Paving a Road

Here is a new piece in 99edo. Since I added an interval similarity score to my software, I am exercising its other various capabilities to see how they work with the new score. Today I have incorporated a scale into the system.

The non-standard Tonnetz diagram above shows the scale I have devised. This diagram shows intervals of 5:4 vertically and 7:4 horizontally. Intervals of 3:2, perfect fifths, are not shown in the diagram.

This scale includes 37 of the 99 pitch classes. Maybe it should be called a subset rather than a scale! How conventional terminology should properly be applied in this kind of unconventional context, I sure don't know enough to have a useful opinion! I'll call it a scale, because it has many properties of a conventional scale like a diatonic scale: it is built from a chain of repeated intervals, it circulates, and it supports comma traversals.

The generating interval for this scale is 16 steps of 99edo, which does not correspond to any very fundamental just-tuned interval; 28:25 looks like the simplest. The circulating quality can be observed in the diagram. A key change in one direction would remove the 0 pitch class and add the 97 pitch class; the other direction removes the 81 pitch class and adds the 83.

This scale supports traversal of a family of commas, formed by combinations of two primitive commas. A comma traversal appears on the diagram as a path from a cell labeled 0 to another cell labeled 0. Since the diagram does not include perfect fifths, a little imagination is required! I have highlighted a cell labeled 58. This is a perfect fifth above the 0 pitch class. One can imagine a third dimension for this diagram, with the next layer above the one shown having a 0 cell right above the 58 cell. So a path from a 0 cell to another 0 cell can be formed from a path from a 0 cell to a 58 cell, just adding one more step of a perfect fifth to climb up to the 0 cell in the next layer.

The short path from a 0 cell to another 0 cell in this diagram corresponds to traversing the comma 3136:3125. There is a path from a 0 cell to a 58 cell directly above it; this corresponds to traversing the comma 393216:390625. These primitive commas can be combined to make paths from the other 0 cells to the 58 cell, corresponding to the commas 2401:2400 and 6144:6125.

This piece has 81 measures, each about 40 seconds long, arranged in a 9x9 matrix. I initialized the system so each measure traversed the comma 6144:6125. I started the thermodynamic simulation with the temperature set moderately high and then gradually lowered it until a phase transition was detected. My intention was that with the constraint of the scale, the initialization, and the moderate starting temperature, the structure of 81 comma traversals would be maintained. But I don't really hear the traversal, nor do I see it when I analyze the score!

So, here is yet another piece in 99edo! From the prior run, I know the temperature of the phase transition. So I initialized the system in the same way, but just set the temperature at the phase transition and let it run a while. With luck, there was enough jostling to create interesting variations, but not too much so the traversal structure could be maintained.

I made a folded score for this second version, where the 81 measures have all been superimposed. This should make clear the general pattern. And indeed, the traversal structure is very clear here. With any luck, it will be apparent to the ear, too!