Permutations and Transformations – Sound Permutations

To play music in the wrong base is essentially a special type of encryption – of cryptography! This idea – i.e. encryption of a message using coordinates that are read in the wrong base – has previously actually been used, however not with musical notes, but with letters of the alphabet.

We are, however, interested in permutations for a quite different reason – because such permutations are the backbone of all operations in the quantum dimension.

Let us get started with the simplest permutation of all – which is “interchanging two items”. Interchanging them once more takes us back to the original state. Now, suppose we add one more item. Then, there are three possibilities for transpositions: 123 can become 213, 321 or 132.

If we operate with one more transposition, there are only two different options, namely 123 can become 231 or 312. The third option would once again give rise to the initial state, without any permutation. In a similar way as in U1-1, a tree structure emerges, that always has two branches to the following generation: either adding one item without additional transposition, or adding an item combined with a transposition.

This is repeated every time further items are added. At the 13th level, 13 faculty – in other words some 6 billion permutations – are classified in accordance with this scheme. Who would have thought that there would be so many different ways of writing musical notes … All the permutations can be understood if we grasp the simple transposition of two items!

Transpositions are, however, not as easy as they seem – as a sneak preview of that, imagine the objects 1, 2 and 3 as coordinate axes – the exchanges then correspond to 90-degree rotations. Similar as in U2-06 Slide 2, consider a banana as object to be rotated.

The 90° rotation around the first axis corresponds to a transposition between coordinate axes 2 and 3. Next, we rotate around the third axis, which corresponds to a transposition between coordinate axes 1 and 2.

The order of operations plays a crucial role. If we first rotate the banana around the third axis, and then around the first axis, the final state is quite different. In other words: rotating operators do not commute – just like the permutations. If we thus interpret the objects that are subject to a permutation as dimensions, we have encountered the basic structure of rotations in any number of dimensions desired.

This is a further piece of the puzzle towards answering the question of whether the future is a permutation of the past – in the quantum dimension, the time evolution is actually described as a complex rotation, which is the core of the so-called Schrödinger equation. The choice of base and the corresponding coordinates is, however, arbitrary. This idea leads to the so-called “gauge principle”, which is the cornerstone of the standard model of elementary particle physics.