A Tale of Two Scales

Here are three new pieces in 43edo:

1. quite conventional
2. less conventional
3. quite unconventional

These are all composed by my highly randomized and highly parameterized software. I had to tweak the program in a variety of ways to coax it to produce pieces that sound reasonably musical. So some of the difference in sound is due to the different tweaks involved. But the fundamental changes were the notions of consonance and the scales.

The first two pieces use the same scale. The tuning system 43edo provides 43 equally spaced notes in each octave. The scale in the first two pieces picks 19 of these notes for use in the composition. The third piece uses a very different subset of 21 notes out of the 43. The first piece differs from the others in the set of intervals considered consonant. The second and third pieces share an extended notion of consonance.

The consonance relationships between the notes of a scale can be visualized using a lattice diagram.

This is the scale lattice used in the first piece. The fundamental consonant intervals are very conventional: perfect fifths (green arrows) and major thirds (blue arrows). This diagram is similar to the circle of fifths, but here I have prioritized the major thirds. Moving four steps along a sequences of green arrows will bring one back around close to where one started, just a blue arrow ahead. This relationship between the green arrows and the blue arrows, between the perfect fifths and the major thirds, is due to the fact that 43edo tempers the syntonic comma. Tempered scales, such as 43edo, are at the foundation of classical European music. Musicians in Europe were experimenting with a variety of tempered scales, including 43edo, in the Baroque era, before the modern 12edo scale became dominant.

This is the scale lattice for the second piece. It uses the same 19 notes per octave, but adds new consonant relationships, based on the 7:4 frequency ratio. The best approximation for 7:4 in conventional 12edo is 10 steps, which gives a frequency ratio of 1.7818. This is quite a bit sharp, one of the reasons that 7:4 is not considered consonant. In 43edo the best approximation for 7:4 is 35 steps, which gives a frequency ratio of 1.7580. This is much close to the pure interval of 1.75, so it is a natural interval to include as consonant.

This is the scale lattice for the third piece. The geometric layout of these diagrams is not fixed: what matters is what notes are present and how they are connected by consonant intervals. But this third scale is indeed very different in those relationships. For example, the 19 notes of the first two scales are connected by a single continuous path of green arrows, of perfect fifths, while in this third scale there are no green paths longer than two steps!

My hope here is that, with three algorithmic compositions based on three different scale lattices, that the sounds of the different scales can start to become apparent.