whole with forward momentum).
So, this is it! Even though the resulting sequence appears to be random, on attentive listening one could notice that there is a strangely balanced feel to it.
top
The Golden Section (a.k.a. Golden Ratio, Golden Mean, Divine Proportion) is a ratio or proportion defined by the number 'Phi' (1.618033988749895... ). It is a kind of universal constant of design, not only observed in the make-up of nature (humans, plants and animals, planetary systems), but also present in architecture, painting, stock market fluctuations. In certain works of art its occurence seems to be purely intuitive. A form containig this proportion "feels" universally balanced.
In my works, the golden sections mark a significant event (change of direction or occurence) in a piece. 'G1' is the main focal point, which correspond to the Golden Section of the entire structure. 'g2's are the golden sections of the two segments divided by 'G1'. More self-affine (fractal???) segments can be defined, theoretically to infinity. In the example below, four iterations of the Golden Section are used:
Golden Section Framework for a 99 measure piece
The significance or impact of an event is proportional to its relationship to the entire structure. In the example above, 'G1' is the most significant event, while 'g4' is the least significant one.
top1. Non-linearity: or fractal, controls the surprise, the unexpected, and is achieved by the Fractal Würfelspiel;
2. Linearity: maintains the flow and carries a listener through a work. This is delivered by recursive (fractal, self-affine) applications of the Golden Section.
These concepts are not treated as opposing factions, but rather like polarities, without absolute positions. When thoughtfully combined, these polarities become a powerful fountain of "life-like" musical material. These are not some dusty old mathematical formulas, but natural processes in action all around and within ourselves.
If traditional music composition is analogous to 'playwriting' and aleatoric/algorithmic composition is analogous to 'genetic engineering', the best analogy to this working method would be 'gardening'. One sows the seeds and watches them grow, trimming and prunning here and there, chosing the right flower-bed for transplanting, and designing the overall layout of the garden, but it is the plants who have to grow themselves.
mp3 examples
All tracks © by Dmitry Kormann. Use of any of these tracks is only allowed with the express permission of the composer.top
the fractal würfelspiel
The Fractal Würfelspiel
This is one of my tools/processes of choice, featuring heavily in pretty much all of my works between 1995 and 2003. Diverse sources of inspiration contributed towards its inception, the most relevant ones being:
1. The Würfelspiel compositions by Haydn and Mozart, where a piece of music was composed of various independent bars to be re-arranged at random. The stylistic qualities of the composer would always remain present, irrespective of the resulting sequence.
2. John Cage's First Construction (in metal), in which the technique of Proportional Durations is developed. This technique involves the application of the same proportions to both phrase lengths and section lengths, thus creating a micro-macrocosmic relationship within a piece:
3. Fractals. These are shapes that exhibit similar (or identical) features at different orders of magnitude. The most popular ones are the Mandelbrot Set:
the Koch Snowflake:
and the Sierpinski triangle:
The following interpretation of the T'ai Chi symbol is most relevant, as it contains in itself the fractal structure, the concept of interlocking polarities, and the ideal of balance (even between peace and chaos):
4. Of equal, if not greater relevance, are the natural growth of structures such as clouds, forests, mountains, reefs, broccoli and ferns.
top
the process
Following is a brief description of a Fractal Würfelspiel at work. In theory, it can be achieved by any means, even pen-to-paper and a blueprint.
Step 1: The creation of initial material, the cells, which could be pretty much anything: an ostinato, a drum loop, a melodic/polyphonic line, a sound sample or (even better) a mixture of all of the above. How much diversity is there in between these elements controls how dramatically broad the resulting work will be. All cells should be of equal length. For the purpose of this explanation, let's say that they will last for 11 bars.
Step 2: Now, let's get these elements and mix them up. Let's arrange them into a specific sequence, which we'll call a 'matrix' from now on. We ca use any sequence we feel like. We could revert to th classics ABA, AABB, ABACABA, etc. But instead of looking over our shoulders all the time, let's combobulate something different. Arranged according to one of my favourite matrix forms, the 3 elements would look like this:
Group ABC: 3 elements (A, B and
C), 11 bars
each = 33 bars total
...meaning: 3 bars of A, 1 bar of B, 2 bars of A, 2 bars of B, and so on. I', particularly fond of matrixes like this in particular because new elements seem to sprout from inside one another. Roughly speaking: we first hear lots of 'A' followed by a little bit of 'B', a little less of 'A' with a bit more of 'B' and a tiny bit of 'C', and so on. We get the sensation of 'A' going away and 'B' growing from within 'A'. The pattern then repeats with 'C' growing out of 'B' and end with 'A' growing out of 'C' to complete the cycle. It ends exactly where the whole sequence would (and could) begin again.
I'm not claiming to have invented this kind of sequence. Stravinsky uses devices such as this in many passages from the 'Rite' in particular. Dennis Hopper and László Kovács also used a similar principle when editing the transitions for 'Easy Rider' (far out, man!).
Step 3: New material (or variation on the same material) is needed in order to create 2 more segments of equal length to the first. We could call the new elements D, E, F, and G, H, I. These are arranged by the same matrix, which will give us:
Group DEF
and:
Group GHI
Step 5: Now that the ABC, DEF and GHI groups have only 11 bars each, let's give them new names. Let's call them 'X', 'Y' and 'Z' respectively. So what do we have now? We have three distinct elements, each of them being 11 bars long. Sound familiar? Yes, that's exactly what we had to begin with, in Step 1! Now, just for fun, let's arrange these elements using exactly the same matrix as before:
If we peek into this new sequence by reverting to the original 3/4 time signature, apparent chaos will begin to show itself:
Step 6: Lather and repeat as needed! New material can be incorporated or developed, on an ever-expanding process. An interesting and rich effect can be achieved by having one or more diverse lines generated by the same process starting at different times. The result can have an effect very similar to Lutoslawski's 'Chain Form' (interlocked self-contained strands of musical material of different lengths and/or starting points binding into a cohesive whole with forward momentum).
So, this is it! Even though the resulting sequence appears to be random, on attentive listening one could notice that there is a strangely balanced feel to it.
top
the golden section
The Golden Section (a.k.a. Golden Ratio, Golden Mean, Divine Proportion) is a ratio or proportion defined by the number 'Phi' (1.618033988749895... ). It is a kind of universal constant of design, not only observed in the make-up of nature (humans, plants and animals, planetary systems), but also present in architecture, painting, stock market fluctuations. In certain works of art its occurence seems to be purely intuitive. A form containig this proportion "feels" universally balanced.
In my works, the golden sections mark a significant event (change of direction or occurence) in a piece. 'G1' is the main focal point, which correspond to the Golden Section of the entire structure. 'g2's are the golden sections of the two segments divided by 'G1'. More self-affine (fractal???) segments can be defined, theoretically to infinity. In the example below, four iterations of the Golden Section are used:
Golden Section Framework for a 99 measure piece
| event | start | g4 | g3 | g4 | g2 | g4 | g3 | g4 | G1 | g4 | g3 | g4 | g2 | g4 | g3 | g4 | end |
| measure | 1 | 14 | 23 | 32 | 38 | 46 | 52 | 57 | 61 | 69 | 75 | 80 | 84 | 89 | 93 | 96 | 100 |
The significance or impact of an event is proportional to its relationship to the entire structure. In the example above, 'G1' is the most significant event, while 'g4' is the least significant one.