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.

Scale Balance

Ah, one more piece in 43edo. Earlier today I was working with a 43edo scale that had 11 pitch classes. I was considering removing one pitch class, to make it better balanced. It really looks like adding one pitch class works better! So this new piece has 12 pitch classes. You could map this to a piano keyboard! It would be mighty strange!

I rearranged the lattice and added the new pitch class. The balance is clear!

Scale Design

Here is a new piece in 43edo. This piece uses a scale with 11 of the 43 pitch classes of 43edo. I designed this scale to support traversal of the comma 12288:12005. A few months ago I posted a piece using a scale with 21 pitches classes out of 43. My idea with this larger scale was mostly to avoid supporting traversal of the syntonic comma. 43edo is a meantone tuning, so it will support traversal of the syntonic comma, i.e. it will support most conventional music. So my little project here is to take a tuning system that supports conventional music, but then to make a scale with it is unconventional. An example of this in conventional 12edo is a whole note scale. A whole note scale includes no perfect fifths or perfect fourths at all!

Instead of making as big a scale as possible that still blocks traversal of the syntonic comma, my idea here was to make a smaller scale that more narrowly focuses on traversing 12288:12005. Another feature of the bigger scale is that it can be constructed with a single generator: it was just a sequence of intervals of size 2. This new scale with 11 pitch classes does have a pretty regular structure: the scale intervals are mostly size 2 and 6, with a single size 3 interval. I have seen where people build scales with very exact mathematical order... the scale I am using here was just something I engineered in an ad hoc way.

Scale lattice diagrams helped me engineer this scale. I was looking at a lattice for the conventional 12edo diatonic scale:

Here the blue arrows show major third intervals and the green arrows show perfect fifths. Most chords are subgraphs connected by these consonant intervals. One could add further arrows for minor thirds, but these are implied by the arrows already in the lattice. A traversal of the syntonic comma appears as a loop on the lattice, e.g. 0, 7, 2, 9, 4, 0. Moving by four perfect fifths arrives at the same pitch class as moving by a major third.

There is an awful lot of good music that can be built from this diatonic scale. The lattice doesn't look so complicated, so my idea here was to make a scale in 43edo whose lattice doesn't look too terribly much more complicated:

This lattice includes red arrows for the interval 7:4. Traversing the comma 12288:12005 appears on the lattice as a loop, e.g. 0. 8, 16, 24, 32, 18, 0.

This scale looks related to 5edo. It has 5 clusters of pitch classes: {40, 0, 2}, {8, 10}, {16, 18}, {24, 26}, and {32, 34}. Hmmm, maybe it would be more symmetric if pitch class 40 was ommitted. But that would leave a pretty big hole in the scale, from 34 to 43, 9 steps. More to explore!