Sloth canons, Tune Smithy Fractal Tunes, and the connection with Per Nørgård's music

Youtube Video

One of the Tune Smithy fractal tunes - it is a sloth canon type melody - if you take every sixth note in the tune you get the original tune again

You get all the Tune Smithy fractal tunes as a bonus feature with the Pro metronome.

You may have seen the fractal tunes which you get in Bounce Metronome and Tune Smithy which are based on an endless strict sloth canon:: Fractal Tunes - hear your rhythms played with an endless continually changing melody line.It's a minor feature in Bounce naturally, since the focus is on rhythms, but it's actually the first of its features to be programmed. When I first wrote Tune Smithy in the late 1990s, it was to find a way to play these sloth canon type melodies on a computer - and Bounce originated as the polyrhythm metronome task in Tune Smithy..

So, anyway, I have just been told that the Danish composer Per Nørgård uses an endless self similar (fractal like) strict sloth canon structure in some of his compositions such as his Symphony number 2. He first discovered his sequence in 1959, so long before I got the idea of making sloth canon sequences for Tune Smithy.

Here is a youtube video of his second symphony: Per Nørgård's Second Symphony

This is his sequence on the on-line encyclopedia of integer sequences The Danish composer Per Nørgård's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation 0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3, 1, ... 

His explanation of how it is constructed: the infinity series - Construction by the projection of intervals

What makes this a sloth canon number sequence is that if you take every fourth number in the sequence you get the original sequencee again. 0, 1, -1, 2,1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3,...

Then of course if you take every fourth of those new numbers (every sixteenth number in the original sequence) you get the same sequence again, and so on. This is a "fractal like" self similar  property.

Interestingly, his sequence is constructed in a different way from the Tune Smithy sloth canons. It is based on a strict sloth canon, but has other properties that the Tune Smithy sloth canons don't have, and if you try to make it from a Tune Smithy seed, it just doesn't work.

I have researched this a bit more and found there are many other "sloth canon" number sequences like this as well, constructed differently from the Tune Smithy sequences, but self similar when you take every nth term in the sequence, so many of those may also have musical potential.

I've written up my findings so far about all this here: Self Similar Sloth Canon Number Sequences