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!