Self Similar Sloth Canon Number Sequences

The Norwegian 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.

Interestingly, his sequence is constructed in a different way from the Tune Smithy sloth canons. It is 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.

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

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

A youtube video of his second symphony: Per Nørgård's Second Symphony

What does it mean to say that these sequences make a sloth canon?
It means that if you take every nth number from the original sequence for suitable n, you get the same sequence again.

So for instance with the fractal tune generated from the 0 1 0 seed in Tune Smithy, if you take every third number in the sequence you get the original sequence again:

This is the Tune Smithy 0 1 0 generated sequence

Number of 1's in ternary (base 3) expansion of n.

So if we underline every third number we get the original sequence again. Numbers highlighted in bold: 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0,....

All the Tune Smithy sloth canon type fractal tunes work that way.

There are several more in the database. So: number of 1's in binary expansion of n (or the binary weight of n) - that's the Tune Smithy sloth canon for the seed 0, 1.

Sum of digits of (n written in base 3) - that's the Tune Smithy sloth canon for the seed 0, 1, 2

Sum of squares of digits of ternary representation of n seed 0 1 4

However there are many self similar, "sloth canon" sequences in the On-Line Encyclopedia of Integer Sequences that can't be made using the Tune Smithy method. For a list of some of them see Some Self-Similar Integer Sequences

This is not the same thing as a fractal sequence
Confusingly, the word "fractal" has already been applied to a different type of number sequence, one which has a different type of self similarity. See Fractal Sequence.

So, not sure what to call this type of sequence. Calling it a self similar sequence doesn't distinguish it from the fractal sequence as that is also self similar in a different way (by removing the first occurrence of each number in the sequence).

For now, let's just call them "sloth canon sequences" because when you turn them into music, you get sloth canons if you add extra instruments to play every nth note, and every n2th note and so on.

Another way of looking at the Tune Smithy sloth canon sequences
The idea here is to generalize the result above that the 0 1 0 seed gives you the sequence of the number of 1's in ternary (base 3) expansion of n

We can find a similar definition for any of the Tune Smithy sequences if we use weighted digit sums - the same idea as a Checksum

We can turn any Tune Smithy seed into a suitable weighted sum such that the weighted sum of the digits of n to an appropriate base gives the nth number in the sloth canon sequence for that seed.

Example to show how it works
So given any Tune Smithy seed, beginning with 0, say 0 2 -1.

The idea is - first you set the base for your number system to the length of the seed, here 3.

Then you have to choose appropriate weights for each digit, so here, first digit 0, it doesn't matter what weight you give it as it always multiplies out to give 0. Then we want to evaluate 1 as 2, as a weighted sum (the second number in the seed), so give it the weight 2. Then we want to evaluate 2 as -1 as a weighted sum, so we give the weight -0.5.

So our weights for the digits are

1, weight = 2 2, weight = -0.5

Now if we find the weighted sum of the digits of n expressed to base 3, we will get the endless tune smithy sloth canon sequence 0, 2, -1, 2, 4, 1, -1, 1, -2,...

So for instance the number 7 (decimal) is 21 (base 3).

The weighted sum of the digits for its digits is 2*-0.5 + 1*2 = 1. Since we are counting starting from 0, then this is the 8th number in the sloth canon sequence, and as we can confirm, the 8th number in this sequence is 1.

Generalization of the Tune Smithy construction
You can make all the Tune Smithy sloth canons using weighted digit sums in this way, because you just need to choose an appropriate base and a weight for each digit. It is easy to seee that the sloth canon construction method for Tune Smithy gives exactly the same result as this weighted sum approach.

Doing it this way though shows that the result isn't limited to integer sequences. It also generalizes to sequences of rationals (fractions) and reals (with infinite decimal expansions like PI) and even complex numbers (using square root of -1) (indeed, for mathematicians, you can also generalize to any field or ring for the weights).

What about the other sloth canon integer sequences
As we see from the The Danish composer Per Nørgård's "infinity sequence" then there are many other sloth canon integer sequences.

Another simple example is this one: Write n in ternary, sort digits into increasing order

So, there are lots of particular examples of that type on this page: Some Self-Similar Integer Sequences

But, are there any other methods of constructing infinitely many sloth canon sequences? Is there any way to classify all the sloth canon sequences?

I don't know the answers to these questions yet. Just posting this here as a reminder so that the information I've found so far is easily available.