Hi There,

This one is fractal by almost any definition - well the rhythm certainly. The pitch also - that's assume you treat the pitch axis much as another space axis:

http://robertinventor.com/software/tunesmithy/tune_smithy_seeds.htm#fractalpitches
The similarity would involve slowing down the time to stretch the time axis, and spreading notes out in pitch (e.g. re-map subdivisions of a whole tone to similar subdivisions of an octave or whatever) to zoom in to the pitch axis.

One could say that pitch is highly non linear because of things like preferred intervals like octaves, fifths etc which we don't have in visual fractals. So though mathematically that is certainly a good fractal, musically it doesn't work quite as visual fractals do.

My sloth canons are fractal using a different kind of a similarity. It doesn't have infinite divisibility of pitch or time. So you can only zoom in a few steps. But you can zoom out a long way (though not for ever as eventually you hit the limits of human pitch perception).

The similarity is that if you play the tune faster, e.g. three times faster, and if you then remove every third note in the tune, then it sounds the same as it did before. Repeat that process any number of times and you still get the same tune.

Also if you take any phrase that occurs in the tune, and then play the tune for long enough, then that phrase will recur. So there is that uniformity and the fractal scaling effect.

Because pitch works so differently from vision, I think this is a better analogue of the visual fractal. You can easily test to see if a tune is fractal in this sense - by looking for a fractal construction of this type. It is usually immediately obvious if you listen to any of the standard sloth canons in Tune Smithy, what the seed phrase is, because you will hear a musical phrase that keeps repeating at different transpositions (often with different step sizes depending on the transposition but musically it sounds like a variant on the "same phrase" each time). Once you identify that phrase, just count how many notes it has, and if it has say seven notes, then eliminate all except every seventh note in the tune. If the result still sounds the same as the original tune - and if you can repeat that process for a large number of steps, then that is the first requirement for it to be one of these sloth canons. Then if you also get the almost periodicity, that any phrase in the tune of any length will eventually repeat, that's the other requirement satisfied.

So it is very well defined mathematically. The only question is whether to call this a fractal or not. The sloth canons that originally came with FTS use the same technique as the Koch snowflake, so a very simple iterative construction. The individual seed phrases are well defined and distinct.

There's another type that FTS can make, the fibonacci fractal tunes. These use Fibonacci rhythms, which is a well defined pattern. An example of this pattern is the sequence of wide and narrow rhombs along a row of a Penrose tiling. Because of the inflation / deflation rules, you can show that you can compose a short plus a long beat to make a new slower version of the long beat at the next level, and use the original long beat as a new version of the short beat at the next level. The result is the same identical rhythm, played more slowly. You get it by omitting notes in the original tune, but this time you use a more elaborate procedure. You look for a long beat followed by a short beat, and you just omit the note that divides those two beats to be left with a single long beat. The result is the same rhythm as before. You can use the same process to go out and out as often as you like with the rhythm.

So long as you are working in the realm of rhythm, where it makes sense in theory to slow down the rhythm by any arbitrary amount, then you can even make the rhythm infinitely detailed like a visual fractal, as the inflation rule can be used both ways either to zoom out for an existing pattern or to zoom in to add more subdivisions to make even faster versions of the same rhythm.

You could pick out the fractal structure there in a fractal rhythm e.g. by making the beats that mark out the slowest rhythm loudest of all, and the fastest beats quietest until they get so quiet you can hardly hear them but exceedingly rapid. In fact I could easily make an example of that using FTS if anyone is interested (no example included with FTS at present because I have only just thought of the idea of doing it while writing this).

The fibonacci rhythm is connected with the Penrose tiling which has a fractal type structure because of the aperiodicity and the inflation rules. The usual way the Penrose tiles are shown don't really bring this out as they are extremely uniform seeming when you zoon out and see them from a distance, not fractal like. But you could bring it out in some way if you could make a 3D landscape out of a Penrose tiling so that it undulates by bringing out the underlying inflation rules that make it up.

I've not seen this suggested before so am thinking about it as I write. But I'm pretty sure it could be done.

Something like this (might be not quite right yet):

Take a penrose large rhomb. Substitute the small and large rhombs as here:

http://tilings.math.uni-bielefeld.de/substitution_rules/penrose_rhomb
Raise the vertex in the middle of the broad rhomb by a large amount say by the length of one of the edges in the tiling. Now substitute again for all the wide and narrow rhombs in the tiling so far. Leave the narrow ones unchanged and raise the centre of the broad rhombs again by the edge length, but this time in the new tiling so the amount it is raised is smaller. Keep repeating the process. Don't bother to try to keep the rhombs flat, just treat the tiling as a texture that you allow to rise and fall to follow the undulations of the landscape you are constructing.

Repeat the process and the result will look pretty much like a fractal. I'd need to do some work to make sure it really is one or maybe you need to modify it in some way. You would even get the infinite detail by going inwards to smaller and smaller tiles as subdivisions of the original tile.

That then would give an infinitely spiky type fractal landscape a bit like a 3D koch snowflake affair. Then in that fractal landsape, the fractal rhythms correspond to the undulations along one of the rows of tiles.

Perhaps I might get out pen and paper and see if I can make this rigorous, for now it is mainly arm waving.

But the 1D fractal rhythms of the fibonacci rhythms - they are very clear. The rhythms anyway because that's just time, - and maybe two dimensional if you include volume as the second dimension. Both are (more or less) linear and straightforward and anything you can do with a visual fractal of this type you can do equally well with a rhythm. So the cantor set type fractal you can do as a rhythm with fluctuating volumes or a fibonacci rhythm ditto. This time the two similiarities are - slowing down the time - and adjusting the volume range and threshold if necessary so that you can hear very quiet notes or small differences in volume.

Pitch is more tricky because the way we hear pitch is so different from volume and time, but I think the sloth canons anyway are a form of audible pitch fractal structure.

Just a few thoughts there. I can make those fractal rhythm examples though with FTS. Maybe I'll give that a go. Got a bit of programming I must do tomorrow but could try it out in a day or two. Also maybe have a go at that 2D penrose fractal idea if I have a bit of time for it :-).

Thanks, hope this helps,

Robert

.

[Non-text portions of this message have been removed]