The Nørgård Palindrome is an ambient electronic music track released recently by Morten Bach and me. It is composed algorithmically and recorded live in studio using a lot of synthesizers. It is the second track of the album, the first being “Lorenz-6674089274190705457 (Seltsamer Attraktor)” which was described in another post.

The arpeggio-like tracks in The Nørgård Palindrome is created from an integer sequence first studied by the danish composer Per Nørgård in 1959 who called it an “infinite series”. It may be defined as

\begin{aligned}
a_0 &= 0, \\
a_{2n} &= -a_n, \\
a_{2n + 1} &= a_n + 1.
\end{aligned}

The first terms of the sequence are

0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, \ldots

The sequence is interesting from a purely mathematical view point, which has been studied by several authors, for example by Au, Drexler-Lemire & Shallit (2017). Considering only the parity of the sequence yields the Thue-Morse sequence, which is a famous and well-studied sequence.

However, we will, as Per Nørgård, use the sequence to compose music. The sequence is most famously used in the symphony “Voyage into the Golden Screen”, where Per Nørgård mapped the first terms of the sequence to notes by picking a base note corresponding to 0 and then map an integer k to the note k semitones above the base note.

In The Nørgård Palindrome, we do the same, although we use a diatonic scale instead of a chromatic scale, and get the following notes when using a C-minor scale with 0 mapping to C:

It turns out that certain patterns are repeated throughout the sequence, although sometimes transposed, which makes the sequence very usable in music.

In the video below we play the first 144 notes slowly along while showing the progression of the corresponding sequence.

In The Nørgård Palindrome, we compute a large number of terms, allowing us to play the sequence very fast for a long time, and when done, we play the sequence backwards. This voice is played as a canon in two, and the places where the voices are in harmony or aligned emphasises the structure of the sequence.

The recurring theme is also composed from the sequence using a pentatonic scale and played slower.

The code use to generate the sequence and the MIDI-files used on the track is available on GitHub. The track is released as part of the album pieces of infinity 01 which is available on most streaming services, including Spotify and iTunes.

Lorenz-6674089274190705457 (Seltsamer Attraktor) is an ambient music track released by Morten Bach and me. It was composed algorithmically and recorded live in studio using a number of synthesizers. This post will describe how the track was composed.

The Lorenz system is a system of ordinary differential equations

\begin{aligned}
\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma(y - x), \\
\frac{\mathrm{d}y}{\mathrm{d}t} &= x(\rho - z) - y, \\
\frac{\mathrm{d}z}{\mathrm{d}t} &= xy - \beta z.
\end{aligned}

where \sigma, \rho and \beta are positive real numbers. The system was first studied by Edward Lorenz and Helen Fetter as a simulation of atmospheric convection. It is known to behave chaotically for certain parameters since small changes in the starting point changes the future of a solution radically, an example of the so-called butterfly effect.

The differential equations above gives a formula for what direction a curve should move after it reaches a point (x,y,z) \in \mathbb{R}^3. As an example, for (1,1,1) we get the direction (0, \rho - 1, 1 - \beta).

In the composition of Lorenz-6674089274190705457 (Seltsamer Attraktor), we chose \sigma = 10, \rho = 28 and \beta = 2 and consider three curves with randomly chosen starting points. The number 6674089274190705457 is the seed of the pseudo-random number generator used to pick the starting points, so another seed would give other starting points and hence a different track.

The curves are computed numerically. Above we show an example of a curve for t \in [0, 5]. The points corresponding to a discrete subset of the three curves we get from the given seed are mapped to notes. More precisely, we pick the points where t = 0.07k for k \in \mathbb{N}.

We consider the projection of curves to the (x,z)-plane. The part of this plane where the curve resides is divided into a grid as illustrated above. If the point to be mapped to a note is in the (i,j)‘th square, the note is chosen as the j‘th note in a predefined diatonic scale (in this case C-minor) with duration 2^{-i} time-units. The resulting track is saved as a MIDI-file.

The composition of the track is visualised in a video available on YouTube. Here, all three voices are shown as separate curves along with the actual track.

The Lorenz system and this mapping into musical notes was chosen to give an interesting, and somewhat linear (musically speaking) and continuously evolving dynamic. Using this mapping, the voices composed moves both fast and slow at different times. The continuity of the curves also ensures that the movement of each voice is linear (going either up or down continuously).

The track is available on most streaming services and music platforms, eg. Spotify or iTunes. The code used to generate the tracks is available on GitHub.

Langton’s Ant is a simulation in two dimensions which has been proven to be a universal Turing machine – so it can in principal be used to compute anything computable by a computer.

The simulation consists of an infinite board of squares which can be either white or black. Now, an ant walks around the board. If the ant lands on a white square, it turns right, flips the color of the square and moves forward. one square If the square is black, the ant turns left, flips the color of the square and moves forward one square.

When visualised, the behaviour of this system changes over time from structured and simple to more chaotic. However, the system is completely deterministic, determined only by the starting state.

In the video above, a simulation with two ants runs over 500 steps and every time a square flips from black to white a note is played. The note to be played is determined as follows:

• The board is divided into 7×7 sub-boards.
• These squares are enumerated from the bottom left from 0 to 48.
• When a square is flipped from black to white, the number assigned to the square determines the note as the number of semitones above A1.

Seven is chosen as the width of the sub-squares because it is the number of semitones in a fifth, so the ants moves either chromatically (horizontally) or in fifths (vertically). In the beginning, they are moving independently and very structured, but when their paths meet, a more complex, chaotic behaviour emerges.