Wednesday, July 3, 2024

Compound Traversal

Here's a new algorithmic composition, in the tuning system 34edo, which divides octaves into 34 equal steps rather than the conventional 12 equal steps.

Changing the tuning system like this does two things. A tuning system makes available some collection of intervals, the building blocks of music. Sometimes this collection of intervals is unconventional. But here I am using very much the same basic intervals in 34edo that conventional musics uses in 12edo, most fundamentally the major third and the perfect fifth, along with the octave. This new composition is not something that would work in 12edo, but the difficulty is not with the basic intervals being used.

The other characteristic of a tuning system that matters musically is how the intervals combine to form more complex intervals. With a tempered tuning system, some complex combinations end up being equivalent to a unison. Such a complex combination is known as a comma; when the tuning system makes it equivalent to unison, then the tuning system is said to temper out the comma.

The conventional 12edo system tempers out the syntonic comma: combine four perfect fifths and a minor sixth and, in 12edo, you will end up three octaves above your starting point. 34edo does not temper out the syntonic comma: the same combination of intervals ends up slightly sharper than three octaves.

The most basic commas tempered out by 34edo are the diaschisma, four perfect fifths and two major thirds, and the W├╝rschmidt comma, eight major thirds and a perfect fourth. This new algorithmic composition is built from traversals of these two commas. A comma traversal, sometimes called a comma pump, is simply a sequence of the intervals that compose a comma. When the comma is tempered out by the tuning system, the comma traversal will return to the starting pitch. Thus one can repeat the comma traversal and the pitch of the music will not drift up or down.

Conventional 12edo does not temper out the W├╝rschmidt comma. That's why this piece would not translate to 12edo. The same sequence of intervals could be used, but in 12edo the piece would drift in pitch.

The piece here is predominantly 27 traversals of the diaschisma. But every third traversal, the sequence is shifted by a major third. With 9 such triple traversals, separated by 8 major thirds, the first and last triples are separated by a perfect fourth. The end of the piece ties back to the beginning.

The way the piece is constructed algorithmically, it starts with just this simple sequence of intervals. The piece is almost 27 minutes long, so each traversal of the diaschisma is about a minute in duration. There are six intervals in the diaschisma. So the initial structure of the piece has pitches being repeated over about ten seconds, then shifting by a perfect fifth or a major third, then another ten second repetition, etc.

The algorithm I use is based on statistical mechanics, with temperature as a key parameter. Consonance is treated as low energy, and dissonance as high energy. At low temperature, pitches will be chosen to be maximally consonant. At higher temperature, more dissonance will be allowed. In a system like this, there will typically be a phase transition, a temperature below which the system will have long range consonance, and above which the overall consonance will break down. Right around the phase transition, there are typically fractal variations. The hope is that such fractal variations in consonance and dissonance will provide musical interest.

This is a plot of the temperature and energy computed in the course of the construction of this piece. The thermodynamic simulation is initiated with the simple comma traversal structure and with low temperature. The temperature is gradually raised; this involves reassigning pitch values to moderately less consonant alternatives. A phase transition is characterized by a rapid rise in energy with a small change in temperature. The composition process ends when this rapid rise is detected.

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