User:Akselai/Quasi-modal construction of fractal scales

A fractal scale is a scale in which each step is recursively divided by some ratio. There is a notion of mode of a fractal scale, which is the scale produced by rotating the dividing ratio. To truly capture the fractal property of these scales, we can go a step further, and create "modes" of a fractal scale beyond the usual definition.

Warning: this is a work in progress.

Suppose X is a word with M^N letters, in which each M characters contain exactly the M distinct letters. Then:

By a n-block of X, one means a length Mⁿ subword of X starting at position kM^N-n for some integer k (here I am using 0-indexing).

By a simple rotation of X, one means rotating all the 1-blocks of X by 1 locally.

By an n-rotation, one means rotating each of the n-blocks by 1 globally, keeping each block fixed.

By a twisted simple rotation, one means rotating the kth 1-blocks of X by k steps locally, where 0 ≤ k ≤ M^N-1.

By a twisted n-rotation, one means (n-1)-rotating the kth n-blocks of X by k steps locally, where 0 ≤ k ≤ M^N-n. (unsure)

A 1-rotation is a simple rotation and a twisted 2-rotation is a twisted simple rotation.

Examples

Consider the word X = abcabcabc, which is the zeroth mode of the 2-fractal scale with 3 divisions. This means each 3 intervals of the scale is divided into a:b:c scale steps in the next iteration.

Performing a simple rotation of this word gives bcabcabca. which is the first mode. Performing the simple rotation twice gives cabcabcab, and thrice gives abcabcabc again.

Performing a 2-rotation of this word gives abcabcabc, which is uninteresting for now.

Performing a twisted simple rotation of this word gives abcbcacab. (Notice that it cycles through the simple rotations of abc.) This means for each 3 consecutive intervals x, y, z of the scale, in the next iteration, x is divided into a:b:c, y is divided into b:c:a, and z is divided into c:a:b.

Now consider Y = abcbcacab. Performing a 2-rotation of this word gives bcacababc, so we have a division pattern that starts with b.