Roll Your Own!

I was visiting the Bright Dawn Buddhist center in California recently and got to talking with Kanon Kubose about software technology. I mentioned my desire to have a web app that would allow people to experiment with alternative tuning systems. Kanon wanted to try using Claude Code. So... in just maybe a week of back and forth, we produced:

microtonal2.vercel.app

It's hardly a polished product... we need a name, for example! But it's already loads of fun to play with!

You can divide octaves into however many equal steps that you'd like, and also pick which primes should be used to define consonant intervals.

You can then build tonnetz-type diagrams whose x and y axes can be chosen from a range of consonant intervals.

The app will then show you a list of commas that the tuning tempers out. That's one of the main advantages of tempered tunings, that one can exploit in various ways commas that have been tempered out. The app will show the path on the tonnetz of a traversal of the comma one selects from the list.

You can then define a scale. A selection of Moment of Symmetry scales is provide, or one can construct an arbitrary scale manually. The scale I constructed manually is actually a Moment of Symmetry scale, generated by the 29\53 step interval corresponding to 35:24. That generating interval is just a bit too esoteric to appear in the app's list that one can pick from!

The composition algorithm, that I use and the app uses, is based on a score for intervals: low cost intervals are more likely to appear in the generated music. The scores are based on the complexity of the just intervals that each tempered interval approximates, and also on how well the tempered interval approximates those just intervals. Anyway you can play with those three sliders to get the interval cost matrix to look like it has promise.

I confess that I have not really played yet with controlling the voices in the composition. Feel free to give it a try!

Next one can specify the general structure of the composition to be produced - the number and topology of the measures. One can also control some of the basic scoring parameters: how wide or narrow should be the pitch ranges of the voices, how large should the pitch jumps of a voice be, etc. Adjusting these to see what the results sound like, that's a lot of the fun here.

The actual composition process is a thermodynamic simulation controlled by a temperature parameter. At a high temperature, pitches are chosen at random with little bias toward consonant intervals etc. At a low temperature, the lowest cost pitch will be picked. At intermediate temperatures, lower cost pitches will be preferred but there will still be a significant chance of a higher cost pitch being picked.

The pitch distribution bar chart on the right side of the screen is a good indicator of the regime that the temperature has driven the system into. With the cost structure that has been chosen, the temperature of 200 created a rather flat distribution of pitches, i.e. the system is in a high temperature regime.

4.8 is evidently a cold temperature, where the composition is dominated by just a few pitch classes.

A temperature of 8.3 has a promising pitch class histogram - not too flat, not too steep!

One can snapshot the system at whatever interesting points along the way, and then see and play a score.

The score at temperature 200 doesn't look utterly random, but nearly so. A snippet of sound: hot sound.

The cold system certainly looks ordered! And sounds it: cold sound.

The intermediate temperature indeed has intermediate structure. The sound, hmmm. It's quite a strange scale, so, hard to say! intermediate sound.

Emergent Structure

Here's a new piece: 270edo scale 16 emergent.

This piece is in a 16 note scale of the tuning system 270edo. The scale is designed to support traversal of the comma 2401:2400.

Lately I have been playing hide-and-seek with comma traversals. The highly randomized algorithm I use is targeted to creating fractal fluctations, in hopes of making interesting music. These fluctuations can conflict with the neat topological structure of a comma traversal. I initialize the system with the traversal, and then apply the thermodynamic randomization. The challenge is to randomize enough to create interesting fluctuations, but not so much as to erase the traversal. Lately most of my attempts have failed to preserve the traversals!

One method that seemed to be working: I would run a long similation, starting with a moderately high temperature and then gradually reducing the temperature and watching for the phase transition which creates the fractal fluctuations. I would record the temperature of the transition, and then run a second simulation, starting again with the traversal as the initialization, but simply running the simulation at the single temperature that I had recorded for the phase transition.

My plan had been to repeat this method with this 2401:2400 traversal. There are 343 measures in the piece, arranged in a 7x7x7 cube. Traversing 2401:2400 involves seven basic steps. I initialized the system to traverse the comma 49 times over the course of the piece. I examined the score after the thermodynamic simulation at moderately high temperature, and did not see any evidence of the traversal having survived. Sure enough, at a much lower temperature, I observed a huge spike in the heat capacity, i.e. a rapid drop in energy, characterizing a phase transition. I expected to see a corresponding spike in the histogram of the pitches in the piece, to see some single pitch coming to dominate. But no dominant pitch emerged! So I checked the score... there is a clear staircase pattern, but it only repeats seven times over the course of the piece! The system at the phase transition has a comma traversal, but it is oriented differently from the one I had planted!

A plot of energy vs. temperature for this simulation run, showing the sudden steepness at the phase transition:

Simplicity

Here's a new piece in 31edo: 31edo scale 9.

This nine note scale is intended to channel the composition algorithm into producing a traversal of the comma 126:125. I can't find any structure like that in the score, or in any of the scores I generated in my various trials. Something is not working as anticipated! One possibility is that this traversal is just too simple, i.e. too easy to jostle away. Another detail I suspect might be causing trouble is the new interval scoring formula I have been using. It scores 7:6 as more consonant than 8:7. This may be providing a downhill pathway to breaking apart the traversal.

Traversal or no, this piece sounds nice, so I thought I should share it. Plus, there is almost always something to be learned from failure!

It's Called Blackjack!

Here is a new diagram for the 21 note scale I posted this morning. I learned a lot about the scale from the effort to untangle the graph!

And here is a new piece... less austere! 41edo blackjack

The kind folks in the facebook microtonal community let me know: this scale is known as blackjack!

21 out of 41

Here is a new piece in 41edo: 41edo scale 21.

The diagram above shows the scale used here. Green arrows are perfect fifths, blue arrows are major thirds, and red arrows are 7:4. The scale is generate by the 4 step interval, which corresponds to a semitone 16:15. A curious feature of this tuning is that six semitones make a perfect fifth, instead of the usual seven.

Interval Cost Function

Here is a new piece: 53edo nxd.

This uses the same scale that I showed in yesterday's post.

For this new piece, I changed the interval cost function. This table shows a large part of the cost function. The composition algorithm prioritizes intervals with low cost. Intervals of the same number of half steps follow diagonals from upper left to lower right. The uppermost such diagonal shows the cost for the half step interval. In an equal tempered scale, these half step intervals would all have the same cost. In this scale, however, the interval from C to Db is not exactly the same size as the interval from Db to D, and so their costs differ.

Marvel Piano

Here's a new piece: 53edo 7x7x7

This is in a 12 note per octave scale that would work quite nicely on a piano!

53edo has much more accurate major thirds than those of conventional 12edo. Moving from C to A# in this tuning moves through two major thirds. This makes the A# quite a bit flatter than conventional tuning. The interval from C to A# is a very sharp approximation of 7:4 in conventional tuning. With this 53edo scale, the 7:4 is just 5 cents off!