Permutations and Transformations – Music Operations

The quantum organ introduced in station U2 – 9 provides a visualization of all possible vibration modes of the wave function of the electron, culminating in a model for the periodic table of elements. While quantum states and the corresponding wave functions are an essential trait of quantum physics, some of their basic properties can also be studied using sound waves… for example, with a usual piano keyboard.

We can apply operations to any melody, i.e. the notes and their temporal progression. If we apply the operator “derivative” on this melody, it sounds like this … while the “integral” leads to this result… How can we apply a “derivative” or an “integral” operator on a piece of music?

To do this, we have to code the music in numbers. First, we label the keys, starting with an arbitrary key. The music – here the melody of “Brother John ” – becomes a sequence of numbers … similar to a piano roll.

Now, we apply the difference operator, which determines the height difference between two adjacent notes of the melody … It starts at zero, then plus two, plus two, then down – minus four, zero – so the same pitch, then plus two, plus two, and so on.

At first, this is just a sequence of numbers, but we can also interpret it as a new melody. In such a way, we obtain the sound of the derivative of Brother John.

The integral operator sums up all numbers…Starting with the first number, -4, then plus -2, gives -6, then plus 0, stays -6, then plus -4, gives -10, then -4 again, gives -14, and so on. This gives us another sequence of numbers that we can interpret again as a new melody – the integral of Brother John!

What happens if we first apply the integral operator and then the derivative operator to “Brother John”? So let’s look again at the differences in the sequence of stages, -4, -2, 0, -4, -4, and so on … This sequence of numbers sounds familiar to us – that’s exactly the initial melody of “Brother John”!

From a mathematical point of view, this is nothing but the fundamental theorem of calculus. But we can also regard this as a special form of encryption and decryption, since – apart from transposition, i.e. a constant shift of the point of reference on the keyboard – no information is lost.

And all these operations are examples for various possible operations on vibrational states, just as operations on wave functions are carried out in quantum physics. Of course, in contrast to usual music, quantum states are not auditible … And we certainly can’t observe them directly. Only by using mathematics, we can find a pathway into the quantum dimension.