On the creation of “The Nørgård Palindrome”

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.

The first 144 notes of Nørgårds’ infinite series mapped to notes in a diatonic scale.

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.