Sloth Canons - Play & Create Tunes as intricate as Snowflakes

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 a sloth canon or canon by augmentation, an example of what Harlam Brothers calls a motivic canon.

To find out more about this right away, skip forward to A sloth canon, or read on to find out a bit about fractals first.

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 - often not identical, but you keep getting patterns similar to the original pattern. The most famous is the Mandelbrot set (opens in new window).

A sloth canon is one possible musical analogue of a visual fractal (perhaps there may be others). All the parts play the same tune, only some faster than others. The faster parts, playing the same tune, then are a musical analogue of the small details that look the same as larger details in a visual fractal. There's also a uniformity to these particular sloth canons, the same material used over and over at all the time scales, which makes it a fractal construction.

For more about this skip forward to A sloth canon.

Or read on to find out more about visual fractals.

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Visual fractals

Visual fractals are common in nature, "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 to show how fractal coastlines are - 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.

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Connection with musical structures

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, often two bars at a time. Then those fall into larger units too. Very often music comes in four bar, eight bar sections, and so on.

So 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.

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Possible connection with natural sounds

Natural sounds like the wind, waves, and water flowing often have variations on many different time scales. I don't know about analysis, but to my ear, crashing sea waves sound fractal in nature, with details within details. Since both the waves and the coastline they crash against are visual fractals (normally), it seems likely that the sound of the waves should also be fractal.

Similarly with wind blowing through trees or mountains, the mountains and trees are fractal, perhaps the fluctuations in the wind strength too, and the sound seems fractal to my ear. Simlarly also for streams flowing. Other things that might be fractal include thunder, and the sound of a fire crackling. It would be interesting to know if anyone has done an analysis of these sounds for fractal content. Do let me know if you have any information about this or good references :-).

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The tunes are based on a sloth canon (or 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 or sloth canon.

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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). If you want to find out more about the details of how this works right away, go to Seeds and Fractals.

The sloth canon is transformed in many ways and it is often hard to pick out, as for instance in the tune at the head of this page.

In fact, often 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.

The one at the top of this pagee 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. To find out more see Seeds and Fractals.

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What to do next

Freeware / Shareware status: This feature is shareware.

To continue reading about the fractal tunes, go on to Seeds and Fractals - which explains how the tunes are made up of seeds, and how the original tunes are based on passing tones.

You are recommended to try out the program first. If you decide to purchase, you need to buy the Play level. If you want to use a music keyboard with FTS as well, or use it to retune your notaiton software or sequencer, you need the Complete level.

To find this feature after you download Tune Smithy:
Look in the Tune Smithy Tasks window for: Fractal composer :or to mainly listen to the tunes: