Construction by 'recursion'

By Jørgen Mortensen

The reason why this method of constructing an infinity series has been included here is because the concept of 'recursion' is much discussed at the moment within fractal geometry.

Recursion means 'refolding' or 'dividing again'. This phenomenon can be observed in the infinity series, which in fact can be created using this principle.
We may regard the expansion from 8 to 16 notes as a process of 'unfolding', see the score sample:
Unfolding takes place in this way: having divided the first 8 notes into 4 equal parts, one then takes notes 3 and 4, which can be regarded as the second quarter of the original whole. Then come notes 1 and 2 - the first quarter of the original 8 notes.

Then come notes 4 and 3, the second quarter again, but now split in the middle and in reverse order. We are now down to the level of individual notes rather than groups of notes. Then new notes are added that have not been used before - first the lowest and then the highest (E and B).
The expansion from 16 to 32 notes takes place in completely the same way:
First, notes 5, 6, 7, 8 (the second quarter of the original)

Then notes 1, 2, 3, 4 (the first quarter)

Then notes 7, 8 and 5, 6 (the second quarter split in the middle and the constituent parts placed in reverse order).

Then notes 8 and 7 (the second half of the second quarter of the original 16 notes in reverse order. We are now down to individual notes).
Then new notes, E flat and C, are added.

Expressed in more general terms, this 'redoubling process' by recursion means:

    • we take the whole section up to the middle and then split it into two parts, which are taken in reverse order;

    • we take a section - only half as long - up to the middle, split it in two, and arrange the parts in reverse order;

    • we repeat this until we reach the level of individual notes;

    • then we add the two new notes, first the lowest and then the highest.
In other words, the row - right down to 4 notes - is constructed by recursion.

This recursion clearly shows that the beginning of the row will reappear again and again in increasingly longer sequences:
Note 6 (= note 1), notes 11 and 12 (= notes 1 og 2),
notes 21 - 24 (notes 1 - 4), notes 41 - 48 (= notes 1-8),
notes 81 - 96 (= notes 1 - 16) notes 161- 192 (= notes 1 - 32), etc.