Toroidal Scales

Here's a new piece of algorithmic music: 99edo scale 25.

This is in the 25 note per octave scale diagrammed above, in the 99edo tuning system. This scale can be generated by the 16\99 interval, which corresponds to a 28:25 frequency ratio. But that's not how I came up with it!

I am working to develop a systematic approach to scale construction, using a toolbox of path-oriented methods. The main idea is to start with a path that traverses some comma, and then to augment this path in various ways until a scale emerges that invites musical composition. Augmentation might be adding another path between two notes in the scale. The new path could consist of the same intervals as an existing path, but simply reordered. Or, the new path could be topologically distinct, so that the old path and the new path form a traversal of some comma. Or, a complete comma traversal could be added to any note in the scale.

My idea with this 25 note scale was start with a traversal of a comma, and then add to each point in the scale a traversal of a second comma. For this piece, I chose the commas 3136:3125 and 6144:6125, both of which are tempered out by 99edo. The first comma can be traversed by a path of seven notes; the second by a path of six notes. For each note in the first path, I added five more notes to make a traversal of the second comma. The second notes of each of these again form a traversal of the first comma, as do the third notes, the fourth notes, etc. This is a very simple property of intervals: if two notes are separated by an interval X, and we raise both notes by the same interval Y, then the raised notes are again separated by the interval X.

Seven traversals of length six could produces as many as 42 notes, but there were many duplicates. The seven traversal produced 23 distinct notes. These were quite nicely spaced, mostly with gaps of 3\99 and 7\99. There were a couple gaps of size 10\99. By adding a couple more notes, bringing the total to 25, a regular pattern of gaps was produced. This same pattern can be generated using a chain of 16\99 intervals. Why the toroidal construction results in the same notes as this simpler method of generation... probably somebody in the microtonal world knows, but I sure don't!

2 3 5 7 11 13

Here is a new piece of music: 270edo scale 23.

This music is in the 270edo tuning system which divides octaves into 270 equal parts. This provides very precise tuning for a wide range of intervals. The music uses a scale with just 23 of the possible 270 pitches in each octave. The diagram above shows the pitches of the scale and their harmonic relationships. Pitches are labeled by integers, from 0 to 269 in each octave, and then starting up with 0 again for the next octave. In ascending order, the pitches in the scale are: 0, 10, 24, 36, 45, 62, 70, 77, 87, 97, 114, 122, 136, 148, 158, 168, 182, 199, 209, 228, 235, 245, and 260. The intervals between the successive pitches vary between 7 and 19 microsteps of the tuning system, so the spacing is reasonably even.

The arrows between pitches show the harmonic relationships. A green arrow represents a 3:2 frequency ratio (the best approximation in the 270edo tuning system), a perfect fifth in conventional tuning. Blue arrows are 5:4 ratio, major thirds in conventional tuning. Red arrows are 7:4, purple are 11:8, and yellow are 13:8; none of these are conventional intervals.

This scale supports traversal of a family of commas. The diagram above shows a traversal of the comma 1001:1000.

This is a traversal of 4096:4095.

This is a traversal of 41503:41472.

It looks to me like these three commas form a basis for all the commas that can be traversed in this scale. With six prime numbers involved in the tuning, the total set of commas tempered out by 270edo should be a five dimensional space. This scale supports traversal of a three dimensional subspace of that full set.

This traversal of 123201:123200 is a combination of the basis set of traversals.

I don't have a systematic way to construct scales like this, where the scales steps are reasonably even, where the graph of harmonic relationships is reasonable dense, and where comma traversals are supported. The set of commas tempered out by a tuning system already tend to have an irregular structure, so these sorts of scales probably won't be very regular either!

2 3 5 11

Here is a new piece: 22edo scale 7

This piece uses a seven note scale in 22edo, as diagramed above. Green arrows represent perfect fifths, 3:2 frequency ratios. Blue arrow represent major thirds, 5:4. Purple arrows represent the unconventional interval 11:8. There are paths in this scale that traverse the commas 100:99 and 55:54.

Leadership

Utah's Governor Cox and Military Installation Development Authority propose Utah playing a leading role in nuclear and artifical intelligence technologies. The form of this leadership seems largely to be a matter of siting nuclear reactors, radioactive waste dump sites, and data centers in the state. Utah's leadership is mainly about cheap resources, tax breaks, and lax regulation. This is a plantation economy that impoverishes the majority, channeling the benefits to a small power elite. This is no kind of true leadership.

Nuclear and artificial intelligence technologies are clearly important, whatever role they might play in the future. Utah could take on a true leadership role. True technological leadership is a matter of the expertise of the people. Expertise can be organized in a variety of ways, in industrial corporations, in governmental agencies, and in civil society groups such as universities.

Nuclear technology is vast and involves expertise in a very broad array of disciplines. Applications include energy production, weapons, and medicine. Nuclear technology continues to be a key source of international conflict. It is a cornerstone of industrial and military power. On the other hand, mining and waste disposal bring disease and ecological degradation, and are thus vectors of disempowerment. Politics is therefore a field of major importance for nuclear technology. Geology is important not just for locating raw materials but also for understanding the pathways by which waste materials might escape their disposal sites over their long half-lives of radioactive toxicity. Biology and public health expertise is important for tracking and managing the risks of exposure to radioactive materials. All of these fields of expertise are invigorated by engagement across a range activities, from research and development to application.

Artificial intelligence is similarly vast. There are computer science and computer engineering challenges involved in making these systems more effective and efficient. There are sociological problems, to understand how these systems are changing the way people behave, are changing the structure of society. There are political problems, to understand how these systems empower some groups and disempower other groups, and how regulation might ameliorate the most negative of these shifts.

Our Utah officials could take bold steps to cultivate real leadership in nuclear technology and artificial intelligence. This means investing in people and organizations, rather than a race to the bottom, selling off assets at discounted prices.

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!