View source for 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''<sup>''N''</sup> letters, in which each ''M'' characters contain exactly the ''M'' distinct letters. Then:

By a '''''n''-block''' of ''X'', one means a length ''M''<sup>''n''</sup> subword of ''X'' starting at position ''kM''<sup>''N''-''n''</sup> 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 ''k''th 1-blocks of ''X'' by ''k'' steps locally, where 0 ≤ ''k'' ≤ ''M''<sup>''N''-1</sup>.

By a '''twisted ''n''-rotation''', one means (''n''-1)-rotating the ''k''th ''n''-blocks of ''X'' by ''k'' steps locally, where 0 ≤ ''k'' ≤ ''M''<sup>''N''-''n''</sup>. (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.