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Endless Canons

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Intro to the musical fractals

Here is one of them:

Gentle crossing

If you want to hear more examples, go to the example tunes pages.

It would be hard to pick it out in that clip because of all the other factors involved, but actually this music is based on musical fractals.

Have you seen those visual fractals? As you zoom in and in on the picture, it looks similar no matter how many times you magnify it - not identical, but you keep getting patterns similar to the original pattern. The most famous is the Mandelbrot set.

They are common in nature too, "almost fractals" you could say because of course eventually there is a limit to how much detail you get to. An example is a tree, it divides into branches, and each branch into smaller branches, and twigs and so on, and a small branch looks quite like the whole tree. Streams behave similarly. Similarly a rocky outcrop on a mountain can look like a miniature mountain and so on. Coastlines are also fractal to the extent that it is hard to tell when looking at a map whether it is say 1 in 1,000 (1 m to a km) or 1 in 1,000,000 (1mm to a Km).

Here is an example - until you look at the scale, or maybe if you are familiar with the style of map, you can tell, but just from the coastline, it is hard to tell whether this map is a map of a tiny spot less than a km across, or (if you just look at the outline) even detailed a map of a tiny pond in a field - or a big ocean hundreds or even thousands of Km wide.


Images produced from the Ordnance Survey Get-a-map service. Image reproduced with kind permission of Ordnance Survey and Ordnance Survey of Northern Ireland.

Location is NX-496-986-GB

Our body is also made up of fractals - the lungs for instance, and the blood supply, brain, they are all fractal in different ways. It is because the lungs are fractal that we are able to breath, as all the intricate detail immensely increases the surface area of the lungs so that they can absorb enough oxygen for our needs.

Music is also fractal in its own way - particularly rhythm. When you play music, it naturally falls into bars. The bars fall into larger units, usually two bars in a call and answer type pattern. Then those fall into larger units too. Very often music comes in four bar, eight bar sections, and so on.

So anyway the idea was to use that kind of structure for the fractally generated melody. So the tune is built up into larger and larger melodic phrases, just as the patterns of beats are made up from larger and larger patterns of bars, two bar groupings, four bar groupings and so on.


The tunes are based on a canon by augmentation

Here is an example to show how they work:

ascending above the clouds - flute

There the fastest moving part is played by the flute.

The same tune with the first part muted, and played at eight times the original tempo, so you hear the second part, the Cor Anglais

ascending above the clouds - Cor Anglais

Repeat the process, with first two parts muted, and eight times faster again:

ascending above the clouds - Flute

Then finally, all the other parts muted, so we are just left with the Marimba

ascending above the clouds - Marimba

Put them together and you get the original tune

ascending above the clouds - flute

- do you hear how all the parts are playing the same tune - but at different speeds?

The marimba only gets to play one note in the clip - at a tempo five hundred and twelve times slower than the last clip, you just haven't heard enough of the tune yet to hear its second note.

Music that works like this is known as a canon by augmentation.


This canon is the basic structure behind nearly all the tunes

This is the basic structure behind nearly all the fractal tunes Tune Smithy makes. (There are one or two based on other principles).

However, it is transformed in many ways and it is often hard to pick out the original augmented canon.

With some of the later developments of the tunes, then the canon isn't really there any more as actual notes, because the notes themselves get transformed and affected by each other or by the various other options used.

But perhaps somehow it still has something to do with the reason this music seems to work, and cohere together, that it has this pattern behind it all, even if very hidden at times.

The one at the top of this page is a case in point as it is transformed by using a polyrhythm structure. The violin, orchestral harp and cello parts are playing seven, five and three beats to a bar respectively. The notes also affect each other, so the instruments don't all play the same tune as each other quite either.

Still it has that fractal feeling to it somehow, giving it a feeling of structure to it at larger and larger scales as you listen to it. Which I think is perhaps what can be so satisfying about listening to this type of fractal music.

The music is basically endless, there is no particular point where it can stop - though you can artificially set a bound on the number of layers of augmentation of the tune.

As to how this canon by augmentation is achieved in Fractal Tune Smithy, and how you can make as many as you like of your own too - that takes on on to seeds. The tunes are built up from a short seed phrase, which is played at larger and larger time scales.


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